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I

I
I

I

75-14,694
BADMUS, Ganlyu Ademola, 1941A FORMATIVE EVALUATION OF THE MATHEMATICS EDUCATION
COMPONENT OF THE EIGHTH CYCLE TEACHER CORPS
PROGRAM AT MICHIGAN STATE UNIVERSITY.
Michigan State University, Ph.D., 1974
Education, teacher training

Xerox University Microfilms,

©

C o p y r i g h t by
GANIYU ADEMOLA BADMUS

1974

Ann Arbor, Michigan 48106

A FO RMA T I V E E V A L U A T I O N OF THE MATHEMATICS
EDUCA T I O N C O M P O N E N T O F T H E EI G H T H CYCLE
T E A C H E R CORPS P R O G R A M A T MICHIGAN
STATE UN I V E R S I T Y
By
Ga n i y u A d e m o l a Badmus

A D I S S E RTATION

Submi t t e d to
M i c h i g a n State University
in par t i a l f u l f i llment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
C o l l e g e of E d u c a t i o n

1974

ABSTRACT
A FORMATIVE EVALUATION OF THE MATHEMATICS
E DUCA T I O N COMPO N E N T OF THE E I G H T H CYCLE
TEAC H E R CORPS PRO G R A M AT MICHIGAN
STATE UNIVERSITY

By
G a n i y u A d e m o l a Badmus

The m a j o r purpose of the study was to pro v i d e both
"intrinsic" and "pay-off"

formative evaluations of the

mathematics curric u l u m and instruction of the eighth cycle
T e a c h e r Corps p r o g r a m at Mich i g a n State Univer s i t y using
internal,
aspect

external and contextual sources.

The int r i n s i c

(1) analyzed and e v a l uated the m a t h e matics content-

me t h o d integrated component;

(2) prov i d e d critical appraisal

of instructional me t h o d and clinical e x p e r i e n c e s .
off a s p e c t evaluated
ments of learning;

(3)

learning;

(4)

learners;

The p a y ­
(5) e n v i r o n ­

(6) comp a r e d the m e t h o d of instru c t i o n

w i t h two similar methods on m a t h e matics a c h i e v e m e n t and
attitude.
T h r e e groups of students w e r e involved in the study.
Students in the T e a c h e r Corps m a t h e m a t i c s e d u c a t i o n p r o g r a m
(interns)

m e t for six hours p e r w e e k Fall term 1973,

hours p e r w e e k W i n t e r and Spring terms of 1974,

They

four

Ganiyu Ademola Badmus
s t u died five learning units w h i c h w e r e laboratory-ori e n t e d
m a t h e m a tics c o n t ent-method integrated taught under m a s t e r y —
learning approach.

A l l interns spent four hours daily in

e lemen t ary school, where they w e r e prov i d e d w i t h clinical
experi e nce supervised by t e a m leaders and faculty m e m b e r s .
T w e n t y - four of thirty interns w h o originally ente r e d the
p r o g r a m w e r e used in this study.

The second group consisted

of twenty-one students randomly selected from volunteers in
Fall,

19 73,

and given conte n t - m e t h o d integrated instruction

s i m i l a r to that of interns but w i t h o u t mastery-approach.
They m e t six hours per w e e k in the laboratory and spent one
hour p e r w e e k on clinical experience in elementary schools.
Th e third group of students, h a d the regular content and
m e t h o d separ a t e d mathematics e d u c a t i o n program.

They w e r e

used for the study during the Fall term w h e n they wer e
h a v i n g the methods course.
requisite

The content course is a p r e ­

for the methods course.

Eighteen students from

this group w e r e included in the study.
Five c r i t e r i o n - r e f e r e n c e d achievement meas u r e s w i t h
r e l i a b i lity estimates ranging from 0.79 to 0.94 were
d e v e l o p ed and used in e v a l u a t i o n of learning;
c o n s t r u c t i o n of the measures

the me t h o d of

insured their content validity.

O t h e r instruments used w e r e Hicks and Perrodin's instrument
for analysis of m a t h e m a t i c a l topics,
B asic M at h e m a t i c a l Understandings,

Dossett's T e s t of

Dutton's Attitude

Inventory, Attit u d e Scales Tow a r d Diffe r e n t Aspects of
M a t h e m a tics devel o p e d by the International Study of

Ganiyu Ademola Badmus
A c h i e v e m e n t in M a t h e m a t i c s ,

Aiken's Enjoyment and Value of

Mathematics S c a l e s .
M u l t i v a r i a t e a n d u n i v a r i a t e a n a l y s i s of v a r i a n c e
w e r e u s e d in a s s e s s i n g the e f f e c t of the c o n t e n t - m e t h o d
i n t e g r a t e d c o u r s e on interns'

p e r f o r m a n c e on the c r i t e r i o n -

m e a s u r e s a n d b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g and a t t i t u d e
of the interns.

T w o - w a y a n a l y s i s of c o v a r i a n c e w a s u s e d to

c o m p a r e the e f f e c t of the three instr u c t i o n s ,
tude,

and m a t h e m a t i c a l a p t i t u d e on t e r m i n a l b a s i c m a t h e ­

m a t i c a l u n d e r s t a n d i n g s a n d attitude.
repeated measures design was
of l e a r n i n g .
(a)

entry atti­

Stepwise

T w o - f a c t o r by o n e - w a y

u s e d to e v a l u a t e e n v i r o n m e n t s

r e g r e s s i o n t e c h n i q u e s w e r e u s e d to

a ssess the c o n t r i b u t i o n o f l e a r n i n g units to the b a s i c

m a t h e m a t i c a l u n d e r s t a n d i n g of the interns,

(b)

determine

the

r e l a t i o n s h i p b e t w e e n a t t i t u d e t o w a r d m a t h e m a t i c s and
interns'
value,

p e r c e p t i o n of m a t h e m a t i c s

enjoym e n t ,

and e n v i r onments.
T h e r e s u l t s of the study

1.

learning,

indicated:

T h e int e r n s m a d e s i g n i f i c a n t ga i n s
c r i t e r i o n - r e f e r e n c e d m e a s ures.

(p < .001)

on the

T h e p e r c e n t a g e of

int e r n s that r e a c h e d m a s t e r y level r a n g e d f r o m
67 p e r c e n t in F r a c t i o n s
2.

to 96 p e r c e n t in Nu m e r a t i o n .

T h e inte r n s s h o w e d s i g n i f i c a n t gains

(p < .0001)

on

test of basic mathematical understanding and atti­
tude t o w a r d m a t h e m a t i c s .

Ganiyu Ademola Badmus
A l l o w i n g for i n i t i a l d i f f e r e n c e s ,

the interns,

after c o m p l e t i n g the m a t h e m a t i c s e d u c a t i o n c o m p o n e n t
of the p r o g r a m s h o w e d s i g n i f i c a n t l y b e t t e r m a t h e ­
m a t i c a l u n d e r s t a n d i n g than a g r o u p of s t u d e n t s

in

the r e g u l a r t e a c h e r e d u c a t i o n program.
The interns h a v e b e t t e r p e r c e p t i o n s of t h e i r l e a r n i n
environments

in m a n y a s p e c t s than o t h e r two g r o u p s

of students.
T h r e e of the l e a r n i n g units a c c o u n t e d for m o r e t h a n
63 p e r c e n t of the interns'

basic mathematical

understandings.
T h e interns'

initial attitude toward mathematics

t h e i r e n j o y m e n t of the i n s t r u c t i o n a c c o u n t e d for
m o r e t h a n 7 3 p e r c e n t of t h e i r t e r m i n a l a t t i t u d e
toward m a t h e m a t i c s .

and

This Thesis is Dedic a t e d
to
Gbadamosi Adebisi,

my father;

A l i m o t u Adufe "Alayo," my mother;
to
S alamotu Lateye, my late step-mother
(my father's senior wife);
to
late Salamo Iya-Ake;
to
Sali m o t u Adetohun, my dear aunt;
to
Sulaiman A. Lawal, my dear friend;
to
Morenike,

my d e a r wife;
to

Ris i k a t Oyin-Ade, my sister;
to
A l l my "Brothers"

and "Sisters"

and to
A l l my children.

ACKNOWLEDGEMENTS

I w i s h to express my a p p r e ciation to Dr. Perry E.
Lanier,

chairman of my doctoral committee,

ment and understanding.

for his e n c o u r a g e ­

Special a p p r eciation is exp r e s s e d

to Dr. Bruce A. Mitchell, my d i s s e rtation director, w h o gave
freely of his time,

talent, wise counsel,

gu iding me through the research project.

and judgment in
His guidance was

a source of perso n a l s a t i s faction a n d encouragement.
I w i s h to thank Dr. W i l l i a m M. F i t z g e r a l d and
Dr. P e t e r A. Lappan, Jr., w h o w e r e mem b e r s of m y doctoral
c o m m ittee and gave hel p f u l suggestions and e n c o u r a g e m e n t
t h r o u g hout my doctoral program.

Special a p p r e c i a t i o n is

ex p r e s s e d to Dr. Lauren G. W o o d b y from w h o s e e x p e rie n c e s I
have g a i n e d a lot.
I w i s h to thank Dr. Gl e n d a La p p a n and Mr.

Paul

K a canek for allowing me to use their students in the study.
Thanks are due to m a n y people w h o have co n t r i b u t e d
to my academic achievements.

Special a p p r e c i a t i o n is

e x p r e s s e d to my old teachers Messrs. M. A. Alii,
Adebambo, A. O. Sanni, A. O. O. Odutola,
O l u n l o y o w h o have demonstrated,

J. A.

and Dr. V. O. S.

m o r a l l y and financially,

un q u a l ified interest in my w o r k and progress.

I w o u l d like to thank the M a t h e matics D e p a r t m e n t
and the College of E d u c a t i o n of M i c h i g a n State University,
and The W e s t e r n State G o v e r n m e n t of N i g e r i a for giving me
financial supports during my graduate study.
Lastly,

thanks are due to all my people to w h o m I

have d edic a t e d this thesis;

to Gbadamosi Adebisi, m y

father --for he has p l a c e d high value on formal e d u c a t i o n and
p a i d dearly for it and X have accepted his value;
A d u f e "Alayo," my mother,
m u c h for me;

to A l i m o t u

the w o m a n who had suff e r e d so

to my sister and b r o t h e r s — for their p r a y e r

that I succeed;

to m y wife and children for the support and

e ndurance they have demonstrated;
r e l a t i v e s — Messrs.

to my dear friends and

S. A. Lawal, A. Adetola,

Bayo O g u n b i , Julius O, Durojaiye,

Sule Odu-ola,

and Moses A. F a t o k i — w h o

supported me du r i n g the h a r d times.

TABLE OF CONTENTS

Page
LIST OF T A B L E S .............................................. viix
C h a pter
I.

THE P R O B L E M ...............................

1

Introduction.
. . . .
Purpose of the S t u d y .................
2
N e e d for the S t u d y ............................
Defini t i o n of Terms W i t h Comments,
. . .
Research Hypotheses
. . . . . . . .
Ass u m p t i o n s of the S t u d y .......................
Scope and Deliminations of the Study.
.
,
II.

REVIEW OF L I T E R A T U R E .....................

4
8
13
17
21

24

I n t r o d u c t i o n ...................................
Influence of School and the Tea c h e r on the
D i s a d v a n t a g e d C h i l d .................
26
Nature of the D i s a d v a n t a g e d and His
Learning P r o c e s s ....................
29
The Th e o r e t i c a l F o u n d a t i o n of the Emer g i n g
Practices in M a t h e matics Curriculum,
Instructional Procedures and Goals for
the D i s a d v a n t a g e d .................
33
Rese a r c h and Evalua t i o n Litera t u r e on
L a b o r a t o r y A p p r o a c h as an Instructional
M e t h o d o l o g y for the D i s a d v a n t a g e d . . .
Bloom's Model of Mas t e r y L e a r n i n g
Pr o m i s i n g I n n o v a t i v e / E x p e r i m e n t a l Field
Ex p e r iences in T e a c h e r Educa t i o n
. . .
Sele c t e d Research and E v a l u a t i o n Literature
on Related (Research) M e t h o d o l o g y of
the S t u d y ............................
72
III.

1

24

42
48
57

D E S C R IPTIVE FEATURES A N D DESIGN O F T H E STUDY.

81

I n t r o d u c t i o n ...........................
Gene r a l C o n t e x t and P r o g r a m Description.

83

v

81
.

Chapter

Page
A n a l y s i s of the M a t h e m a t i c s C o n t e n t in the
M a t h e m a t i c s E d u c a t i o n C o m p o n e n t of the
P r o g r a m ............................................ 102
D e s c r i p t i o n of the M a t h e m a t i c s M e t h o d s
I n t e g r a t e d w i t h M a t h e m a t i c s C o n t e n t and
109
C l i n i c a l E x p e r i e n c e .........................
A n A p p r a i s a l of the I n s t r u c t i o n a l Method.
.
118
S a m p l e s ............................................124
M e a s u r e s and I n s t r u m e n t a t i o n ...................... 126
D e s i g n o f S t u d y ..................................... 137
A n a l y s i s of D a t a ..................................... 143
M e t h o d o l o g i c a l A s s u m p t i o n s and L i m i t a t i o n s
of the S t u d y ..................................... 146
S u m m a r y ............................................151

IV.

P R E S E N T A T I O N A N D A N A L Y S I S OF D A T A ................... 153
A n a l y s i s of the M a t h e m a t i c s C o n t e n t and
M e t h o d s of the L e a r n i n g U n i t s ...................154
E v a l u a t i o n of L e a r n i n g ..............................158
E v a l u a t i o n of L e a r n e r s ..............................164
E v a l u a t i o n of I n s t r u c t i o n ...................... 171
Q u a n t i t a t i v e E v a l u a t i o n of E n v i r o n m e n t s
of L e a r n i n g .........................................181
I n t e r p r e t a t i o n and D i s c u s s i o n of F i n d i n g s
.
184
A s s e s s m e n t of C o n t r i b u t i o n of L e a r n i n g
U n i t s to the B a s i c M a t h e m a t i c a l
U n d e r s t a n d i n g s of Int e r n s ..................
19 3
A s s e s s m e n t of C o n t r i b u t i o n of D i f f e r e n t
A s p e c t s of A t t i t u d e T o w a r d M a t h e m a t i c s
and S c h o o l L e a r n i n g to G e n e r a l A t t i t u d e
T o w a r d A r i t h m e t i c ............................. 198
S u m m a r y of F i n d i n g s ............................. 201

V.

SUMMARY,

CONCLUSIONS,

AND RECOMMENDATIONS

.

.

205

205
S u m m a r y .................................... ...
C o n c l u s i o n s .................................... 213
I n t e r p r e t a t i o n and D i s c u s s i o n of F i n d i n g s
.
216
R e c o m m e n d a t i o n s .................................
224
BIBLIOGRAPHY

...............................................

2 28

Chapter

Page

A P P E N D ICES
Appendix
A.

Set of Five Pre- and Post-Test Forms of
the C r i t e r i o n - R e f e r e n c e d Measures .
.

,

.

252

A Test of Basic M a t h e matical Understandings
F o r m A (Pre-Test) and F o r m B (Po s t - T e s t ) .

*

271

C.

Dutton A r i t h m e t i c Atti t u d e Inventory.

.

301

D.

A t t i t u d e Scales To w a r d Different Aspects
of M a t h e m a t i c s .................................... 303

E.

E n j o y m e n t and Value of Mathematics Scales

,

.

308

F.

Raw Scores of Tea c h e r Corps Interns, G^.

.

.

310

G.

Raw Scores of the "Comparison Groups" on
All M e a s u r e s ....................................... 311

B.

.

.

LIST OF TABLES

Table
1.
2.
3.

Page
Suggested Topics for the Mat h e m a t i c a l
Pr e p a r a t i o n of Elementary School Teachers

. .

106

Reliability Coefficients for Pre- and PostC r i t e r i o n - R e f e r e n c e d A c h i e v e m e n t Measures

• »

132

C o r r e l a t i o n Co e f f i c i e n t Between Pre- and PostT e s t Scores of the Students in Regular
Methods Co u r s e (Education 325E) on ItemSampled C r i t e r i o n - R e f e r e n c e d A c h i e vements
. .

133

4.

A C o m p a r i s o n of M a t h e m a t i c a l Topics Cov e r e d by
the Tea c h e r C o r p s , the Reg u l a r Elementary
Educa t i o n P r o g r a m and the TTT Experimental
P r o g r a m ............................................. 157

5.

Means and Standard Deviations of Pre-test,
Post-test, and G A i n Scores on the Five
C r i t e r i o n Measures for the Interns
.
.

. . 161

6.

M u l t i v a r i a t e Anal y s i s of Interns on Differences
B etween Pre- and P o s t -Scores on the Five
C rite r i o n M e a s u r e s ................................... 161

7.

N u m b e r and P e r c e n t a g e of Interns that Reached
Mas t e r y Level on L e a r n i n g Units
..............

163

Means a n d S t a n d a r d Deviations of Pre- and P o s t ­
test Scores of Interns on Dossett's and
Dutton's Tests ...................................

16 6

8.

9.

M u l t i v a r i a t e and Univar i a t e Analy s i s of V a r i a n c e
of Interns on Gains Doss e t t and Dutton's
T e s t s ................................................. 167

10.

Grade Levels of Interns Where Attit u d e s were
D e v e l o p e d ............................................. 169

viii

Table

Page

11.

Interns*

12.

S u m m a r y o f A n a l y s i s of C o v a r i a n c e for the
G r o u p s on the T e s t of B a s i c M a t h e m a t i c a l
U n d e r s t a n d i n g ....................................

174

G r o u p s M e a n Scores on the T e s t o f B a s i c
..................
Mathematical Understandings

175

13.

F e e l i n g s A b o u t A r i t h m e t i c in G e n e r a l

.

170

14.

S u m m a r y of the A n a l y s i s of C o v a r i a n c e for the
Scores of G r o u p s on A t t i t u d e T o w a r d
Arithmetic
........................................ 179

15.

G r o u p s M e a n Sc o r e s o n A t t i t u d e T o w a r d
A r i t h m e t i c ............................................ 180

16.

S u m m a r y of M u l t i v a r i a t e A n a l y s i s of V a r i a n c e
(Repeated M e a s u r e s Design) on P e r c e p t i o n of
Learning Environments
.........................

183

M e a n Scores of G r o u p on P e r c e p t i o n on
.........................
Learning Environments

184

17.
18.

C o r r e l a t i o n C o e f f i c i e n t s B e t w e e n Post-test,
in B a s i c M a t h e m a t i c a l U n d e r s t a n d i n g , A t t i t u d e
T o w a r d A r i t h m e t i c and O t h e r V a r i a b l e s
( M e a s u r e s ) ............................................ 195

19.

A n a l y s i s of V a r i a n c e for O v e r a l l R e g r e s s i o n
o f P o s t - t e s t in B a s i c M a t h e m a t i c a l U n d e r ­
s t a n d i n g s and O t h e r V a r i a b l e s ...................... 197

20.

A n a l y s i s of V a r i a n c e for O v e r a l l R e g r e s s i o n
of Post-test Attitude Toward Arithmetic
a n d O t h e r V a r i a b l e s ................................. 200

ix

CHAPTER I
THE P R O B L E M
Introduction
In the early nine t e e n sixties there w a s a tumult
about the difficulty of training teachers
w h o s e pupils were descr i b e d as
ally different."

"disadvantaged" or "cultur­

M c G e o c h and Copp

pr oject at Teachers College,

for urban schools

(1963)

reported a pilot

Colum b i a Univer s i t y ca l l e d the

"Teaching Corps," w h i c h w a s a p r o g r a m desi g n e d to train
special teachers for the disadvantaged.

Goldberg

(1963)

d e s c r ibed a hypo t h e t i c a l m o d e l of a successful teacher of
the disadvantaged.

Haub r i c h

(1965)

e f f e c t i v e training of teachers
c h i ldren while Rivlin

(1962)

gave guidelines

for

for cultur a l l y d i s a d v a n t a g e d

h a d ear l i e r outl i n e d a v a r i e t y

of d e s i r e d m o d i f i c a t i o n s in e x i s t i n g modes of teach e r
p r e p a r a t i o n for large city schools.

Ausubel

(1964)

dis­

c u ssed the r e v e r sibility of the cognitive and m o t i v a t i o n a l
effects o f cultural d e p r i v a t i o n and i m p l i cations for
t e a ching the c u l t ural ly depri v e d child.
In a n a t i o n - w i d e eff o r t to give children from "low
income"

families b e t t e r educational o p p o r tunities and to

1

2
improve the qual i t y of teacher education programs

for both

c e r t i f ied teachers and i n e x p e r i e n c e d teacher i n t e r n s ,
T e a c h e r Corps was e s t a b l i s h e d by Congress in 196 5.
Today Tea c h e r Corps projects exist in over 150
school districts,

5 prisons,

and 17 juvenile institutions.

They o perate in c o o p e ration w i t h about 85 colleges and
universities.
income areas,

The proj e c t gives school districts in low
their communities,

the chance to w o r k together.

and nearby universities

Its p h i l o s o p h y embodies

m odels and guidelines of M c G e o c h and C o p p
(1963),

Rivlin

(1962), Haub r i c h

a competency-based,

(1965),

field-based,

(1963), Go l d b e r g

and others.

community-based,

It is

bi l i n g u a l

a nd b i c u l t u r a l program.
Purpose of the Study
The purpose of this study is to pro v i d e a formative
e v a l u a t i o n of the m a t h e m a t i c s e d u c a t i o n compo n e n t of
e ighth cycle T e a c h e r Corps P r o j e c t at M i c h i g a n State U n i ­
v e rsity w i t h the Lansing, M i c h i g a n school district.
F o l l o w i n g the model of Scriven
and C u n n i n g h a m
"intrinsic"

(1973),

(1967),

Sanders

this study w i l l focus on b o t h the

and "pay-off" parts of formative e v a l u a t i o n of

the p r ocess and p r o d u c t of the m a t h e m a t i c s c u r r i c u l u m and
instruction.

Internal,

external,

and contex t u a l sources

are used.
Specifically,
g a t i o n sought:

the intri n s i c aspect of the in v e s t i ­

3
1.

To analyze and evaluate the mathematics content in
the mathematics e d u c ation component of the p r o g r a m
and to assess w h e t h e r they m e e t the mathemati c a l
need of the interns.

2.

To provide a critical appraisal of the instructional
methods and the clinical experiences in the
program.
The p a y - o f f aspect of the study sought:

3.

To evaluate the effect of the instruction as
p r e s c r i b e d by the mathematics education component
of the Teacher Corps P r o g r a m on the interns in
rela t i o n to specified competencies and to assess
if the interns achieved a degree of mastery over
these competencies.

4.

To evaluate the eff e c t of the instruction on the
basic m a t h e m a t i c a l knowledge of the interns.

5.

To evaluate the effect of the instruction on the
interns'

6.

attitudes toward mathematics.

To assess the contribution of diffe r e n t units of
m a t h e m a t i c s i n s t r uction to the general mathem a t i c a l
knowledge of the interns.

7.

To assess the c o n t r ibution of different aspects of
attitude toward m a t h e matics and school learning to
the general attitude of the interns to w a r d m a t h e ­
matics .

4
8.

T o c o m p a r e the m a t h e m a t i c a l u n d e r s t a n d i n g and
a t t i t u d e t o w a r d m a t h e m a t i c s of the inte r n s in this
p r o g r a m w i t h the m a t h e m a t i c a l u n d e r s t a n d i n g and
a t t i t u d e s of s t u d e n t s e n r o l l e d in the r e g u l a r
t e a c h e r - e d u c a t i o n program.

The p u r p o s e is to

determine,
(a)

the e f f e c t of m e t h o d of i n s t r u c t i o n
l e v e l s ) and e n t r y a t t i t u d e

(three

(three l e v e l s ) upon

the m a t h e m a t i c s a c h i e v e m e n t at the e n d of
instruction;
(b)

the e f f e c t of m e t h o d of i n s t r u c t i o n
levels and m a t h e m a t i c a l a p t i t u d e
levels)

(three

(three

o n a t t i t u d e t o w a r d m a t h e m a t i c s at the

e n d of i n s t r u c t i o n .
9.

T o c o m p a r e the e f f e c t o f three m e t h o d s of
i n s t r u c t i o n on d i f f e r e n t asp e c t s of a t t i t u d e

(five

levels).
10.

T o use the resu l t s of the i n v e s t i g a t i o n to m a k e
specific recommendations

for r e p l i cation,

diffus i o n ,

a n d i n s t a l l a t i o n of a c h a n g e of proced u r e ,

as w e l l

as r e f i n e m e n t of its o v e r a l l design.
N e e d for the Study
While

"lower-class"

and

"disadvantaged"

are n o t

n e c e s s a r i l y s y n o n y m o u s , the l i t e r a t u r e g e n e r a l l y v i e w s the
m i d d l e - c l a s s t e a c h e r in r e l a t i o n to l o w e r - c l a s s a n d d i s ­
a d v a n t a g e d c h i l d r e n , the r e s u l t s s e e m to c o i n c i d e .

Many

5
e ducators

like Be c k e r

(1952), Davis

Bowman, Liddle, Matthews,
Hickerson

(1948), Havighurst,

and Pierce

(1966), Riessman

(1962),

(1962), Arnez

(1966),

agree that the teachers

middle -class attitudes and values are in conflict w i t h
those of lower-class or d i s a d vantaged students,

and t h e r e ­

fore antithetical to the focal concern of the child r e n and
y o u t h they serve.

Be c k e r

(1952), Davi d s o n and Lang

and Rosenthal and J a c o b s o n

(1960),

(19 6 8) have shown that teachers'

expectations influence the aspirational level a n d learning
of the child and these expectations tend to v a r y inversely
w i t h the child's socio-economic class.

Unfortunately

children perceive and fulfill these lower expectations,
c o n f irmed Clark
berg

(1965).

(1962), Vontress

Davidson and Lang

(1963), and Sexton

(1960), Frieden-

(1964)

maintain

that m o s t of the teachers of the d i s a d v a n t a g e d dislike
a n d/or dist r u s t the children and teachers replace their
main function of teaching by an emphasis on discipline.
Clark
(1967)

(1965), Ri v l i n

(1962,

1965), Landers

(1964),

Groff

repo r t e d that m a n y new teachers are unwil l i n g to

a ccept appointmen t to teach the d i s a d v a n t a g e d w h i l e
e x p e r i e n c e d teachers tend to seek transfers.
In response to these sensitivities,

the Nati o n a l

S c ience F o u n d a t i o n funded an SMSG c o n f erence on Math e m a t i c s
E d u c a t i o n in the Inner City Schools in March,

1970.

One of

the five posi t i o n papers w a s p r e p a r e d and p r e s e n t e d by
W o o d b y on "A Survey of E x i s t i n g Projects W h i c h A t t e m p t to
A t t e n d to Innercity Problems in Mat h e m a t i c s Education."

6
T h i s p a p e r e x a m i n e d the forces a n d issues t h a t led to the
f u n d i n g o f the p r o j e c t s .

There was

also a p a n e l

focusing

o n p e d a g o g y a n d the l a b o r a t o r y a p p r o a c h as p o s s i b l e p a r t i a l
s o l u t i o n to the p r o b l e m s .
In his r e a c t i o n paper,
the p r o j e c t s
themes

Forbes observed that although

s u r v e y e d d i f f e r in m a n y ways,

like i n d i v i d u a l i z a t i o n ,

objective-oriented programs,

diagnosis

suc c e s s

and p r e s c r i p t i o n ,

for e x p e r i e n c e s of

s t u d e n t s , s t u d e n t s i n v o l v e m e n t in learning,
image,

t h e r e are c o m m o n

s t u d e n t self-

teacher-training and development and these

h a v e r e l e v a n c e for e d u c a t i o n o f all children.

themes

The p a r t i c i ­

p a n t s a l m o s t u n a n i m o u s l y a g r e e d t h a t p r i o r i t y in i n n e r c i t y
e d u c a t i o n s h o u l d be g i v e n to p r e s c h o o l ,
e arly p r i m a r y y e a r s

k i n d e r g a r t e n and

a n d that little v a l u e w o u l d be

a c c o m p l i s h e d u n l e s s the p r o g r a m i n c l u d e s g r e a t e m p h a s i s
on d e v e l o p i n g a p p r o p r i a t e a t t i t u d e , i n s i g h t s , and the
u n d e r s t a n d i n g of d i f f e r e n t c u l t u r e s

among teachers

in the

innercity s c h o o l s .
T h e l a s t r e m a r k of the p r e c e d i n g p a r a g r a p h c o u p l e d
w i t h w h a t is a l r e a d y k n o w n a b o u t e x i s t i n g m i d d l e — class
innercity teachers possibly
" s p e c ial p r o g r a m s . "

T h e c h a n g e d goals w e r e d i r e c t e d

toward giving young people
opportunities

led to c h a n g e the go a l s of

from poverty backgrounds new

to o b t a i n a c o l l e g e e d u c a t i o n and to m a k e

h i g h e r e d u c a t i o n r e s p o n s i v e a n d r e l e v a n t to t h e i r s p e c i a l
educational needs

{Astin e t al.,

1972).

The students

r e c r u i t e d into th e s e c o m p e n s a t o r y p r o g r a m s w e r e c o n s i d e r e d

7
"disadvantaged"

or

"high risk" on the a s s u m p t i o n t h a t they

lack the r e q u i s i t e m o t i v a t i o n and a c a d e m i c sk i l l s to seek
a n d s u c c e s s f u l l y p u r s u e a c o l l e g e educat i o n .

Accordingly

s u c h p r o g r a m s h a v e f o c u s e d on c o r r e c t i n g th e s e m o t i v a t i o n a l
a n d a c a d e m i c deficits.
C o n s i d e r i n g the a m o u n t of m o n e y e x p e n d e d by the
g o v e r n m e n t and the a m o u n t of t i m e , e n e r g y and h u m a n
resources

s u p p l i e d b y the u n i v e r s i t i e s

in s u c h progr a m s ,

t h e i r e x i s t e n c e r a i s e s s o m e is s u e s n o t only a b o u t the
general educational experiences

that lead to the s u c c e s s f u l

d e v e l o p m e n t of l a t e n t t a l e n t b u t also a b o u t the b a s i c
p r e m i s e s u n d e r l y i n g the a d m i s s i o n criteria.
to p a v e the w a y

In an a t t e m p t

for n e e d e d a n s w e r s to g e n e r a l q u e s t i o n s

of the form:
C a n these p r o g r a m s h e l p the u n d e r p r e p a r e d , s p e c i a l l y
a d m i t t e d s t u d e n t s to m a k e e d u c a t i o n a l a n d so c i a l
a d j u s t m e n t n e c e s s a r y to c o m p l e t e a c o l l e g e e d u c a t i o n ?
T o w h a t e x t e n t do h i g h e r e d u c a t i o n a l p r o g r a m s for the
disadvantaged serve their clients?
W h a t t y p e s of
p r o g r a m s * c o m p o n e n t s s h o w the g r e a t e s t p r o m i s e ?
This study examines

the f o l l o w i n g issues:

T o w h a t e x t e n t do es the m a t h e m a t i c s i n s t r u c t i o n for
the d i s a d v a n t a g e d i n t e r n s s e r v e the i n t e r n s ?
w h i c h of
t he c o l l e g e e n v i r o n m e n t s and e x p e r i e n c e s f a c i l i t a t e
the m a t h e m a t i c a l g r o w t h o f t h e s e d i s a d v a n t a g e d i n t e r n s ?
A s s u m i n g (i) C a r r o l l * s thesis, (ii) B l o o m ' s t h e o r y of
M a s t e r y l e a r n i n g are valid, (iii) recent, B e g l e (1971)
r e s e a r c h r e p o r t w h i c h seems to j u s t i f y the t h e o r i e s and
(iv) r e c e n t p u b l i c a t i o n b y Astin, e t al. (1972) that
b e i n g s o c i o - e c o n o m i c a l l y d i s a d v a n t a g e d is n o t in itself,
a s e v e r e h a n d i c a p to the s t u d e n t o n c e h e gets to the
c o l lege, is it p o s s i b l e to d e s i g n a m a t h e m a t i c s
i n s t r u c t i o n for t h e s e i n t e r n s t h a t w i l l b r i n g t h e i r
b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g and t h e i r a t t i t u d e
t o w a r d m a t h e m a t i c s to the s a m e level as o t h e r
p r o s p e c t i v e e l e m e n t a r y t e a c h e r s in o t h e r p r o g r a m s ?

8
In the m a s t e r y learning model,

all students are h e l p e d to

achieve a c r i t erion mas t e r y of the learning at hand.

The

focus is not on separa t i n g students into grade c l a s s i f i ­
cations but rather on h e l p i n g students reach the ma s t e r y
level.

Ries s m a n

(19 6 3) conte n d e d that the d i s a d v a n t a g e d

child is typically a phys i c a l learner,

and the phys i c a l

learner is gener a l l y a slo w e r learner and this slowness
s hould not be e q u a t e d w i t h stupidity.
characteristics of the slow learner,

D i s c u s s i n g the
Schulz

(1972)

remarked

that cultural d i f f erences and deficient cognitive f unctioning
are m a j o r influences on the b e h a v i o u r and a c h i e v e m e n t of
slow learners.

In the words of Pikart and W i l s o n

"Research o n the development,

(1972),

use and v a l i d a t i o n of the

m a s t e ry learning model for slow learners in m a t h e m a t i c s is
an obvious need."
Defini t i o n of Terms With Comments
1.

Criterion-Referenced M e a s u r e ;

"One that is c o n ­

s t r u c t e d to y i e l d m e a s u r e m e n t s that are directly i n t e r ­
p r e t a b l e in terms of s p e c i f i c p e r f o r m a n c e standards."
2.

E c o n o m i c a l l y D i s a d v a n t a g e d — children and adults

from home and/or community b a c k g r o u n d w h e r e a m a j o r i t y of
the residents lack adequate financial income thus r e s u l t i n g
in s u b s t a n d a r d living conditions.
3.

E d u c a t i o n a l l y D i s a d v a n t a g e d - -individuals

from home

a n d/or c o m m unity b a c k g r o u n d lacking cultural assets

9
n e c e s s a r y for no r m a l school a c h i e vement thus pla c i n g the
i n d i v iduals at a dis a d v a n t a g e grade level-wise.
4.

Experimental G r o u p s :

was a group of thirty

junior e l e m e n t a r y e d u c a t i o n majors sele c t e d for the T e a c h e r
Corps p r o g r a m w h o p a r t i c i p a t e d in the ei g h t h cycle p r o g r a m
at M i c h i g a n State University.
r e f e r r e d to as "Interns."

The mem b e r s w e r e usually

The group p a r t i c i p a t e d in a

s p e c i ally desig n e d mat h e m a t i c s i n s t ructional program.
w a s a g r o u p of twenty - t w o students

G2

in the r e g u l a r t e acher

e d u c a t i o n p r o g r a m randomly sele c t e d from students w h o
r e g i s t e r e d for M a t h e m a t i c s
given a mathematics
o f G-^.

201 in 1973 fall.

T h e g r o u p was

i n s t r uctional p r o g r a m sim i l a r to that

G^ was a group of students in the r e g u l a r t e acher

e d u c a t i o n p r o g r a m e x p o s e d to the regular mathematic s
i n s t r u c t i o n w h i c h was d i f f e r e n t from those g i v e n to G ^ and

5.

F o r m a t i v e E v a l u a t i o n — is the process of judging a

fluid proc e s s or p r o d u c t that can be revi s e d in form.

The

results of such ev a l u a t i o n studies are given to persons
d i r e c t l y invol v e d in the p r o c e s s or in d e v e l o p i n g the
product.
6.

F o r m a t i v e P r o c e s s / I n t e r i m E v a l u a t i o n — is the type

of e v a l u a t i o n w h i c h provi d e s p e r i o d i c feedback to p e r s o n s
r e s p o n s i b l e for i m p l e m e n t i n g plans,

or p r o c e d u r e s or

d e v e l o p i n g a p r o d u c t that is n o t y e t fully assembled.

It

10
has three objectives:

(1) to de t e c t or p r e d i c t defects

in p r o c e d u r a l de s i g n or its i m p l e m e n t a t i o n during i m p l e ­
m e n t a t i o n stages,
decisions,
occurs.

and

It is,

7.

(2) to provide i n f o r mation for p r o g r a m e d

(3) to m a i n t a i n a record of p r o c edur e as it
thus,

concerned w i t h p r o g r a m improvement.

F o r m a t i v e Pro d u c t E v a l u a t i o n —

and Cunningham,

19 73)

(following Sanders

is the e v a l u a t i o n of the p r o d u c t as

it has b e e n put together strictly for feedback to the
developer.

D e s c r iptive and con t e n t analyses techni q u e s as

d e s c r i b e d under int e r i m formative evalua t i o n activities
are e x t r e m e l y important at this point.

K n o w ledge about the

e x t e n t to w h i c h val u e d objectives are achie v e d w i t h a
p l a n / p r o d u c t are important.

What Anderson

(1969)

termed

"field test"

is an e x c e l l e n t example of this type of

evaluation.

Borich

(1971)

has also s u g g e s t e d a con c e p t u a l

m o d e l for formative p r o d u c t evaluation.
p r o d u c t w i t h a sample of subjects

V a l i d a t i o n of a

from the ta r g e t p o p u l a t i o n

or a feasibility study o f a p l a n for e d u c a t i o n a l change are
the m o s t frequently found formative p r o d u c t e v a l u a t i o n
studies in the literature.
The sources of external and internal inform a t i o n
listed u n d e r formative interim / p r o c e s s e v a l u a t i o n activities
are a ppli c a b l e to formative p r o d u c t e v a l u a t i o n acti v i t i e s
also.

T h e o b j e c t under scrutiny at this p o i n t w i l l b e the

e n t i r e a s s e m b l e d product, however,
nents.

ra t h e r than its c o m p o ­

C o n t e x t u a l i n f o r m a t i o n is of ut m o s t i m p o r t a n c e at

11
this point.

T h e f o r m a t i v e p r o d u c t e v a l u a t i o n s h o u l d test

the p r o d u c t in the c o n t e x t w i t h i n w h i c h it is i n t e n d e d to
function.

T h e c o l l e c t i o n of c o n t e x t u a l i n f o r m a t i o n in

interim/process

formative evaluation cannot be accomplished

in i s o l a t i o n from a p a r t i c u l a r set of o b j e c t i v e s o r a
p a r t i c u l a r p r o d u c t si n c e by d e f i n i t i o n the role of c o n t e x t
is to spec i f y the limits of the product.
"The task of the f o r m a t i v e evaluator,

therefore,

is

to e s t a b l i s h w h e t h e r p r e d i c t e d r e l a t i o n s h i p s b e t w e e n
context,

internal,

a n d e x t e r n a l i n f o r m a t i o n holds.

Is it

the c a s e , for i n s t a n c e , t h a t s t u d e n t s w i t h s p e c i f i e d e n t r y
behaviours

(context)

learn m o r e m a t h e m a t i c s

(external)

a programed test using hierarchial sequencing

from

(i n t e r n a l ) ?

A n a n a l o g o u s q u e s t i o n in the p r o p o s e d study is,
"Is it the case that p r o s p e c t i v e e l e m e n t a r y s c hool
t e a c h e r s w i t h low s o c i o - e c o n o m i c b a c k g r o u n d — l o w - a c h i e v e r s
in m a t h e m a t i c s - - ( c o n t e x t )

c a n learn m a t h e m a t i c s

level of c o m p e t e n c y as s t u d e n t s
nal)

from a mastery

to the same

in r e g u l a r p r o g r a m

learning mathematics

c o n d u c t e d in a l a b o r a t o r y s e t t i n g

(exter­

instruction

(internal)?"

The f o r m a t i t i v e e v a l u a t o r o f t e n is n o t s a t i s f i e d
if he o b s e r v e s s h a r p d i f f e r e n c e s in the e f f e c t i v e n e s s of
two programs,

h e likes to find o u t w h y the d i f f e r e n c e

o c c u r s if h e is to give c o m p l e t e i n f o r m a t i o n c o n c e r n i n g
p o s s i b l e r e v i s i o n s o f the m a t e r i a l .
m a t i o n is,

t h o u g h n o t always,

Explanatory infor­

n e e d e d for f o r m a t i v e p r o d u c t

12
e v a l ua t i o n work,

it can be critical and should not be

overlooked.
8.
1967)

Interim Intrinsic E v a l u a t i o n " - ( f o llowing Scriven,
is the evalua t i o n of transactional or mean s - t o - t h e -

ends p r o g r a m characteristics.

It is an interim/process

formative evaluation w h i c h relies on internal information,
b o t h descriptive and critical appraisal.

Examples of

activities that fell into this category are the analysis of
the content of the p r o g r a m components or the appraisal of
instructional strategies w h i c h are w e l l i l l u strate d by
M o r r i s s e t t and Stevens
(1968), and Eash

(1968,

(1970).

e x c e llent discussion of

1971), Tyler and Klein

Stake

(1970)

has prov i d e d an

the use of professionals in such

ev a l uation studies.
9.

Lear n i n g U n i t :

a c q uired under a single
10.

The m a t h e m a t i c a l objectives to be
set of learning conditions.

Mastery L e a r n i n g — is an i n s t r uctional strategy

w h i c h proposes that under appropriate instructional
conditions virtually all students can learn m o s t of w h a t
they are taught.

The sequential steps in the strategy are

s p e c i fication of objectives,
level

(sc o r e ) , unit teaching,

feedback,

d e s i g n a t i o n of the m a s t e r y
formative tests,

diagnosis of areas of deficiency,

procedures,

re-test.

In Bloom's model,

immediate

alternate

an attempt is made

to alter the amount of time a stud e n t spends in st u d y i n g a

13
task and thus bring about a level of learning d e t e r m i n e d to
be m a s t e r y of the task.
11.

Pay-off E v a l u a t i o n —

(following Scriven,

1967)

is

interim/process formative evalua t i o n w h i c h relies on
external information.

This is the m o s t common type of

evaluation activity in instructional development;
for some people,
evaluation.

this type is the only

indeed,

"real" type of

The methods used to collect external i n f o r ­

m ation for pay - o f f evaluations are e x c e l l e n t l y illust r a t e d
by M e t f e s s e l and M i c h a e l

(1967), Markle

(1971), Goo d w i n and Sanders
12.

(1970), A b e d o r

(1971).

P o v e rty-area S c h o o l — elemen t a r y school in w h i c h a

m a j o r i t y of the children are from e d u c a t i o n a l l y and
e c o n o mically d i s a d v a n t a g e d environments,
13.

The D i s a d v a n t a g e d — the individual w h o comes from a

home and/or community en v i r o n m e n t that is lacking e c o n o m i ­
cally and educationally.
Research Hypotheses
The following hypothesis w i l l be tested to assess
the e ffect of the i n s t r uctional p r o g r a m on the a c h i e v e m e n t
of interns on the p r e s c r i b e d mat h e m a t i c a l competencies
(integrated content a nd methods).
level of significance w i l l be used.

In each case the

.05

14
A.

The p o s t - t e s t means score w i l l exceed the p r e - t e s t
means score of the interns

(G^) on the criterion-

r efere n c e d measures.
The univariate hypotheses associated w i t h this
m u l t i v a r i a t e hypothesis are:
The m e a n post- t e s t score of the interns w i l l be
h i g h e r than the m e a n pre- t e s t score on the criterionr e f e r e n c e d meas u r e s in:
a.

measurement,

b.

numeration,

c.

addition and s u b t r a c t i o n of w h o l e numbers,

d.

m u l t i p l i c a t i o n and divi s i o n of whole numbers,

e.

fractions.
The

following two hypotheses w i l l be tested to

assess changes in the interns'

b a s i c mat h e m a t i c a l kn o w l e d g e

and attitude toward arithmetic:
B 1.

The p o s t - t e s t mean score of the interns w i l l e x c e e d
their pre- t e s t m e a n score on Dossett's test of
m a t h e m a t i c a l understanding.

B 2.

The p o s t - t e s t m e a n score of the interns w i l l exceed
their pre- t e s t m e a n score on Dutton's a r i t h m e t i c
a ttitude inventory.
The following hypotheses w i l l be tested to compare

the interns and students e n r o l l e d in the reg u l a r teacher
e d u c a t i o n program.

The pur p o s e is to determine

e f f e c t of m e t h o d of i n s t r uction
a ttitude toward m a t h e matics

(three levels)

(three levels)

(1) the
and entry

upon the

15
m a t h e matics a c h i e v e m e n t at the e n d of instruction,

and

the e ff e c t of m e t h o d of i n s t r u c t i o n

and

m a t h e m a t i c a l aptitude

(three levels)

(three levels)

(2)

on the attit u d e toward

m a t h e matics at the end of instruction.
C 1.
(a)

W h e n a linear a d j u s t m e n t is m a d e for the effect of
v a r i a t i o n due to d i f f e rences in p r i o r m a t h e m a t i c a l
aptitude,

as m e a s u r e d by Dossett's pre-test,

there

w i l l be no s i g n i f i c a n t d i f f e r e n c e in mathem a t i c s
achievement,

as m e a s u r e d by Dossett's post-test,

b et w e e n the met h o d s of instruction.

(b)

T h a t is,

there w i l l be no t r e a tment effect.

W h e n a linear

a d j u s t m e n t is made for the eff e c t

v a r i a t i o n due to d i f f e rences in p r i o r m a t h em a t i c a l
aptitude,

as m e a s u r e d by Dossett's pre-test,

there

w i l l be no s i g n i f i c a n t d i f f e r e n c e in m a t h e ma t i c s
achievement,

as m e a s u r e d by Dossett's post-test,

b e t w e e n the entry attitudes.
T h a t is, there w i l l be no atti t u d e effect,
(c)

W h e n a linear

a d j u s t m e n t is m a d e for the eff e c t of

v a r i a t i o n due to d i f f e rences in p r i o r m a t h e m a t i c a l
aptitude,

as m e a s u r e d b y Dosse t t ' s pre-test,

there

w i l l be a c o n s t a n t d i f f e r e n c e in m a t h e m a t i c s
achievement,

as m e a s u r e d by Dossett's post-test,

b e t w e e n the met h o d s of i n s t r u c t i o n at all levels
entry of attitude.

16
That is, there w i l l be no treat m e n t by a t t i t u d e
interaction.
C 2.
(a)

W h e n a linear adjust m e n t is m a d e for the e f f e c t of
va r i a t i o n due to differences in p r i o r attitu d e
tow a r d mathematics,
test,

as m e a s u r e d by Dutto n ' s p r e ­

there w i l l be no s i g n i f i c a n t d i f f e r e n c e in

attitude toward mathematics,

as m e a s u r e d by

Dutton's p o s t - t e s t bet w e e n the meth o d s of in­
struction.
That is there w i l l be no t r e a tment effect.
(b)

W h e n a linear a d j u stment is m a d e for the ef f e c t
v a r i a t i o n due to d i f f erences in p r i o r attitud e
to w a r d mathematics,
test,

as m e a s u r e d by Dulto n ' s

pre­

there w i l l be no s i g n i f i c a n t d i f f e r e n c e in

attitude toward m a t h e m a t i c s as m e a s u r e d by D u l t o n ' s
post-test,

b e t w e e n the entry attitudes.

T h a t is there w i l l be no attitude effect.
(c)

W h e n a linear a d j u s t m e n t is m a d e for the e f f e c t of
v a r i a t i o n due to d i f f e rences in p r i o r atti t u d e
tow a r d mathematics,
test,

as m e a s u r e d by Dulton's p r e ­

there w i l l be a cons t a n t d i f f e r e n c e in

attitude toward mathematics,

as m e a s u r e d by

Dult o n ' s post-test, b e t w e e n the methods of i n ­
struc t i o n at all levels entry of attitude.
That is,
interaction.

there w i l l be no treat m e n t by a t t i t u d e

17
The following hypotheses w i l l be tested to compare
the interns and other two groups of students in the regular
teacher education p r o g r a m on five different aspects of
attitudes.
D.

(a)

There w i l l be no s i g n i ficant difference betwe e n the
m ean-scores of the three methods instruction
(groups)

on Husen's Atti t u d e Scales.

That is,
(b)

there w i l l be no treatment m a i n effect.

There will be no s i g n i ficant difference betw e e n the
m e a n - scores of the three attitude levels groups on
Husen's Attitude Scales.
That is,

there w i l l be no entry attitude m a i n

effect.
(c)

There w i l l be a constant differ e n c e in attitudes,
as m e a s u r e d by Husen's A t t i t u d e Scales, b e t w e e n
the methods of instruction at all levels of entry
attitude.
That is, there w i l l be no treat m e n t by attitude
interaction.
As s u m p t i o n s of the Study
The m a t h e m a t i c s e d u c a t i o n component of the Te acher

Corps p r o g r a m at the M i c h i g a n State U n i v e r s i t y includes
the following assumptions

(which are not n e c e s sarily unique

to the d i s a d v a n t a g e d ) :
1.

T h a t there are differences b e t w e e n the skills
requ i r e d to teach in low-income schools and

18
m i d d l e - c l a s s s c h o o l s , b u t this does n o t imp l y that
u n i q u e p r i n c i p l e s of l e a r n i n g are i n v o l v e d in the
two d i f f e r e n t settings.
2.

T h a t the p r o g r a m for i n n e r city e d u c a t i o n s h o u l d
include great emphasis on developing appropriate
a t t i t u d e s , insig h t s , a n d u n d e r s t a n d i n g amon g
teachers

3.

in i n n e r city schools.

T h a t low a c h i e v e r s in m a t h e m a t i c s m a y be c l a s s i f i e d
i n t o three groups,

viz:

those w h o h a v e i n t e l l e c t u a l

d e f i c i e n c i e s , those w h o have c u l t u r a l d e f i c i e n c i e s ,
a n d those w h o h a v e b o t h i n t e l l e c t u a l and c u l t u r a l
deficiencies

(and p e r h a p s a d i f f e r e n t g r o u p w h o are

neither culturally nor intellectually deficient but
are s l o w e r in l e a r n i n g ) .

It is i m p o r t a n t to c o n ­

s i d e r these d i f f e r e n c e s a m o n g low a c h i e v i n g p u p i l s
in teaching,

both

for i n d i v i d u a l i z i n g the c u r r i c u l u m

a n d for d e v e l o p i n g s e l f - i n s t r u c t i o n a l mater i a l s .
4.

T h a t m a n y c h i l d r e n f r o m l o w - i n c o m e areas are
a s s u m e d to be l o w - a c h i e v e r s b e c a u s e they are
c u l t u r a l l y d e f i c i e n t and two factors are of i m p o r ­
t a n c e in m o t i v a t i n g such children,

viz:

(a)

the

n e e d to d e v e l o p a t e c h n i q u e of c h a n g i n g c h i l d r e n ' s
b e h a v i o u r in o r d e r for s c h o o l l e a r n i n g to take
place,

and

(b)

the n e e d for f l e x i b i l i t y in a p p r o a c h

to t e a c h i n g d e p e n d i n g upon the v a l u e sys t e m s o f the
homes

f r o m w h i c h these c h i l d r e n come.

19
5.

That differences

in v a l u e s , p r i o r e x p e r i e n c e s and

environments among children from various i n c o m e ,
ethnic,

a n d r a c i a l s u b - g r o u p s are so g r e a t that,

teachers n e e d s p e c i a l t r a i n i n g in o r d e r to a p p l y
the p r i n c i p l e s of t e a c h i n g a n d fash i o n the
instructional procedures
6.

for e a c h group.

T h a t the f o l l o w i n g t h r e e a s s u m p t i o n s o f t e n m a d e
w i t h r e s p e c t to the p u p i l / a d u l t of low a b i l i t y m u s t
be rejected:
a.

T h a t the p r o g r a m for the p u p i l / a d u l t of low
abil i t y s h o u l d be formed on drill!

b.

T h a t the l o w - a b i l i t y c h i l d / a d u l t s h o u l d n o t be
r e q u i r e d to t h i n k !

c.

T h a t any p r o g r a m for l o w - a b i l i t y s t u d e n t s
should involve

7.

little or n o reading,

T h a t a n y o n e w h o tea c h e s m a t h e m a t i c s s h o u l d
know mathematics,

(b)

like m a t h e m a t i c s ,

Cc)

(a)
continue

to learn m a t h e m a t i c s , (d) be able to c o m m u n i c a t e
w e l l w i t h the learner,
learning p r o c e s s .

and

(e) u n d e r s t a n d the

In a d d i t i o n , t e a c h e r s of low

a c h i e v e r s n e e d s p e c i a l k n o w l e d g e of the p s y c h o ­
l og i c a l a n d s o c i o l o g i c a l b a c k g r o u n d s o f the
children.
8.

That particularly

for the

for m a t h e m a t i c s comes
physical w o r l d •

l o w - a c h i e v e r s , the n e e d

f r o m e x p e r i e n c e s in the

The emphasis

in e l e m e n t a r y s c h o o l

20
s h o u l d be upon the d e v e l opment of the exp e r i e n t i a l
b a c k g r o u n d enab l i n g symbolic activity and abstract
reasoning.
9.

That Ausub e l ' s three-part teaching strategy of
culturally d i s a d v a n t a g e d children in a laboratory
setting is both a theoretically and p e d a g o g i c a l l y
sound approach.

10.

T h a t Bloom's mo d e l of mas t e r y learning w h i c h
eme r g e d from the w o r k of Carroll

(196 3), and

suppo r t e d by the ideas of M o r r i s o n
(1966), Ski n n e r
Anderson

(1959),

(1954), Suppes
and Gl a s e r

(1926), B r u n e r

(1966), G o o d l a d and

(1968)

is a useful

strategy in d e s i g n i n g the in s t r u c t i o n of the
(disadvantaged)

p r o s p e c t i v e teachers o f the d i s ­

advant a g e d .
11.

That little is k n o w n about disa d v a n t a g e d children's
c apacity to learn, b u t the limits may be
w h a t they now learn and,

far b e y o n d

if prop e r l y taught,

can

learn m o r e m a t h e matics than ordinarily.
12.

T h a t teachers w i t h P o f f e r b e r g e r - N o r t o n
characteristics

(in chapter II)

(195 6)

affect students'

attitude and a c h i e v e m e n t positively, moreover,
tea c h e r - i n i t i a t e d t e a c h e r - s t u d e n t p e r s o n a l i n t e r ­
actions do s i g n ificantly influence students'
achievement.

21
Scope and Delimitations of the Study
To provide i n f o r m a t i o n on the forces that influence
a student*s achievement,

it is neces s a r y to w o r k w i t h i n a

framework that offers a w i d e range of p o t e n t i a l l y relevant
variables w h i c h reflect theory and prac t i c e of teaching and
learning.

EPIC E v a l u a t i o n Ce n t e r in Tucson, A r i z o n a has

d e s igned such framework.

A n adaptation of it to the

p r o p o s e d study is shown below:

Behaviour
Psychomotor Domain
Affective Domain
Cognitive Domain
K

o
H

e

u

£H

Content
Method

U)
55

Organization
Facilities
Cost

n
o

in

i— i

POPULATION

22
The structure is comp o s e d of three sets of variables
— instruction,

population,

and b e h a v i o u r — and it has b e e n

m o s t useful as a heuristic device to reveal combinat i o n s of
variables

leading to a more complete description and

analysis of the instructional program.

Analysis of v a r i ­

ables is generally limited only by the nature and scope of
the p r o g r a m and the desire

for simple or complex analysis.

The forces affecting programs results are obviously
pr o duced through the i n t e r action of variables on each of
the dimensions.

Consequently,

the m a j o r limitations of

this study are:
1.

The study intends to evaluate only the ma t h e m a t i c s
educa t i o n compo n e n t of the Teac h e r Corps program.

2.

Evalua t i o n is confined solely to:
a.

Anal y s i s and critical appraisal of the content,
m e t h o d and o r g a n i z a t i o n of i n s t r u c t i o n —
internal information;

b»

Cognitive and affective b e h a v i o u r of the
interns at interim/process s t a g e — external
information;

c.

Cognitive and affec t i v e behaviours of the
interns at the entry and pro d u c t s t a g e s —
contextual information.

([a] and

[b] have b e e n termed "intrinsic"

off" evaluations r e s p e ctively by Scriven
and

and "pay­
(1967)

[c] has b e e n gener a l l y termed "contextual"

e v a l u a t i o n .)

23
3.

The study does n o t evaluate the e f f e c t of
i n s t r u c t i o n on t e a c h i n g b e h a v i o u r of the interns.

4.

The study does n o t evaluate the e f f e c t of the
p r o g r a m on the b e h a v i o u r s o f the pu p i l s ta u g h t by
the interns,

nei t h e r does it e v a l u a t e the ef f e c t

of the p r o g r a m o n the remai n i n g p a r t of the p o p u ­
lation .
5.

The facilities

and cost aspects of the ins t r u c t i o n

and the p s y c h o m o t o r do m a i n of b e h a v i o u r shall not
be i n v e s t i g a t e d .
6.

The i n s t r uments to be used in the s t u d y shall have
the inhe r e n t l i m i t ations as d i s c u s s e d by Glen n o n
(1949).

CHAPTER II
R E V I E W OF L I T E R A T U R E
Introduction
Distinguishing between

" c u r r i c u l u m static" and

" c u r r i c u l u m d e v e l o p i n g , " Wright,

et al.

tl9 71)

stressed

that m o d e r n conc e p t s o f c u r r i c u l u m favor d y n a m i c c u r r i c u l u m
d e v e l o p m e n t w h i c h is c o n t r o l l e d b y p h i l o s o p h i c a l ,
logical,
Foshay

and s o c i o l o g i c a l

(1961)

c urric ulum.

forces.

psycho­

A d e c a d e b e f o r e this,

d i s c u s s e d the d i f f i c u l t y of h a v i n g a "balanced"
In p a r t i c u l a r , three sources o f e l e m e n t a r y

s c h o o l m a t h e m a t i c s have b e e n i d e n t i f i e d :
1.

the n a t u r e of the learner, w h i c h m a y b e r e f e r r e d
to as the e x p r e s s e d n e e d s - o f - t h e - c h i l d t h e o r y of
curriculum.
This p r o v i d e s p s y c h o l o g i c a l b a s i s for
curriculum theory,

2.

the n a t u r e of his a d u l t society, w h i c h m a y b e
r e f e r r e d to as the n e e d s - o f - a d u l t society, s o c i a l
u t i l i t y , i n s t r u m e n t a l i s m or s o c i o l o g i c a l b a s i s for
c u r r i c u l u m theory,

3.

the n a t u r e of the c o g n i t i v e a r e a - m a t h e m a t i c s , w h i c h
may be r e f e r r e d to as the s t r u c tural, the p u r e
m a t h e m a t i c a l , or the logi c a l t h e o r y o f curric u l u m .
This p r o v i d e s the l o g i c a l , o r pure m a t h e m a t i c a l
basis for c u r r i c u l u m t h e o r y .

E a c h has the p o t e n t i a l

to c o n t r i b u t e s i g n i f i c a n t l y to a

w e l l — d e s i g n e d c u r r i culum.

"Any u n i l a t e r a l a u t h o r i t a r i a n

24

25
v i e w o f the c u r r i c u l a r b a s i s o f the p r o g r a m is an e x t r e m i s t
v iew,"

ar g u e d G l e n n o n and C a l l a h a n

(1970).

In o r d e r to

h a v e a c l e a r p e r c e p t i o n of a b a l a n c e d theory of c u r r i c u l u m ,
a c l e a r p e r c e p t i o n of each of these e x t r e m i s t the ories is
important.
T he m a j o r o b j e c t i v e s o f the T e a c h e r Corps are to
s t r e n g t h e n the e d u c a t i o n a l o p p o r t u n i t i e s a v a i l a b l e to
c h i l d r e n from l o w - i n c o m e

families and to a s s i s t c o l l e g e s

a n d u n i v e r s i t i e s a n d local school d i s t r i c t s to b r i n g a b o u t
b a s i c changes in the w a y s
used.

in w h i c h t e a c h e r s are t r a i n e d a n d

The tea c h e r int e r n s are b e i n g t r a i n e d to t e a c h in

the p o v e r t y - a r e a schools.
se l v es m e m b e r s

M a n y of the int e r n s are t h e m ­

of m i n o r i t y or l o w - i n c o m e g r o u p s .

m i n o r i t y a n d "l o w - i ncome" o f int e r n s

can p r o v i d e c h i l d r e n

w i t h m o d e l s o f a c h i e v e m e n t and scholarship.
the first p a r t of the r e v i e w e x a m i n e s
matics

These

Consequently,

the type o f m a t h e ­

c u r r i c u l u m and i n s t r u c t i o n w h i c h w i l l m e e t the n e e d s

of the d i s a d v a n t a g e d w h i l e the s e c o n d p a r t e x a m i n e s recent,
i m p o r t a n t and s u b s t a n t i v e m e t h o d o l o g i c a l d e v e l o p m e n t s w h i c h
are h e l p f u l in d e s i g n i n g the s t u d y
revise,

in o r d e r to r e f i n e ,

and e x t e n d w h a t are a l r e a d y k n o w n in the f i e l d of

m a t h e m a t i c s e d u c ation.
Spec i f i c a l l y ,

the r e v i e w of l i t e r a t u r e p e r t i n e n t to

the s tudy has b e e n o r g a n i z e d u n d e r s e v e n categorie s :
1.

the i n f l u e n c e o f the s c h o o l and the t e a c h e r o n
the d i s a d v a n t a g e d child,

26
2.

r e s e a r c h on the n a t u r e of the d i s a d v a n t a g e d a n d his
learning p r o c e s s ,

3.

the t h e o r e t i c a l
matics
goals

4.

f o u n d a t i o n s of the e m e r g i n g m a t h e ­

curriculum,

instructional procedures and

for the d i s a d v a n t a g e d ,

r e c e n t e m p i r i c a l s t u d i e s w h i c h s e e m to j u s t i f y some
of the c i r r i c u l a r a n d i n s t r u c t i o n a l pr a c t i c e s ,

5.

Bloom's model of mastery
m e t h o d of i n s t r u c t i o n ,

l e a r n i n g as an e m e r g i n g

its theory,

practice and

r e s e a r c h findings,
6.

promising,

innovative/experimental field-experiences

in t e a c h e r e d u c a t i o n ,
7.

Related

(Research)

Methodology.

I n f l u e n c e of the S c h o o l and the T e a c h e r
on the D i s a d v a n t a g e d C h i l d
"The e s s e n c e of e d u c a t i o n a l h i s t o r y , " claims G ross*
"is to e n a b l e e d u c a t o r s

to learn f r o m the m i s t a k e s of the

p a s t a n d use t h e m i n t e l l i g e n t l y to u n d e r s t a n d the d e v e l o p ­
m e n t of the p h i l o s o p h y o f p r e s e n t e d u c a t i o n a l i n s t i t u t i o n s . "
The

l i t e r a t u r e c i t e d u n d e r t h e n e e d for the s t u d y t o g e t h e r

w i t h the p r e s e n t d i s c u s s i o n h e l p s

to i l l u m i n a t e the

p h i l o s o p h y of the p r o g r a m a n d h e n c e a c c o u n t for the
philosophical

forces o n the cu r r i c u l u m .

* P r o £ e s s o r C a r l G r o s s m a d e the s t a t e m e n t at the
i n t r o d u c t o r y lec t u r e on E d u c a t i o n a l History:
P l a t o to
Locke, W i n t e r T e r m 1974, M i c h i g a n S t a t e U n i v e r s i t y .

27
Reis s m a n

(19 6 3) maintains that there is a g r e a t

deal of evidence that the depr i v e d children and their
adults have a much more positive attitude toward ed u c a t i o n
than is generally believed.

One factor that obscures the

recognition of this is that w h i l e depri v e d individuals v a l u e
education,

they dislike the school,

they are aliena t e d from

the school and they resent the teachers.

Reiss m a n believes

that it is not the d i s a d v a n t a g e d w h o have capitulat e d to
t heir environment, but the teachers and hence w h e n c o n ­
si d ering attitude of the disadvantaged,

attitude toward

ed u c a t i o n and a t t i t u d e toward the school m u s t be considered
separately.

Deutsch

(1967)

claimed that the school only

reinforces the negative responses of children from deprived
backgrounds.

Gor d o n and W i l k e r s o n

(19 66)

ar g u e d that two

f a c t o r s — low m o t i v a t i o n and low s e l f - e s t e e m - - h a n d i c a p the
d i s a d v a n t a g e d child in his acad e m i c d e v e lopment and they
cited studies w h i c h indicate that the m o t i v a t i o n of d i s ­
a d v a n t a g e d children not only is "likely to be lower but is
likely to be dire c t e d toward goals i n c o n s i s t e n t w i t h the
de mands and goal of formal education.
further,

..."

that the typical c u r r i c u l u m is i n c o n gruent w i t h

the child's social experiences,

and this incongruence,

to g ether w i t h lack of motivation, makes
a c h i e vement or success" unlikely.
(1968)

They w e n t

"normal school

Fantini and W e i n s t e i n

suggest that the condition of b e i n g d i s a d v a n t a g e d

cuts across all segments of society;
failure is erroneous,

the idea of human

only i n s t i tutions fail.

Exami n i n g

28
the relation between a student's s e l f - concept and his
experiences in school,

Deutsch recognized that the latter

m a y "either reinforce invidious self-concepts acquired from
the e n v i r o n m e n t or help to d e v e l o p — or even in d u c e — a
negative self-concept.

The school contributes

further to

these negative self-images bec a u s e it fails to stimulate
or create interest in the child at the same time that it
regulates his behaviour.
students improve,

Vane

(1966)

notes that few

once they have e s t a b lished a poor a c h i e v e ­

m e n t record early in their career.

Oak l a n d

(19 70)

remarked

that it seems likely that the lack of neces s a r y antecedent
exper iences causes the child to fall further behind as the
c u r r i c u l u m builds upon abilities he has not acquired.
Deutsch

(1967)

maint a i n s that,

as the age increases it becomes m o r e and more difficult
for these d i s a d v a n t a g e d children to develop compensatory
m echanisms, to respond to special programs, or to make
the p s y c h o l o g i c a l re-adjustments required to overcome
the cumulative effects on their deficits.
In a n a t i o n-wide study on d i s a d v a n t a g e d students recently
p u b l i s h e d by Astin, e t al.

(1972)

it was reported!

. . . It w a s e n c o u r a g i n g to find that the high school
achievements and college progress of these d i s a d v a n ­
taged students* did n o t differ substantially from those
students from more advant a g e d backgrounds.
It w o u l d
seem that b e i n g soci o - e c o n o m i c a l l y d i s a d v a n t a g e d is
not, in itself, a severe hand i c a p to students once they
get to college.
*For the pur p o s e of the analysis of data collected,
they def i n e d "disadvantaged" students as those w h o s e family
incomes was b e l o w $6,000 per y e a r and w h o s e parents h a d not
c o m p l e t e d h i g h school.

29
Go l d b e r g
W o l f and W o l f

(1967), Stodolsky and Le s s e r

(1962)

(1967),

poi n t e d o u t that teachers o f t e n start

w i t h zeal and energy and w i s h to do an effec t i v e job, but
are t hwarted by "reality" and "cultural shock" and are
u nable to fulfill their p r o f e s s i o n a l r e s p o nsibilitie s and
therefore become frustrated,
the w r o n g attitudes,

indifferent,

added O r n s t e i n

angry and "learn"

(1968a).

Nature of the Disadvantaged* and
His Lear n i n g Process
Roueche and W h e e l e r

(1973)

define the "disa d v a n ­

taged" as "the social strata h a v i n g least access to h i g h e r
education."

They rema r k e d that in other context it m a y be

an e u p h e m i s m for low-achieving students or simply for
e c o n o m ically p o o r students.
v a r i o u s l y d e s c ribed as

T h e y o b s e r v e d that this group is

"socially disadvantaged,"

e d u c a t i o n a l l y underprepared,"

"culturally deprived,"

"socio—e c o n o mically deprived,"
"developmental students,"
advantaged,"

"high-risk

"opportunity deprived,"

"socially and culturally d i s ­

"chronically poor,"

"poverty-stricken,"

"culturally alienated."
The p o p u l a t i o n s among the "disadvantaged" vary from
each other in a nu m b e r of ways,
(1967)

Go r d o n

(196 5) and No a r

o b s e r v e d that they can have such co m m o n c h a r a c t e r ­

istics as low e c o n o m i c status,

low social status,

low

*The w o r d "disadvantaged" w i l l h e n c e f o r t h be taken
to q u a l i f y both chil d r e n and y o u t h (young men and women)
w h e r e v e r no s p e c i f i c a t i o n is made for distinction.

30
education achievement,

m a r g i n a l o r n o em p l o y m e n t ,

limited

p a r t i c i p a t i o n in c o m m u n i t y o r g a n i z a t i o n s ,

limited immediate

potential

food and sleep,

for u p w a r d m o b i l i t y ,

too l ittle p e r s o n a l attention,

too little

too l i t t l e s e l f - r e s p e c t and

s e l f - c o n f i d e n c e , too little r e a s o n to try and too l ittle
h a p p i ness.
blacks,

The p o p u l a t i o n s c o n s i s t p r i m a r i l y of A m e r i c a n

Mexicans,

P u e r t o Ricans,

S o u t h e r n r u r a l / m o u n t a i n whites.

A m e r i c a n Indians,

T h e i r c h i l d r e n c o m e to

and leave s c h o o l d i s a d v a n t a g e d to the d e g r e e
c u l t u r e has

and

that t h e i r

f a i l e d to p r o v i d e t h e m w i t h the e x p e r i e n c e s

t y p i c a l o f c h i l d r e n a n d y o u t h that A m e r i c a n s c h o o l s and
colleges

are a c c u s t o m e d to teaching.

Traditional methods

tend to w i d e n the g a p as the d e p r i v e d s t u d e n t stays in
schools.

Th e r e are s o m e s p e c i f i c e n v i r o n m e n t a l

personal characteristics
inter-related)

factors,

and p a r t i c u l a r e x p e r i e n c e s

(all

t h a t h a v e a d e t r i m e n t a l e f f e c t on the

learning process

(treated as the d e p e n d e n t variable)

s t u d i e s reviewed.

in

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

v a r i a b l e s are self- c o n c e p t ,

m o t i v a t i o n a n d the c o m p o n e n t s

o f v a r i a b l e l e a r n i n g are language,

c o g n i t i o n and p e r c e p t u a l

style.
The socio-economic class

in w h i c h a c h i l d is

s o c i a l i z e d p r o v i d e s e x p e r i e n c e s w h i c h m a y i n f l u e n c e his
academic achievement.

R e v i e w i n g the res u l t s o f n u m e r o u s

studies, Gordon

c o n c l u d e d that h o m e s o f low SES

(1965)

fail to p r e p a r e the c h i l d for l e a r n i n g b e c a u s e the y
appropriate stimuli

like books,

toys,

lack

and i n s t r u c t i o n a l

31
e q u i p ment.

T h e a b s e n c e of v i s u a l stimuli,

an e x c e s s of noise,

li m i t s c o n c e n t r a t i o n .

together with
This

led D e u t s c h

(196 7) to r e p o r t that a p e r s o n for low SES lacks the
e x p e r i e n c e t h a t e n a b l e s h i m to

"manipulate and o r g a n i z e the

v i s u a l p r o p e r t i e s of his e n v i r o n m e n t and thus p e r p e t u a l l y
to o r g a n i z e a n d d i s c r i m i n a t e

the n u a n c e s of his e n v i r o n ­

m e n t ."
A m o n g others, M o n t a g u e
(1964)

(1964),

D e u t s c h and B r o w n

s h o w e d t h a t SES c o r r e l a t e s w i t h i n t e l l i g e n c e which,

a c c o r d i n g to H u n t

(1961),

is a f u n c t i o n of the p r o c e s s o f

p e r s o n a l i n t e r a c t i o n w i t h e n v i r o n m e n t r a t h e r than a
p r o d u c t of g e n e t i c
Schaefer

(1961)

factors.

Ricsin

(1961),

T h o m p s o n and

d e s c r i b e h o w e a r l y s t i m u l a t i o n affects the

d e v e l o p m e n t of p r o p e r n e u r a l s t r u c t u r e s w h i l e Solomon,
e t al.

(1961)

c o n t e n d t h a t an i m p o v e r i s h e d e n v i r o n m e n t

r e d u c e s a p e r s o n ' s d i s c r i m i n a t o r y and m a n i p u l a t i v e
and his d e s i r e
be necessary

for e x p l o r a t o r y b e h a v i o u r w h i c h is t h o u g h t to

for p r o b l e m solving.

c o m p e t e n c e in v a r i o u s
"has

abilities

languages,"

"Failure to acq u i r e
reported Ausubel

(1964),

l i m i t e d the d i s a d v a n t a g e d c h i l d 's abil i t y to a d v a n c e

f r o m c o n c r e t e to a b s t r a c t r e a s o n i n g " ; "hence he c a n n o t
handle symbolic

l a n g u a g e and c o n c e p t s ," a d d e d S l a u g h t e r

(19 69).

(19 6 5) w e n t

Gordon

f u r t h e r to say t h a t the

i n a b i l i t y to r e a s o n by i n d u c t i o n a n d to a p p l y or t r a n s f e r
knowledge through

l i n k i n g of c o n c e p t s is c h a r a c t e r i s t i c of

the t h o u g h t p r o c e s s of d i s a d v a n t a g e d youth.

32
Gordon

(1965)

s u g g e s t e d that lo w e r - c l a s s h o m e s are

d e f i c i e n t in the n e c e s s a r y i n t e r a c t i o n w i t h p a r e n t s that
activates
(1967)

the chil d ' s

interest and motivation.

Deutsch

e m p h a s i z e d that the m i d d l e —class c h i l d rece i v e s

greater intellectual stimulation,

for w h i c h he is rewarded,

than does the l o w e r - c l a s s child, w h o s e p a r e n t s s e l d o m
s u b j e c t h i m to the p r e s s u r e of a formal a d u l t - c h i l d
l e a r n i n g situation.'*’
Dave

It is n o t the status p e r s e , a r g u e d

(1963), p a r e n t - b e h a v i o u r is the c r i t i c a l v a r i a b l e

(which does n o t m a k e the home e n v i r o n m e n t stimulating.)
Fifer

(19 64)

found that, w i t h i n e t h n i c groups,

c h i l d r e n of

d i f f e r e n t s o c i a l cla s s e s d i f f e r e d in their p e r f o r m a n c e on
a n u m b e r of tests.
(196 8)

Epps

(1970),

Rosenberg

(1965), A s b u r y

found that p a r e n t a l b a c k g r o u n d affects the c h i l d ' s

s e l f - c o n c e p t and s e l f - a s p i r a t i o n s .

While Goff

(1954)

r e p o r t e d t h a t c h i l d r e n of low SES feel m o r e i n a d e q u a t e in
sc h o o l t h a n others,

Epps

(19 70)

h y p o t h e s i z e d t h a t e v e n at

the time they e n t e r school, such c h i l d r e n h a v e l i t t l e self2
c o n f i d ence,
w h e r e a s E d w a r d s and W e b s t e r (196 3) f o u n d that
positive self-concepts
a n d a s piration.

are r e l a t e d to a c a d e m i c a c h i e v e m e n t

Coleman,

all f a ctors co n s i d e r e d ,

et al.

(1966)

reported,

t h a t of

the d e g r e e to w h i c h a p e r s o n p e r ­

c e i v e s h i m s e l f as b e i n g able to c o n t r o l his e n v i r o n m e n t and
his f u ture is the m o s t c r i c u a l to a c h i evement.

1op.
2

op.

c i t . , p.
cit.

49.

Hal l

(19 69)

33
reported a study which supported Coleman's

conclusi o n .

H a l l ' s study goes a s t e p f u r t h e r to s h o w t h a t conve r s e l y ,
the sense o f p o w e r l e s s n e s s a n d i n a b i l i t y to c o n t r o l o n e ' s
d e s t i n y , c h a r a c t e r i s t i c of p e r s o n s

f r o m lower s o c i o ­

e c o n o m i c groups a n d c u l t u r a l l y d e p r i v e d h o m e s , redu c e s
m o t i v a t i o n and leads to u n f a v o u r a b l e s e l f - c o n c e p t s , t h e r e b y
inhibiting

learning potential.

Clift

(1969)

has s u m m a r i z e d

the c h a r a c t e r i s t i c traits of c u l t u r a l l y d i s a d v a n t a g e d y o u t h
under three categories— personality,

cognitive

func t i o n s

an d e d u c a t i o n a l values.
The T h e o r e t i c a l F o u n d a t i o n o f the E m e r g i n g
P r a c t i c e s In M a t h e m a t i c s Cu r r i c u l u m ,
Instructional Procedures a n d G o a l s
for the D i s a d v a n t a g e d
Gordon and Wilkerson
all o u r c u r r e n t e f f o r t s

(1966)

r e m a r k e d that d e s p i t e

(on e d u c a t i n g the d i s a d v a n t a g e d ) ,

e f f e c t i v e a p p r o a c h to t e a c h i n g the d i s a d v a n t a g e d ha s n o t
b e e n found.

Goldberg

(19 64), H a b e r m a n

(1963), W e b s t e r and L u n d

(1969)

(19 64), P a s s o w

and Wilkerson

(1964,

1966),

a l l c l a i m t h e r e is n e e d for m u c h s t u d y on t e a c h e r b e h a v i o u r
c h a r a c t e r i s t i c s and the r e l a t i o n s h i p b e t w e e n the t e a c h e r s
of the d i s a d v a n t a g e d and t h e i r students.
Lesser

(1967)

Stodolsky and

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

w a n t to s u c c e e d b u t fail b e c a u s e b e h a v i o u r t e c h n i q u e s h a v e
not been developed
Daniel

"which p r o v i d e d e s i r a b l e o u t c o m e . "

(1967), G o l d b e r g

(1967),

and W i l k e r s o n

(1964,

1966)

c o n t e n d t h a t w e m u s t l e a r n w h a t works, w i t h whom, w i t h w h a t

34
variables; w i t h o u t this knowledge, we cannot succeed and
will jump from one approach to another.
In the area of general psychology, Gor d o n and
Wilkerson

(1966)

discu s s e d the i n t e r actionist and p r o j e c t i v e

theories of behaviour.

The former sees behav i o u r p a t t e r n s

as being shaped by interaction of individual w i t h his
e n v i r o n m e n t w h i c h acts to foster or impede the p s y c h o l o g i c a l
and intellective d e v e l opment necessary for learning.

This

i n t e r a c t i o n i s t theory implies that compensatory e d u c a t i o n
and e nrich m e n t of the programs can affect,

in some fashion,

the disabilities associ a t e d w i t h disa d v a n t a g e d status.

The

p r o j e ctive theory assumed that individual b e h a v i o u r is
b a s e d on cer t a i n p r e d e t e r m i n e d patterns w h i c h are a c t i v a t e d
by e n v i r onmental stimuli; e n v i r o n m e n t a l forces m a y a f f e c t
be h a v i o u r a l forms,

but,

in general, b e h a v i o u r is co n t r o l l e d

by drives that are "genetically est a b l i s h e d and bound."
The theory thus emphasizes the intractability and p e r m a n e n c e
of early characteristics implying that no amount of social
action can improve performance.
The p r o p onents of the projective theory m u s t have
re c eived a great shock from Brun e r ' s

(1962)

hypothesis:

Any subj e c t can be ta u g h t e f f e c tively in some i n t e l l e c ­
tually ho n e s t form to any child at any stage of d e v e l o p ­
m ent

1Jerome S. Bruner, The Process of Educa t i o n
(Cambridge:
Harv a r d U n i v e r s i t y Press, 1962), p. 33.

35
C a rroll

(1963)

t i m e - to-mastery model gave a second b l o w to

propo n ents of projec t i v e theory.
thesis B l o o m

(1969)

Descri b i n g Carroll ' s

said,

. . . Implicit in this formulation is the assump t i o n
that given enough time, all students can conceiv a b l y
attain m a s t e r y of a learning task. . ..
Adler

(1957)

rejected projec t i v e

theory in its entirety.

He r e a ffi r m e d his posi t i o n again in 1972.

He d e s c ri b e d the

theory as one of the p r i n c i p a l causes of p o o r learning by
those w h o m it stigmatizes as genetically inferior.
e x p l ained that pupil failure is taken for granted,

He
and so

n e i t h e r teacher nor c h i l d is required to exert the e f f o r t
that m i g h t p r e v e n t failure.

Adler

(1972) m a i n t a i n s that

though the i n t e r a c t i o n i s t theory

points to social

ences that do e x i s t and that can

be corrected,

influe-

it only

tells the half truth and it has b e e n harmful b e c a u s e it is
g e n e rally c o m b i n e d w i t h the tacit assumption that ch i l d r e n
w h o have been s u b j ected to these r e t a rding social influences
cannot be expe c t e d to learn in school.

He contends that

any theory that reduces our e x p e c t a t i o n of w h a t chil d r e n
can do inevitably reduces their level of ac h i e v e m e n t and
b e comes a se l f - f u l f i l l i n g prophecy.
Silberman
theory is harmful,
expectation,

(1970)

and A d l e r

(1972)

beli e v e the I.Q.

it affects n o t only the teacher's

as shown by J a c o b s o n and Rosenthal

b u t also the pupil's e x p e c t a t i o n of himself.
teacher b a c k from teaching,
learning.

(1968),

It holds the

and holds the pupil b a c k from

In the w o r d s of teachers:

36
W e think the children are n o t capable of learning and
w e teach them l e s s . T h e n they learn less and become
less capable of learning.
C l e a r l y by 196 4 the USOE and the N C T M h a d lost
confidence in the p r o j e c t i v e theory and they jointly
spons o red a conference held in Washington,

D.C.

25-27,

in mathematics.

H.

1964 to discuss the "low achievers"

L . Phillips,

on March

specia l i s t in m a t h ematics at the Of f i c e of

Ed u c a t i o n gave five reasons why the USOE and the N C T M were
co n c erned a b o u t low achievers in mathematics.

Five p o s i t i o n

papers w e r e p r e s e n t e d on s o c i ological and psyc h o l o g i c a l
factors in low ach i e v e m e n t and six other papers were
invited on the then current p r o m i s i n g practices.

Am o n g

other r e c o m mendations that e m e r g e d from the conferenc e was
a g u i d e-line

for teaching m a t h e matics to the low achiever

(appendix C of the r e p o r t ) , w h e r e it is r e c o m m e n d e d that a
laboratory set t i n g is especi a l l y e f f e ctive for low achievers,
Woodby

(1965).
Ag a i n in April,

196 4, through the interest of the

C o o p e r a t i v e Research b r a n c h of the Uni t e d States Of f i c e of
E d u c a t i o n funds w e r e made available by the USOE to SMSG for
an e x p l o r a t o r y Co n f e r e n c e on M a t h e matics E d u c a t i o n F o r
B e l o w A v e r a g e Achie v e r s held in Chicago,
A p r i l 10 and 11, 1964.
conference.

Illinois,

on

Seven papers w e r e p r e s e n t e d at the

T w o of the papers of p a r t i c u l a r relev a n c e to

d i s a d v a n t a g e d are

"Psychological Issues In the D e v e l o p m e n t

of M a t h e m a t i c s C u r r i c u l a for Socially D i s a d v a n t a g e d
Children" by H. Be i l i n and L. G. Gotkin and "Mental

37
D e v e l o p m e n t a n d L e a r n i n g of M a t h e m a t i c s in S l o w — le a r n i n g
C h i l d r e n " by G l o r i a F. L e i d e r m a n .
T h e s u m m a r y o f the C o n f e r e n c e G e n e r a l D i s c u s s i o n
p r o b a b l y s t i m u l a t e d b y t h e s e two p a p e r s p u t e m p h a s i s on
four a r e a s :
1.

Mental Development:

T o p r o c e e d f r o m conc r e t e

g r a d u a l l y to abstract,
makes mathematics

and h a b i t s of t h o u g h t w h i c h

instruction possible was

suggested

for research.
2.

Distinction between

low achievers:

Four d i s t i n c t

cla s s e s w e r e identified:
a.

children who

have intellectual deficiencies,

b.

children who

have c u l t u r a l d e f i c i e n c i e s ,

c.

children who

have both deficiencies,

d.

c h i l d r e n w h o are n e i t h e r c u l t u r a l l y n o r
i n t e l l e c t u a l l y d e f i c i e n t b u t are slow learners,
a l t h o u g h they m a y n o t be less i n t e l l i g e n t than
the a v e r a g e group,

they take

lo n g e r time to

process informa t i o n .
I n d i v i d u a l i z a t i o n of c u r r i c u l u m a n d d e v e l o p m e n t of
s e l f - i n s t r u c t i o n a l m a t e r i a l n e c e s s i t a t e d the
d i s t i n c t i o n a m o n g t h e s e groups.

They emphasized

t h a t in o r d e r to m o t i v a t e t h o s e c l a s s i f i e d as
c u l t u r a l l y defici e n t ,
" docility"
place,

we should

(a) d e v e l o p

in o r d e r for s c h o o l l e a r n i n g to take

(b) be

f l e x i b l e in a p p r o a c h to t e a c h i n g

38
d e p e n d i n g u p o n the v a l u e sys t e m s of the h o m e s f r o m
w h i c h these c h i l d r e n come.
3.

Action Programs were suggested
the d e f i c i e n c i e s
our knowledge

in culture.

and h u n c h e s

for c o m p e n s a t i n g

for

T h e s e are b a s e d u p o n

as to w h a t and h o w to

teach.
4.

D e m o n s t r a t i o n and R e s e a r c h P r o g r a m s on m o t i v a t i o n ,
h e l p i n g c h i l d r e n to c h a n g e g o a l s
strategies
teach.

and a t t i t u d e s ,

for t e a c h i n g a n d w h a t m a t h e m a t i c s

to

T h e y r e j e c t e d v i g o r o u s l y three a s s u m p t i o n s

o f t e n m a d e w i t h r e s p e c t to p u p i l o f low ability,
viz:
a.

t h a t the p r o g r a m for the p u p i l of l o w - a b i l i t y
s h o u l d be

b.

f o u n d e d on drill,

t h a t the l o w - a b i l i t y c h i l d s h o u l d n o t b e
r e q u i r e d to t h i n k ,

c.

t h a t any p r o g r a m for l o w - a b i l i t y s t u d e n t s
should involve

little or no r e a d i n g .

T h e y e a r 1964 a p p e a r e d to b e r e m a r k a b l e

in the

h i s t o r y o f m a t h e m a t i c s e d u c a t i o n of the d i s a d v a n t a g e d .
David P . Ausubel

(1964)

p u b l i s h e d a p a p e r , "How r e v e r s i b l e

are the c o g n i t i v e a n d m o t i v a t i o n a l e f f e c t s of c u l t u r a l
deprivation?

Implications

d e p r i v e d child."

for t e a c h i n g the c u l t u r a l l y

A u s u b e l d r a w s on r e s e a r c h to a i d in his

a s s e s s m e n t of the c o n s e q u e n c e s of c u l t u r a l d e p r i v a t i o n on
the d e v e l o p m e n t o f v e r b a l a n d a b s t r a c t i n t e l l i g e n c e ,
w e l l as o n m o t i v a t i o n

for a c a d e m i c a c h i e v e m e n t .

as

He b a s e d

39
his argument on the "critical periods" hypothesis that
"there are optimal periods of readiness for all kinds of
c ognitive development."

The corollary to this hypothesis

is that individuals w h o fail to acquire these skills at
appropriate times are forever handicapped in attaining
them.

To Ausubel,

the theory does not p r o c l a i m that a

person cannot acquire these intellectual skills or subject
m atter contents at times other than the critical period;
rather, he contends,

there is a considerable loss of "years

of opportunity w h e n reasonably economical learning could
have occurred if attempted,

but did not."

The consequences

is a learning deficit w h i c h hampers both current and future
intellectual development.

A u s u b e l believes

that the

e n v i r on m e n t a l l y induced r e t a r dation in verbal intell i g e n c e
is somewhat reversible.

He examines theoretical bases and

r esearch evidence in this area and possibilities of
reversing such retardation.

In discussion of educational

implications of this theory for the culturally depr i v e d
child, Aus u b e l concludes w i t h a three- p a r t teaching
strategy that emphasizes,
1.

selec t i o n of learning mater i a l s g e a r e d to the
learner's readiness state,

2.

c o n s o lidation of all ong o i n g learning tasks b e f o r e
introducing new o n e s , and

3.

d e v e l o p m e n t and use of struct u r e d materials to
facilitate sequential learning.
Jencks and Riesman

language"

(1968)

have called "in d r a matic

for bold new approaches in mee t i n g the needs of

40
the d i s a d vantaged students.

They beli e v e it is a m i s t a k e

to try to teach u n s o p h i s t i c a t e d students traditional
academic subjects by traditional academic methods.

Such

students must progress step by step from their natural
culture in w h i c h they are immersed.

T h e y argued that m a n y

skills can be taught by using materials drawn from this
po p u l a r culture,

"assuming the teacher is familiar w i t h it

and has some a p p r e ciation of it."

One of the consequences

of this call was the SMSG conference on the Mathema t i c s
Ed u c a t i o n of the Inner City Schools held in P h i l a d e l p h i a in
March,

1970.

This confer e n c e was discussed in one of the

pr e c eding s e c t i o n s .
Bl o o m

(1968)

states the social and econ o m i c

imperatives for exten d i n g higher e d u c ation opp o r t u n i t y to
all.

He maintains

that the b a s i c p r o b l e m is to det e r m i n e

how the largest p r o p o r t i o n can learn e f f e c tively those
skills and subject mat t e r regarded as essen t i a l for their
own d e v e l o p m e n t in a com p l e x society.

The B l o o m

(1968)

M a s t e r y mo d e l appears to be one of the m o s t promising
developments on the cur r e n t scene.

C o m m e n t i n g on M a s t e r y

learning and its i m p l i c a t i o n for C u r r i c u l u m Development,
Cronbach

(1969)

e x p r e s s e d h o w e n t h u s i a s t i c he was once

about the Carroll's model and h o w they

(his group)

h a d to

a b andon the whole not i o n of time to m a s t e r y or time to
reach criterion or rate of learning.

He repor t e d that they

did this because learning is m u l t i - d i m e n s i o n a l in a l a b o ­
ratory,

a n d it is far m o r e m u l t i - d i m e n s i o n a l in t h e school

41
for as one t h i n g is b e i n g taught,
h a p p e ning.

a lot o f things are

He c o n c l u d e d that things

for w h i c h w e can

c l e a r l y use a t r a i n i n g m e t h o d o l o g y d e s i g n e d to b r i n g p e o p l e
to a p e r f o r m a n c e c r i t e r i o n a p p a r e n t l y are lim i t e d to s t a t i c
k n o w l e d g e and a l g o r i t h m s .
T o cause l e a r n i n g is the k e y c o n c e p t in the B l o o m 1s
model.

New p r o g r a m s w i l l n o t b e d e v e l o p e d ov e r n i g h t .

i n d i v i d u a l i n s t r u c t o r rem a i n s

the k e y to any e f f e c t i v e

p r o g r a m for the d i s a d v a n t a g e d .
educational programs
an d W i l k e r s o n

After studying compensatory

at a v a r i e t y o f i n s t i t u t i o n s ,

Gordon

(1966)^ d e f i n e d four g e n e r a l themes o r

o b j e c t i v e s of s u c h progr a m s ,
hu m a n i t a r i a n ,

The

the f i r s t two of w h i c h are

the t h i r d is a r e s e a r c h o b j e c t i v e a n d the

fourth is a v a r i a t i o n of s o c i a l lifting;

viz.,

1.

h e l p i n g the d i s a d v a n t a g e d to d e v e l o p t h e i r p o t e n t i a l
and p r o v i d i n g t h e m w i t h e q u a l o p p o r t u n i t i e s ;

2.

a s s i s t i n g in the e l i m i n a t i o n of a c a d e m i c d e f i c i ­
en c i e s ;

3.

s t u d y i n g the eff e c t s o f the p r o g rams;

4.

a c h i e v i n g a d i v e r s i f i e d s t u d e n t body.

and

T h e f i r s t t h r e e have b e e n i n c o r p o r a t e d into the p r o p o s e d
study b u t r e s t r i c t e d to the m a t h e m a t i c s e d u c a t i o n c o m p o n e n t
of this program.

^op.

cit.,

pp.

148-49

42
Re sea r c h and E v a l u ati on Li ter a t u r e on Laborat ory
A p p r o a c h as an Instruct ion al M e t h o d o l o g y
for the D isa d v a n t a g e d
As it has been p o i n t e d out se veral times in p r e c e d i n g
sub-sections,

the w o r d

"disadvantaged"

refers to children

as w e l l as adolescents cultur all y and socio-ec ono mic all y
disadvantaged.

Research has shown that they are slow-

learners and co nsequently low-achievers b ecause they have
cultural d e f i c i e n c e s .1

P ikart and W i l s o n

research on ins tru cti ona l progr ams
general.

(1972)

rev iewed

for slow-le arn ers in

This review together w i t h several reports on

teaching of the dis a d v a n t a g e d i ndi c a t e that the mos t
pr omi s i n g p rac t i c e in the m a t h e m a t i c s t eaching to the
d i s a dv ant age d is the mas ter y lea rning m o d e l via laboratory
approach.

A s ummary of the app r o a c h was given by A l l e n C.

F r i e b e l .2
In 19 71, the m ath ema tic s edu ca t i o n community in the
Un i t e d States gave favorable pu bli c i t y in s upport of the
laboratory approa ch to teaching m a t h e m ati cs to elementary
school ch ild ren and pr o s p e c t i v e e lem ent ary school teachers.
The De ce m b e r 1971 issue of the A r i t h m e t i c T e a c h e r was

Ri c h a r d W, Schulz, " Characteristics and Needs of
the Slow Learner," T h e Slow L e a r n e r , 35th Yearbook, NCTM,
1972, pp. 1-25.
A l s o R iessman (1963).
2
A l l e n C. Friebel, "Mathematics Experi enc e," in
T e ac h i n g the D i s a d v a n t a g e d C h i l d , e d i t e d by S. W. T ied t
(Oxford Uni ver s i t y fre'ss, 1968) , pp. 16 5-9 4.

43
d e v o t e d to m a t h e m a t i c s

l abo rat ori es.

T h r e e of these

a r t i c l e s w i l l b e b r i e f l y disc uss ed:
Discussing

"The M a t h e m a t i c s L a b o r a t o r y

m e n t a r y an d M i d d l e Sch o o l s ," B a r s o n

(19 71)

for E l e ­

n o t e d t h a t it

is i m p o s s i b l e to gi ve a u n i v e r s a l d e f i n i t i o n of a "Math.
L a b ." due to its v ar i o u s uses an d s t y l e s of o r g a n i z a t i o n .
He g ave s e v e n c h a r a c t e r i s t i c s o f a "good" m a t h - l a b
distinguished between

four types of m a t h - l a b — d e c e n t r a l i z e d /

c l a s s r o o m l a b , c e n t r a l i z e d l a b , t e a m - r o o m lab,
movable

and

roving/

l a b — a n d c o n c l u d e d w i t h the m a j o r p u r p o s e s /

o b j e c t i v e s of m a t h - l a b viz. m o t i v a t i o n ;

enrichment;

a r t i c u l a t i o n w i t h the r e g u l a r m a t h e m a t i c s program;

and

r e v i e w , reinforcement and remediation.

Embank

a t t e m p t e d to a n s w e r s e v e n t e e n q u e s t i o n s

on w h a t ? w h y ?

when? how?
R o b e r t E.

as r e l a t e d to the m a t h e m a t i c s
Reys

(1971)

discussed

Using Manipulative Materials."
lative m a t e r i a l s

(1971)

lab ora tor y.

"Considerations

for T e a c h e r s

He d i s t i n g u i s h e d m a n i p u ­

f r o m t e a c h i n g aids an d c i t e d r e f e r e n c e s

for

s e l e c t i o n and uses of m a n i p u l a t i v e m a t e r i a l s and g a v e a
list of nine,

non-independent,

n o n - e x h a u s t i v e s t a t e m e n t s on

l e a r n i n g the ori es w h i c h f o r m the b a s i c

foundation underlying

the r a t i o n a l e

for u s i n g m a n i p u l a t i v e m a t e r i a l s

in l e a r n i n g

mathematics.

He c o n c l u d e d t h a t t h o u g h the r a t i o n a l e seems

e d u c a t i o n a l l y s o u n d , r e s e a r c h in this a r e a has n o t b e e n
c o n c l u s i v e in s u p p o r t i n g / r e f u t i n g t h e v a l u e of m a n i p u l a t i v e
materials.

44
Fe instein

(19 72) ob se r v e d that little is kno wn

about inner-city youth or how they learn m a t h e m a t i c s , the
characteristics of good teachers in urban schools and
related educatio nal problem.

He re viewed the need to

improve the preparat ion of teachers from C U P M p o i n t of
view a n d e m p h asi zed that it w o u l d be m o s t unwise to impose
additional qualific ati ons on prospe cti ve teachers of ghetto
youth w h e n it is already d iff icult to attract capable
teachers to inner-city schools.

He strongly re com men ded

di sc o v e r y - o r i e n t e d activity for training of p r o s p e c t i v e
teachers of inner-city youth using laboratory approach.
Rouse
matics

(19 72)

dis cussed the m i s c o n c e p t i o n of m a t h e ­

laboratory by some teachers.

He e m p h a s i z e d that

the term "math lab" is n o t totally descriptive of the
instructional strategy that it represents,

for the t erm

suggests only the p res e n c e of special e q u i p m e n t

(as some

teachers take i t ) , but provides no clue to the special
me tho dol ogy w h i c h comprises the e ssence of this ap proach to
instruction.

He d i s t i ng uis hed D i e n e s 1 co n c e p t of it from

that of Nu ff i e l d M a t h e ma tic s T e a c h i n g P r o j e c t to illustrate
that there exists no single,

universal c oncept of w h a t

constitute an ideal "math-lab" program.
ch ara cte riz ing princi ple s of the str ategy
Dienes*

terms)

He then gave seven
(using some

d isc u s s e d the relati ons hip bet w e e n the

principles and Piaget's Cog ni t i v e Th e o r y and d i s t i n g u i s h e d
the laboratory approach from the w e l l - k n o w n sym bol ic
di sco ver y method.

45
Vance

(1969)

i n v e s t i g a t e d a l a b o r a t o r y p r o g r a m that

m i g h t b e t y p i f i e d as a s e p a r a t e d program.,.

It f u n c t i o n e d as

an a d j u n c t to t he r e g u l a r c u r r i c u l u m in s e v e n t h and e i g h t h
grades.

T e s t s of achieveme nt,

r ete nti on,

a nd t r a n s f e r

r e v e a l e d t h a t the s t u d e n t did l e a r n n e w m a t h e m a t i c a l ideas
in the l a b o r a t o r y s e t t i n g , a l t h o u g h they le a r n e d s l i g h t l y
less
c l ass

than a s e c o n d e x p e r i m e n t a l g r o u p t a u g h t the w h o l e situation.

W h i l e a t t i t u d e s in m a t h e m a t i c s a m o n g the

e x p e r i m e n t a l gr oup s w e r e n o t s i g n i f i c a n t l y d iff er e n t ,
s t u d e n t r e a c t i o n w as m o r e

f a v o u r a b l e to the l a b o r a t o r y

s e t t i n g than to the c lass s e t t i n g .
s t u d y r e c e i v e d the ir m a t h e m a t i c s
separated

geometry

instruction

in this
(Math 201)

fro m the m e t h o d s in str uct ion .

Wilkinson
developed

Th e g r o u p

(19 70),

lab ora tor y un its

u s i n g a n i n t e g r a t e d approach,
to t e a c h topics in m e t r i c

to s i x t h grade pupils.

that the s t u d e n t s

Analysis of data indicated

ta ugh t b y the l a b o r a t o r y m e t h o d d i d as

w e l l as o n the g e o m e t r y a c h i e v e m e n t p o s t - t e s t as s t u d e n t s
in t he c o n t r o l class i n s t r u c t e d b y a c o n v e n t i o n a l t e a c h e r t e x t b o o k ap proach.

W i l k i n s o n a l s o r e p o r t e d t h a t the

laboratory approach did not significantly affect p u p i l s '
attitude

t o w a r d m a t h e m a t i c s b u t t hat the m e t h o d a p p e a r e d

to be m o r e e f f e c t i v e w i t h s t u d e n t s of m i d d l e or low
intelligence.

G r o u p G 2 in the p r o p o s e d s t u d y w i l l u se an

i n t e g r a t e d a p p r o a c h s i m i l a r to this.

Johnson

(1970)

in a

y e a r long study to d e t e r m i n e the e f f e c t i v e n e s s of u s i n g
activity-oriented

lessons

to t e a c h n u m b e r theory,

g eom etr y,

46
a n d measurement:,
mathematics

and r a t i o n a l n u m b e r s

f oun d th at

in s e v e n t h g r a d e

(1) p e r f o r m a n c e of s t u d e n t s

ta u g h t

e x c l u s i v e l y by the a c t i v i t y - a p p r o a c h w a s i n f e r i o r to that
of s t u d e n t s r e c e i v i n g t e x t - b o o k - b a s e d o r a c t i v i t y - e n r i c h e d
i nst r u c t i o n ,

(2)

laboratory

lessons in the s t u d y of m e a s u r e ­

m e n t a n d g e o m e t r y w e r e p a r t i c u l a r l y e f f e c t i v e for l ow and
m i d d l e - a b i l i t y s tud ents,

and

(3)

attitude measures

to r e v e a l any s i g n i f i c a n t d i f f e r e n c e s

fa i l e d

among treatment

groups.
Wasylyk

(1970),

using integrated program,

organized

a m a t h - l a b to t e a c h m e a s u r e m e n t c o n c e p t s a n d sk i l l s
ab il i t y n i n t h g r a d e stu dents.

F i r s t t h e s t u d e n t s w o r k e d in

small groups using concrete materials,
di s c u s s i o n s ,

there w e r e class

p r o b l e m l a b o r a t o r y ses sio ns,

s p e c i f i c purpose.

to low-

each with a

R e s u l t s of the s t u d y i n d i c a t e d t h a t the

a c h i e v e m e n t of s t u d e n t s

in the l a b o r a t o r y g r o u p w a s s i g n i f i ­

c a n t l y h i g h e r than those in th e c o n t r o l g r o u p t a u g h t the
s a m e t op i c s in t e a c h e r - d i r e c t e d s e t t i n g .
was

fo u n d t hat the s t u d e n t s

in the

In a d d i t ion ,

it

laboratory group

e x h i b i t e d s i g n i f i c a n t l y h i g h e r a t t i t u d e to w a r d s m a t h e ­
matics

t h a n d i d c o n t r o l group.

Students'

preference

this l a b o r a t o r y m e t h o d w a s s t r o n g l y ind ica ted .

This

for
seems

to i n d i c a t e a s t r o n g s u p p o r t f or the u se of this a p p r o a c h
to d i s a d v a n t a g e d s t u d e n t s .

T h e a p p r o a c h is a l m o s t i d e n t i ­

cal w i t h t h a t of G r o u p G 1 in the p r o p o s e d study.
m a n y of t h e m as h a v i n g a w e e k b a c k g r o u n d .

I see

47
Hollis

(19 71)

r e p o r t e d a s t u d y w h i c h is n o t e w o r t h y

for the p e r i o d of study,

c a r e f u l s a m p l i n g an d analysis.

Th e i n v e s t i g a t o r c o n c l u d e d t h a t l a b o r a t o r i e s o r g a n i z e d to
p r o v i d e p e r s o n a l a n d i n d i v i d u a l i z e d a s s i s t a n c e are m o s t
h e l p f u l to learners

that are e i t h e r a c a d e m i c a l l y o r c u l ­

turally disadvantaged.

W i l d e r m a n and K r u l i k

(197 3), w h o

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

laboratories

at the u n i v e r s i t y level a n d p u b l i c

s c h o o l s in the U n i t e d S t a t e s , r e p o r t e d two s tu d i e s c a r r i e d
o u t by S c h i p p e r t

(196 5) and H o w a r d

(19 70).

Schippert found

t h a t the i n n e r - c i t y p u p i l s w h o m a n i p u l a t e d a c t u a l m o d e l s o r
presentations of mathematical principles showed signifi­
c a n t l y h i g h e r a c h i e v e m e n t o n m e a s u r e s of sk ill s t h a n did
pupils

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

v e r b a l an d w r i t t e n d e s c r i p t i o n s of t h o s e p ri n c i p l e s .
Howard used mathematics

l a b o r a t o r y e x p e r i e n c e s to f a c i l i ­

tate a h i e r a r c h y of n e e d e d c o n c e p t s w i t h e n v i r o n m e n t a l l y
an d a c a d e m i c a l l y d i s a d v a n t a g e d rural c h i l d r e n •

S uch

e x p e r i e n c e s r e s u l t e d in b o t h a c h i e v e m e n t and a t t i t u d i n a l
gains.
In summary,
suggests

the r e s e a r c h and e v a l u a t i o n

literature

t hat l a b o r a t o r y a p p r o a c h e s can be u s e d p r a c t i c a l l y

an d v e r y e f f e c t i v e l y w i t h c u l t u r a l l y / e n v i r o n m e n t a l l y /
academically disadvantaged s t u d e n t s .

H o w e v e r , the de g r e e

of e f f e c t i v e n e s s of u t i l i z a t i o n d e p e n d s on the o r g a n i ­
zations .

F u r t h e r m o r e , l a b o r a t o r y a p p r o a c h e s are n o t a

48
panacea, b u t appear to be an ef fective in str uct ion al
m e t h od olo gy in a teacher's repertoire.
Bloom's M o d e l of Ma ste ry L e a r n i n g
Over the years ed ucators have b e l i e v e d that only a
few

(between 20% and 30%)

can learn,

w h a t the schools have to teach.

to any g r e a t extent,

One idea b e g i n n i n g to

shape educatio na l views and pra ctices is m a s t e r y learning.
The b a s i c assumpt ion is that all, or almost all,
can learn w e l l
approached,

(1)

students

if i n s t r u c t i o n is sy ste m a t i c a l l y

(2) if students are p r o v i d e d w i t h ad equate help

w h e n and w h e r e they have learning difficulties,
are given suf fic i e n t time to a chieve mastery,

(3) if they

and

(4) if

there is some clear cr ite r i o n of w h a t consti tut es mastery.
This b a s i c idea was e m p h a s i z e d by the J e s u i t Schools be for e
the sevente ent h century, Co men i u s in the se ven t e e n century,
P e sta l o z z i in the e i g h t e e n t h century,

He rba rt in the

ninetee n century, W a s h b u r n e and his W i n n e t k a P l a n in the
twentie th century and M o r r i s o n in 1920s and 30s
1922; Morrison,

1926).

(Washburne,

The idea did not resurf ace until

the late 1950s and e arly 1960s as w h a t is known as P r o g r a m m e d
In struction
Atkinson,
(Bloom,
learning

(Skinner,

1968).

1968),

1954;

Suppes,

1966; Glasser,

B loom's m o d e l of M as t e r y

1968;

learning

de vel o p e d from Car roll's m o d e l of school

(Carroll,

1963), has p ro v e d to be v e r y effective.

It has been d e m o n s t r a t e d in Un i t e d States and K o r e a that
the c l a s s r o o m p r o c e dur es of this m o d e l is ca pable of m a k i n g

49
75 per c e n t of students achieve a high level
of m as t e r y

(Chung, e t al.,

1970; Bloom,

(80% or higher)

1968).

Theory:
Carroll's m o d e l

(196 3) is a c o n c e p t u a l i z e d model

b a s e d on five v a r i a b l e s — -aptit ude for p a r t i c u l a r kinds of
learning, qu a l i t y of instruction,
instruction,

perseverance,

ability to u nde rst and

and opportunity.

Carroll

(1963)

defines apt itu de as the a m o u n t of time req uir ed by the
learner to attain m a s t e r y of a lea rning task.

Bloom

(1971)

a rgued that Car ro l l ' s d e f i n i t i o n of aptitude implies that
g i v e n e nough time,

all students can conceiva bly attain

ma s t e r y of a learning task and that if Car rol l is right,
then learning m a s t e r y is t h e o re tic all y available to all,
w e can find the means of hel pin g every student.

if

He

su pported C a r r o ll' s view of aptitude w i t h the w o r k of
Gl ass er
(1961).

(196 8), A t k i n s o n
Car r o l l

(1963)

(19 6 7), B l o o m

(1964), and Hunt

defines the qua l i t y of ins tru cti on

in terms of the degree to w h i c h the presentation,

explana­

tion, and or dering of el ements of the task to be learned
ap pro ach the o p t i m u m for a g ive n learner.

Ca rro ll

(1963)

explains how the ab ility to u n d e rs tan d i nst ruc tio n i n t e r ­
acts w i t h the m e t h o d and type of instruction.

He defines

the a bility to u n d e r s t a n d i n s t r u cti on as the ability of
the learner to u n d e r s t a n d the n ature of the task he is to
learn and the p r o c e d u r e s he is to fo llow in le arning of
the task.

C ar r o l l defines p e r s e v e r a n c e as the time the

50
le arner is w i l l i n g to spend in learning.
"opportunity," C arr o l l means
C arroll

(1963)

Finally, by

"time a llowed for learning."

c omb i n e d the five variables to define the

degree of learning of the i***1 individual and the tth task
as a funct ion of the ratio of the amount of time the ind i­
vidual

(learner)

act ually spends on the learning task to

the total amount of time he needs.
(Carroll,

1962)

He de m o n s t r a t e d

that the n u m e rat or of this fraction w i l l

be e qual to the sma l l e s t of the three quantities:
tunity, p e r s e v e r a n c e and aptitude
time)

oppor­

(all de fin ed in terms of

and the d e n o m i n a t o r is the time n e e d e d to learn after

a d j u s t m e n t for q u a l i t y of in stru cti on and ability to u n d e r ­
stand instruction.
Thus,

Degree of _
learning

The

time allowed, perseverance, aptitude
Y
[quality of instruction, ability to understand instruction

"function"

is n o t u sed in m a t h e m a t i c a l sense.

If the

qu al i t y of i n s t r u c t i o n a nd the ab ility to und ers t a n d
in s t r u cti on w e r e o p t i m a l the time n e e d e d for i nst ruc tio n
w o u l d be minimized.

Imp lied in the d e f i n i t i o n of the

de gre e of learning is the h y p o t h e s i s :
E v e r y b o d y can learn to m a s t e r y level if he spends the
am o u n t of time he needs to ma s t e r the task.
Bl oom*s model

(1968)

is a t r a n s f o r m a t i o n of

C a r r ol l's c o n c e ptu al m o d e l into an ef fective w o r k i n g model.
Ha v i n g s u p p o r t e d Car ro l l ' s view of a pti t u d e with some

51
st u d i e s

as d i s c u s s e d above,

B l o o m r e a s o n e d t hat if a p t i t u d e s

w e r e p r e d i c t i v e o f the rat e at wh ich ,
the level to wh i c h ,

a n d no t n e c e s s a r i l y

a s t u d e n t c o u l d learn a g i v e n task,

it

s h o u l d h a v e b e e n p o s s i b l e to fix the d e g r e e of l e a r n i n g
e x p e c t e d of s t u d e n t s at so me m a s t e r y
ca lly m a n i p u l a t e

the r e l e v a n t i n s t r u c t i o n a l v a r i a b l e s in

C a r r o l l ' s m o d e l s uch
a t t a i n e d it.

level an d to s y s t e m a t i ­

t h a t all or a l m o s t all s t u d e n t s

B l o o m a s s e r t e d t h a t if s t u d e n t s w e r e n o r m a l l y

d i s t r i b u t e d w i t h r e s p e c t to a p t i t u d e

for a s u b j e c t a n d if

they w e r e p r o v i d e d u n i f o r m i n s t r u c t i o n in t e r m s o f q u a l i t y
an d l e a r n i n g time th en a c h i e v e m e n t at the s u b j e c t ' s
c o m p l e t i o n w o u l d be n o r m a l l y d i s t r i b u t e d .
conditions

U n d e r suc h

the c o r r e l a t i o n b e t w e e n a p t i t u d e m e a s u r e d at

the b e g i n n i n g of the i n s t r u c t i o n a n d a c h i e v e m e n t m e a s u r e d
at the e n d of t h e i n s t r u c t i o n w i l l be r e l a t i v e l y h i g h
(typically a b o u t + 70).

Conversely,

if s t u d e n t s are

n o r m a l l y d i s t r i b u t e d w i t h r e s p e c t to ap titude,

b u t the k i n d

a n d q u a l i t y o f i n s t r u c t i o n a n d l e a r n i n g time a l l o w e d are
made appropriate
learner,

to the c h a r a c t e r i s t i c s a n d n e e d s of e a c h

the m a j o r i t y of s t u d e n t s w i l l a c h i e v e m a s t e r y of

the subject.

The

c o r r e l a t i o n b e t w e e n th e a p t i t u d e and

achievement should approach
Carroll's model Conant
su c c e s s o f a p u p i l
four f actors

zero

(1964)

(Bloom,

19 68).

From

a r g u e d t h a t th e d e g r e e of

faced w i t h a ta sk in s c h o o l d e p e n d s on

two o f w h i c h r e s i d e in the i n d i v i d u a l and the

o t h e r two s t e m f r o m e x t e r n a l c o n d i t i o n .

Th e t wo i n d i v i d u a l

e l e m e n t s b e i n g the a p t i t u d e a n d p e r s e v e r a n c e w h i l e

the

52
e x t e r n a l e l e m e n t are the q u a l i t y of i n s t r u c t i o n and the
a m o u n t of time a vailable.

In B l o o m ’s s t r a t e g y an a t t e m p t

is m a d e to find w a y s o f a l t e r i n g the time i n d i v i d u a l
s t u d e n t s n e e d for l e a r n i n g as w e l l as to fin d w a y s of
p r o v i d i n g w h a t e v e r time is n e e d e d b y eac h student.

The

s t r a t e g y finds s o m e w a y of s o l v i n g th e i n s t r u c t i o n a l
p r o b l e m as w e l l as the s c h o o l o r g a n i z a t i o n a l p r o b l e m s ,
i n c l u d i n g that of time.
Practice:
The operating procedures
n o t static.

for m a s t e r y

l e a r n i n g are

A c t i v e r e s e a r c h c h a n g e s p r o c e d u r a l s tra teg ies .

The major operating procedures

that h a v e b e e n f o u n d m o s t

u s e f u l in d e v e l o p i n g a nd c a r r y i n g o u t m a s t e r y l e a r n i n g
strategies will be d e s c r i b e d .
Th e suc c e s s of the s t r a t e g y r est s on the a c c e p t a n c e
of its b a s i c a s s u m p t i o n t hat a l m o s t all s t u d e n t s ca n learn
to a h i g h

level.

administrators,

The acceptance stimulates
as w e l l as st ude nts .

the teachers,

It provides

a touch­

s t o n e for the s o l u t i o n of m o s t p r o c e d u r a l p r o b l e m s
e n c o u n t e r e d d u r i n g a s t r a t e g y ' s d e v e l o p m e n t a n d / o r its
i m p l e m e n t a t i o n b y s e a r c h i n g for a c t i o n s w h i c h are
to p r o m o t e

the l e a r n i n g o f all,

Its a c c e p t a n c e a l s o h e l p s

n o t j u s t some,

l ikely

students.

justify modification of grading

p o l i c i e s a nd p r a c t i c e s so t h a t all s t u d e n t s w h o a t t a i n
m a s t e r y can be a p p r o p r i a t e l y r e w a r d e d for t h e i r effort s.

53
Its c o o p e r a t i v e i n s t e a d of c o m p e t i t i v e e f f e c t w i l l b e
d i s c u s s e d in c h a p t e r I I I .
Mastery

le ar n i n g a p p r o a c h p r o d u c e s b e s t r e s u l t s

in

s u b j e c t s p o s s e s s i n g s o m e and f r e q u e n t l y all of the f o l l o w i n g
characteristics:
1.

Subjects

that r e q u i r e e i t h e r m i n i m a l p r i o r l e a r n i n g

or p r e v i o u s l e a r n i n g w h i c h m o s t le arn ers a l r e a d y
possessed.

F o r ex amp le,

fi r s t grade a r i t h m e t i c or

a b e g i n n i n g a l g e b r a course.
le arn ers

in such a s u b j e c t dep e n d s

the q u a l i t y of instruc tio n,
provides

C l e a r l y the su c c e s s o f
a g r e a t d e a l on

a nd m a s t e r y a p p r o a c h

i n s t r u c t i o n o f o p t i m a l q u a l i t y to ea ch

learner.
2.

Subjects

th at are s e q u e n t i a l l y learned.

Suc h

subjects contained well-defined units whose

lea r n i n g

is c u m u l a t i v e in that the l e a r n i n g of any units
builds upon
3.

the l e a r n i n g of all p r i o r units.

S u b j e c t s w h i c h t e n d to be c l o s e d and e m p h a s i z e
c o n v e r g e n t thinking.

By "closed"

subjects Blo o m

(1971)

means

is a fi n i t e set of i deas and b e h a v i o u r s

s u b j e c t s w h e r e there
to be

l e a r n e d and

a b o u t w h i c h the re is a c o n s i d e r a b l e a g r e e m e n t a m o n g
curriculum makers
En gli s h ,
Bloom

and te ach e r s .

Mathematics,

(1971)

Sci ences.

E x a m p l e s o f this are
Following Guilford

(1959),

defines subjects which emphasize convergent

t h i n k i n g as those in w h i c h s t u d e n t s are t a u g h t to o b t a i n

54
"right answers," or "good solutions" through "appropriate
thought processes" or "accepted p r o b l e m - s o l v i n g modes."
Though m ast ery learning has d e m o n st rat ed its
relativ ely p ositive effects on subjects p o s s e s s i n g the
above charac ter ist ics

(like early courses in Mathematics,

English,

Rea d i n g and S c i e n c e s ) , it has also w o r k e d for

subjects

like p h i l os oph y

(Moore, Mahan,

po s s e s s i n g other charac ter isti cs

and Ritts,

1968)

than those di scu s s e d

above.
Basic in the pr ac t i c e is the p r o b l e m of de fin ing
w h a t is m e a n t by m astery on the a c h i e ve men t
test.

(summative)

Ori gin a l l y B l o o m and his associ ate s set the level

re quired for a grade of A in a non - m a s t e r y class as the
de fin i t i o n of m a s t e r y for the m a s t e r y classes.

Recently,

m o r e ob jec t i v e e m p i r i c a l st an d a r d se tti ng p roc ed u r e s have
been developed

(Block,

19 70).

T h e em pir ica l w o r k suggests

that if students learn 80 to 85 p e r c e n t of the skills in
each unit,

then they are likely to ex hib it m ax i m a l pos iti ve

co gnitive and aff ective d e v e l o p m e n t as m e a s u r e d at the
subject's completion.

This w o r k further suggests that

be sid es b e i n g an u n r e a l i s t i c e x p e c t a t i o n in terms of
st u d e n t and teacher time and effort to require or eve n
en cou rag e students to learn all or n e a r l y all

(90 to 95%)

of e a c h unit it m a y have m a r k e d n ega t i v e co nse que nce s for
students int e r e s t in and attitudes t oward the learning
(Sherman,
(1973)

1967;

Bornmuth,

1969; Block,

1970).

Mi lli man

d i s c u s s e d five factors w h i c h can be c o n s i d e r e d in

55
se tting a p a s s i n g-s cor e and test lengths on domainr e fer e n c e d measures.

This is very useful in d e s i g n i n g

s u mmative tests.
Th oug h summat ive ev a l u a t i o n can assess student
achieve men t at the end of instruction,
the t e a c h ing -le arn ing process.

it cannot help guide

More central to the ma ste ry

learning strategies then is the d ev e l o p m e n t of feed-back
and corrective pr oce d u r e s at various stages or parts of the
learning process.

W hile a v ariety of feedback pro ces ses

like workbooks, quizzes,

homework,

etc.,

are possible,

formative e v a l u a t i o n de sig n e d to be an integral p art of the
te ach ing - l e a r n i n g p rocess has b e e n m o s t useful
1969) .

Such instruments are brief,

diagnostic,

(Airasian,
and co n­

st ruc ted to de termine w h a t each stu d e n t ha d learned in a
p a r t i c u l a r unit,

chapter,

still needs to learn.

or p a r t of the course and w h a t

The instruments n o t u s e d to judge

or guide the stu dent b ut are of v alu e in p r o v i d i n g feedback
to b o t h s tud ent and te ache r on w h a t aspects or ele ments of
learning unit still ne ed to be mastered.
failure of ma s t e r y

The success or

learning st rategy depends on the degree

of eff ici enc y of these formative tests in p i n p o i n t i n g the
learning needs of the st u d e n t and s u p p l e m e n t a t i o n of the
original i n s t r u cti on

(Bloom, et al.,

1971).

Since there

are no k now n m eth ods

for going from a s tud ent's incorr ect

formative test responses to sp ec i f i c l earning co rre cti ve
he needs,

a w i d e va rie ty of i n s t r uct ion al c o r r ect ive s is

ge ner a l l y mad e available so that the s tu d e n t can dis cov er

56
those b e s t s uited to his cha ract e r i s t i c s and needs.
correcti ve s are small group p r o b l e m sessions,
tutoring,

and a l t e r n a t i v e l earning m a t e r ial s

native textbooks, workbook s,
visual methods,

tapes,

Such

i ndividual
(like a l t e r ­

p r o g r a m m e d instruction,

audio­

a cad e m i c games and puz z l e s and r e­

teaching) .
Re se a r c h Fi ndings on Use of
M a s t e r y L ear n i n g Con cep ts
a n 5 Strategies
The results

from o v e r 45 m a j o r studies c arr i e d under

school conditions in dicate that m a s t e r y

learning has m a r k e d

ef fects on st udent co gni t i v e and a ffe ctive d e v e l o p m e n t and
their le arning rates

(Block,

1971; Peterson,

1972).

Ma ste ry

learning p r o c e d u r e can e nable four-fi fth s of students to
reach a level of a c h i e v e m e n t w h i c h less than o n e - f i f t h
at tain u n d e r conventional,
procedures.

uniform, g r o u p - b a s e d i nst ruc tio nal

The add iti o n a l time n e e d e d for this is 10 p e r ­

cent to 20 p e r c e n t of the no r m a l class time.

The strategies

s e e m to be e s p e c i a l l y e f f e c t i v e for those s tudents w h o
typical ly have h a d p r o b l e m s in le ar n i n g under o rdi nar y
in str uct ion al conditions.

R ese a r c h is repeat edl y d e m o n ­

s t ra t i n g that i n d i v i d u a l d i f f e r enc es in achievement,

time,

or rate of learning is la rgely a fu nction of the pr e p a r a t o r y
or prior i n s t r u c t i o n a l a ppr oa c h e s and that for subjects
w h e r e m o s t of the st udents h a v e ac hi e v e d the p r e - r e q u i s i t e
learnings, m a s t e r y p r o c e d u r e s a ppear to be able to almost
e lim i n a t e the eff e c t s of ind ivi d u a l d i f f e r e n c e s o n level

57
of a c h i e v e m e n t w h i c h indicates that u nder ideal c onditions
in div idu al differen ces in school learning ap proach a
va nis h i n g po i n t

(Bloom,

19 73).

M a s t e r y m e t h o d s also

pr oduce m a r k e d l y gr eate r i nterest in and be t t e r attitudes
toward the m a t e r i a l le arned than m o r e c onventional
approaches.

T hey s e e m to help m o s t students ove rco me

feelings of d e f e a t i s m and p a s s i v i s m b r o u g h t to learning.
Th eir p o w e r f u l affective c ons equ enc es may be a t t r i b u t e d to
m a n y factors,

e s p e c i a l l y the coo per ati ve r ath er than

competitive le arning conditi ons

(Johnson an d Johnson,

19 74)

p e r s o n a l i z e d att en t i o n to each stude nt' s learning problem,
success ful and rew ar d i n g lea rning experie nce s and the use
of c ert ain correctives w h i c h add a p e r s o n a l - s o c i a l aspect
to the learning.
e a rl i e s t units

Finally,

m as t e r y approaches of the

in a school su bje ct appears to facilitate

the learnin g of the sub seq uen t units,
learning units
Merril, Barton,

e s p e c i a l l y w h e r e the

are se quen tia lly arr a n g e d
and Wood,

(Block,

1970;

19 70).

Promising Innovative/Experimental Field
E x p e r i e n c e s In T e a c h e r E d u c a t i o n
E a r l i e r in 19 72 H a t f i e l d a cting for the Na ti o n a l
C o u n c i l of T ea chers of M a t h e m a t i c s C o m m i s s i o n o n T e a c h e r
E d u c a t i o n re qu e s t e d over 100 m a t h e m a t i c s tea c h e r educators,
teachers,

and s u p e r v iso rs to ide n t i f y a few of the m a j o r

qu estions or issues they saw in m a t h e m ati cs
p r eparation.

teacher

Here are six of the p a r a p h r a s e d re p o r t of

their responses on some school b a s e d issues and problems:

58
1.

There is a g r e a t push for more field ex per ien ces
in t eacher e d u c a t i o n today.
I am sure at least
certain aspects or kinds of field experi enc es are
good.
B u t if other i n s t i tut ion s are like mine,
their students too of ten g e t p l a c e d for field
ex periences in q u i t e conventional, m e d i c o r e s i t u ­
ations.
Such experi enc es tend to d evelop teachers
w h o adhere p r e t t y m u c h to the status quo; and they
do not he lp m u c h to de v e l o p innovative, originaltype teachers.
I think there is merit in some of
the kinds of s i m u l a t e d experie nce s w h i c h hav e been
d e v e l o p e d over the p a s t several years.
W h i c h ones
and for w h a t purposes are not really clear.
There
seems to be a div ersity of o p i n i o n about them.

2.

H ow can the st uden t t eaching exp eri e n c e be improved?
St ud e n t teachers us ually m o d e l their teachin g after
the c ritic teacher.
This is fine w h e n the critic
is a s uperior teacher, b u t all too o f t e n teachers
w h o are m er e l y average serve as critic teachers,
and it seems as if the cycle of m e d i c o r i t y p r o d u c i n g
more m e d i o c r i t y is perpetuated.

3.

W h a t structural changes can b e institu ted so that
the m a t h e m a t i c s e duc a t o r w o r k i n g w i t h p r e s e r v i c e
e l e m en tar y e d u c a t i o n m a j o r s in a u n i v e r s i t y - c e n t e r e d
methods course can coo rdi n a t e his efforts w i t h the
real life situations that exist out in the schools?

4.

P rov iding early and s ome wha t c o n s i s t e n t pu b l i c
school c l a s s r o o m i n v o l v e m e n t for p res er v i c e teachers.

5.

How to facilitate i n t e rac tio n between school p e r ­
sonnel and college staffs?
Is it p o s s i b l e to get
co llege teachers on a re gular basis in the p u b l i c
schools?
I surely think that school of e d u c a t i o n
pr ofe sso rs o u g h t to be in t hat c l a s s r o o m teaching.
W e are p l a n n i n g on u s i n g a team app roa ch for this
bu t finding lots of opposition.
It is n o t just in
sc h e d u l i n g — m a n y are si m p l y afraid of a large class
of unruly h i g h school stu dents w i t h all the pro blems
they h a v e — esp eci all y in the inner city areas
(Hatfield, 1972).

6.

In spite of fewer jobs for m a t h e ma tic s teachers,
there seems little e vid e n c e that the b e s t peo ple
are b e i n g hired.
Indeed, it seems that m a n y pe opl e
are still be i n g h i r e d to teach m a t h e m a t i c s w h o are
n o t q u a l i f i e d in any way.
How can h i r i n g pr actices
be imp roved to help . . .

59
Th o u g h there are m a n y rationales
innovations in field experience,

this report w i l l focus on

pr ograms w h i c h use field experience,
of te aching as a career
courses

for the ex ist ing

(1) as an e x p l o r a t i o n

( s c r e e n i n g ) , (2) as part of m e t h o d

(and formation of style and p h i l o s o p h y ) , (3)

internships,

and

(4)

in

to foster be t t e r u n d e r s t a n d i n g be twe en

the c ollege and pu b l i c school per so n n e l w i t h gr eat est
emphasis on
1.

(2).
C l a s s r o o m Experien ces Be for e A d m i s s i o n to Tea che r

E d u c a t i o n P r o g r a m s .— C oup l e d w i t h issues

(2) and

(4) above

is the p r o b l e m scr ee n i n g of future teachers in general.
Co nan t

(1963)

b e l i e v e d that future teachers sh o u l d be

se lected from the top 30 p e r c e n t of high school graduates
by 19 74 and su ggested a high school p r o g r a m that w o u l d help
such p r o s p e c t i v e teachers study w i t h profit and w i t h o u t an
e x c e ss ive d em a n d on time and e nergy of the p rog ram .

This

is a w a y of sc ree n i n g for b e t t e r aca demic p r e p a r a t i o n of
teachers,

if implemented.

Ana lo g o u s to this is s cre eni ng

for p r o f e s s i o n a l p r e p a r a t i o n of teachers.

Many y o u n g m en

and w o m e n do n o t dis covery until their senior y e a r that
they could h a v e c o n c e nt rat ed o n te aching in the el e m e n t a r y
or the s eco nda ry school,
pr og r a m s at all.

or, do n o t go into teaching

Do ugherty

(1973)

r epo r t e d a p r o g r a m w h i c h

pr ovi d e s exp eri enc es that w i l l de t e c t p ersons who w i l l not
be happy o r success ful in teaching.

The M i c h i g a n State

U n i v e r s i t y has also ini tiated a sele c t i o n procedure, pri or

60
to ad mission to teacher education,
direction.

that is he l p f u l in this

The m o s t im portant cri ter ion used in this

selection process is d e m o n s t r a t e d ability, on the part of
the applicant,

to w o r k e f f e c t i v e l y w i t h children.

admission to the program,

Prior to

each student is re quired to spend

a m i n i m u m of 60 hours in a pu b l i c school c l a s s r o o m service
as a teacher's aide.

An e v a l u a t i o n of his p e r s o n a l q ual ities

in relation to the expect ati ons of the teacher role, w ith
special emphasis on the p r o s p e c t i v e teacher's ability to
relate to youngsters,

is m a d e by b o t h the pu b l i c school

teacher w i t h w h o m the student has w o r k e d and the univer sit y
represen tat iv e w h o has obs e r v e d h i m as a t eacher aide.
The EXCEL p r o j e c t of Tea chers College,
University,

Muncie,

Ball State

Indiana re po r t e d by the A m e r i c a n

A s s o c iat ion of Col leg es for T e a c h e r E d u c a t i o n
involves the student w i t h c l a s s r o o m activities
freshmen through his senior year.

(19 73)

also

from his

This allows h i m ample

time to decide if tea ching w o u l d be a suita ble career for
him.

This p r o j e c t w o n the d i s t i n g u i s h e d a c h i e ve men t in

1973.

The A s s o c i a t i o n

U n i v er sit y Southeast,

(1974)

also rep orted on the Indiana

N e w Al b a n y Te ach er E d u c a t i o n p r o g r a m

wh ich has early and cont in u i n g field exper ien ces i n c o r p o ­
rated into the program.
2.

C l a s s r o o m Experien ces as Part of M e t h o d C o u r s e s .—

Del P opo lo

(19 70)

ob served often the student t eac her

arrives upon the student t eaching scene completely

61
unheralded?
alike,

a m i d s t th e c u r i o u s

he gropingly

attempts

glances

of pupils

and staff

to f i n d h i s c l a s s r o o m in a

maze of corridors.
In a t t e m p t
educators have

to a l l e v i a t e

also observed,

this p r o b l e m whi c h other

Anderson and Boop

reported a restructured secondary professional
of s t u d e n t t e a c h i n g o f B u t l e r U n i v e r s i t y
which used

to be essentially

University's
principal

conventional

in I n d i a n a p o l i s

the s a m e as t h e M i c h i g a n

student

teaching program.

State
The

fiv e b l o c k s

full days of o b s e r v a t i o n organ i z e d

of time w i t h

the s e q u e n t i a l n u m b e r o f d a y s

within each progressive block being
more gradual
The

semester

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

i n troduction of twelve
into

(197 2)

i n t e n t is

introduction

i n c r e a s e d to p r o v i d e

into school

to p r o v i d e m o r e g r a d u a l

student teaching.
continuity between

c o l l e g e c l a s s r o o m t h e o r y a n d a p p l i c a t i o n i n th e p u b l i c
school

assignment by providing

the s t u d e n t s

could

relate during

segment of classroom work
Th e c a l e n d a r

fr a m e s o f r e f e r e n c e

the g e n e r a l m e t h o d s

in the u n i v e r s i t y

f or the p r o f e s s i o n a l

to which

classroom.

semester has

thi s

form:

62

Week

Monday

Tuesday

-

1

Wednesday

Thursday

Friday

C o l l e g e Cla:ssr oom

2

College Classroom

3

College Classroom

4

Public
Schools
Obs. #2

College Classroom

College Classroom

5

College Classroom

6

Public Schools
1
_
.
, _ ,!
Paraprofessional Actxv.

7

Public Schools
Limited Participation

8

College

Classroom

Public
Schools
Obs. #1

College Classroom
Public Schools
Obs. #3 a n d #4

„ , , ___ „ „
_____
College Classroom
^
College Classroom
Public Schools
EXTENSIVE PARTICIPATION

Pull Time Student Teaching

9-16

A n e v a l u a t i o n o f d ata, u s i n g E d w a r d s
Preference

Schedule,

Personal

Minnesota Teacher Attitude

Inventory

and other constructed o p i n i o n a i r e s , revealed:
a.

A substantial increase in k n o w l e d g e of students
by the student teachers.

b.

A substantially greater professional
o f th e s t u d e n t t e a c h e r s .

c.

A gre a t e r c o n t i n u i t y of school
ences .

d.

A b e t t e r p r o v i s i o n for c o o p e r a t i o n in s u p e r v i s o r y
activities b e t w e e n the u n i v e r s i t y supe r v i s i o n and
c o o p e r a t i n g t e a c h e r s ( A n d e r s o n a n d B o o p , 1972).

involvement

and college experi­

63
T h i s d e s i g n has b e e n
original

in e x i s t e n c e

for tw o y e a r s

findings have been substantiated,

the r e s u l t s w e r e n o t

confirming

j u s t d u e to H a w t h o r n e ' s

to the u n i v e r s i t y

for the i m p l e m e n t a t i o n o f

that

effect.

A v a i l a b i l i t y of a s u f f i c i e n t n u m b e r o f s c h o o l s
(close en ough)

and the

around

is a n e c e s s a r y

condition

this program.

i

The Ohio State University
Science

and Mathematics

teaching experiences

It w a s

most

in t h a t

times,

labelled the methods

divorced

ers,

and formal research

and mathematics
teaching
school

studies

levels during

(B losser,

lacking
frame-work,

and 1970-71

1972).

are actively

five quarters

observ­
Science

i n v o l v e d in
and senior high

of t h e i r j u n i o r a n d

in c o l l e g e .

of the p r o g r a m d u r i n g

the

as i n d i v i d u a l s

three quarters

is a c c o m p l i s h e d b y s p e n d i n g

one-to-one

as

instructors,

junior high school

Learning about pupils

This

courses

in 1969-70,

students,

undergraduates

at elementary,

senior years

after the faculty

from children and practical

Its m o d i f i c a t i o n s

were based on evaluations by

the

(Blosser,

t h e y ar e t a u g h t i n a t h e o r e t i c a l

totally

experiences.

throughout

junior and senior years

first i n i t i a t e d in 1968-69

and s t u d e n t s h a v e
reality

E d u c a t i o n is on e w h i c h i n v o l v e s

in t h e p u b l i c s c h o o l s

undergraduate student's
1972).

u n d e r g r a d u a t e p r o g r a m in

tutorial

of the junior year.

They

the emphasis

j u n i o r ye a r .

two half-days pe r w e e k on

then broadened

v i d u a l as a m e m b e r o f a s m a l l

of

is

group

to t h a t of the i n d i ­
in t h e s e c o n d q u a r t e r

spend other

t i m e s in

64

method-courses.
mathematics
science

They spend

laboratory

education.

course or

The

(inner-city,

third quarter on either
laboratory activities

two schools

suburban)

in c o n t r a s t i n g c o n ­

a n d the s t u d e n t s

ar e i n v o l v e d

in a h a l f - d a y

teaching assignments,

five days a week,

the q u a r t e r .

In t h e s e c o n d q u a r t e r

the s t u d e n t s

p a t e in a f u l l d a y t e a c h i n g
on e of the

two h a d

Staropoli

in

f i r s t q u a r t e r o f t he s e n i o r y e a r

is d i v i d e d i n t i m e b e t w e e n
text

the

for

partici­

a s s i g n m e n t in a s i n g l e s c h o o l ,

in p r e c e d i n g q u a r t e r .
and Heitzmann

(1973)

reported a similar

p r o g r a m b e i n g i m p l e m e n t e d at U n i v e r s i t y o f D e l a w a r e ,

Newark

for e i g h t m e t h o d c o u r s e s .
Lancaster
College,
theory

(19 73)

B o s t o n b a s e d on

reported a p r o g r a m at Emerson
the r a t i o n a l e o f a c o u n t e r p o i n t

that direct experience gives

that theory

an a t t e m p t

more manageabl e pieces.

S t u dents are ex p o s e d

Memorial Elementary School,

school w h i c h utilizes

teaching-teams.

t e a c h i n g p r a c t i c u m i n the
things

from freshman

in c o m m o n w i t h

first cycle of it was

junior year.
those

Sowell and Hodgin
St a r t " p r o g r a m i n s t u d e n t

(19 72)

into

to t e a c h i n g

to s e n i o r y e a r
a non— graded

They have

formal

The p r o g r a m has

already described.

just completed

and

The

to b r e a k d o w n t h e l e a r n i n g

as s t u d e n t a i d e s

in th e V e t e r a n s

many

to t h e o r y

in t u r n g i v e s m e a n i n g to e x p e r i e n c e .

program reflects

experiences

relevance

The

i n 1973.

r e p o r t e d o n a "H e a d

t e a c h i n g in m a t h e m a t i c s .

subjective evaluation procedure was

used.

Balka

A more

(1974)

65
reported on early experiences
school mathematics

in w h i c h

i n th e t e a c h i n g of s e c o n d a r y

a survey of the attitudes

comments of the under-graduates
teachers

a nd

preservice mathematics

response was high l y positive.
Colles

and Pagni

(19 73)

reported a study which was

a field-based method course which
they b e l i e v e d

that the m e thods

p r i o r to t h e i r s

taught in methods

did not carry over

t his h a s r e s u l t e d i n f r u s t r a t i o n ,
eventual

they designed because

loss o f a p o t e n t i a l l y

to a c t u a l

courses

teaching and

discouragement and

good

teacher

from the

profession.

T h e i r p r o g r a m is v e r y s i m i l a r

ab ove .

c a r r i e d o u t an e v a l u a t i o n w h i c h u t i l i z e s a

They

comprehensive
(1)

feedback s y s t e m c o n t a i n i n g three elements:

encourage high school students

i n s i g h t o n t he p r o s p e c t i v e
taped

to t h o s e d e s c r i b e

to r eac t, w h i c h p r o v i d e s

teacher's behavior,

(2)

video­

l e s s o n s w e r e r e v i e w e d b y t h e i n s t r u c t o r a n d th e

st u d e n t ,

w h i c h e n a b l e d th e s t u d e n t

to se e his

d e f i n i t i o n w i t h his perspectives,

and

Interaction Analysis

used

category was

d a t a a b o u t th e t e a c h i n g b e h a v i o r s

r o l e in

(3) m o d i f i e d F l a n d e r s
to o b t a i n o b j e c t i v e

as r e l a t e d to e l e m e n t s

of t he s t u d e n t ' s p e r s o n a l i t y .
Despite

its o b v i o u s

of its o b s e r v a n c e ,
educational
to e a r l y
student

there are

literature

objective

l i m i t e d o b j e c t i v e d a t a in the

about the effects

field experience
teaching.

value and increasing reports

of and reactions

for e d u c a t i o n s t u d e n t s p r i o r

A study which represents

appraisal of students'

a step

to

toward

a n d c o o p e r a t i n g t eac her s'

66
reactions

to o n e p r e - s t u d e n t t e a c h i n g p r o g r a m of m e t h o d s

courses combined with
Gantt and Davey

field experiences was

(19 73).

The study involved

reported by
40 j u n i o r

under-graduate elementary majors

in the U n i v e r s i t y of

Maryland*s College of Education,

5 methods

an d

18 t e a c h e r s

at 2 elementary

schools.

faculty-members,
F o u r o u t o f the

29 t h r e e — h o u r s e s s i o n s w e r e d e v o t e d to e l e m e n t a r y
classroom experience.

school

T h e a n n o u n c e d o b j e c t i v e o f the

school-based phase of

the m e t h o d s

c o u r s e was:

To p r o v i d e o b s e r v a t i o n a l a n d t e a c h i n g e x p e r i e n c e for
the s t u d e n t in e l e m e n t a r y l a n g u a g e arts, r e a d i n g , a n d
s o c i a l stu di e s :
e m p h a s i s s h o u l d n o t b e f o c u s e d o n the
r e q u i r e m e n t of t e a c h i n g s k i l l s , b u t r a t h e r o n f a m i l i a r i
zation w i t h pupil and behaviour.
Gantt and Davey
evaluative

f o r m in this

(19 73)

utilized a three-part

study:

Part A reported how con­

f i d e n t the s t u d e n t f e l t a b o u t h i s a b i l i t y
stressed in methods

course

teaching experience;

to his

to a p p l y

ideas

forthcoming student

part B measured

t h e e x t e n t to w h i c h

the f i e l d e x p e r i e n c e w a s p e r c e i v e d as a v a l u a b l e p a r t of
methods

course;

comment

fu l l y o n any of his r e a c t i o n s

was

part C offered

t he s t u d e n t o p p o r t u n i t y
to t h e p r o g r a m .

reported that feelings of confidence

for s t u d e n t t e a c h i n g w e r e e x p r e s s e d b y

strong mandate

students

as a

input,

an d c r i t i c a l g r o u p d i s c u s s i o n s

seems

apparent.

direct

and a

in s u p p o r t o f t he p r e - s t u d e n t t e a c h i n g

field-experience

It

about readiness

c o n s e quence of the c o m b i n a t i o n of theoretical
field experiences,

to

67
3.

I n t e r n s h i p s .— T w o p r o g r a m s h a v e b e e n s e l e c t e d to

illustrate

t his e x p e r i e n c e — o n e is d e s c r i b e d in d e t a i l

w h i l e o t h e r is d o c u m e n t e d .
The Michigan State University

in c o o p e r a t i o n w i t h

over sixty different p u b l i c school systems
State have

e s t a b l i s h e d the E l e m e n t a r y

Students preparing

Intern Program

for elementary school

Michigan State University

in M i c h i g a n
(EIP).

t e a c h i n g at

a re g i v e n a c h o i c e at t h e e n d of

t h e i r s o p h o m o r e y e a r of e i t h e r f o l l o w i n g th e t r a d i t i o n a l
on-campus

route or enrolling

tw o y e a r s

of p r e p a r a t i o n .

a ten-week

summer

d u r i n g h is

sophomore

March,

he m o v e s

mentary school

which

a n d j u n i o r ye a r s .

to an o f f - c a m p u s
teaching methods

From september-

internship
and student

six-month period.

ce n t e r .

During this

time,

requirements

in l i b e r a l a r t s

During his
student becomes

He returns

he completes

session

and m a j o r a n d m i n o r

areas.
the

an i n t e r n t e a c h e r w i t h a s a l a r y of a p p r o x i ­
the r e s p o n s i b i l i t y

D u r i n g thi s y e a r ,

intern consultant who has
with about

to t he

his work

fourth c a l e n d a r year of study,

$5,000 per year w i t h

classroom.

Ele­

t e a c h i n g are

fo r s p r i n g q u a r t e r a n d a f i v e - w e e k s u m m e r
fol l o w s .

fi n a l

The student choosing EIP attends

toward general education requirements

mately

fo r t h e i r

s e s s i o n at M i c h i g a n S t a t e U n i v e r s i t y

i n t e g r a t e d d u r i n g this
campus

i n th e E I P

h e is s u p e r v i s e d b y

the

as a f u l l - t i m e a s s i g n m e n t w o r k i n g

fi ve o r s i x i n t e r n s .

s p e n t in c l a s s r o o m s t u d y

for a

One evening a week

a n d a t t h e e n d o f t he y e a r ,

is
the

68
student qualifies
c er t i f i c a t e .

for a b a c h e l o r ' s d e g r e e an d a t e a c h i n g

T h e h e l p p r o v i d e d for these b e g i n n i n g tea chers

is b u i l t i nto the p r o g r a m o n a s e l f - f i n a n c i n g basis.
A l t h o u g h i n t e r n t eac her s are no t p a i d full sa laries,

the

sc h o o l s y s t e m pays t he same a m o u n t for t heir s e r v i c e s as
they w o u l d for a r e g u l a r l y c e r t i f i e d b e g i n n i n g teacher.
Th e s a l a r i e s o f the in t e r n c o n s u l t a n t s are p a i d f r o m the
d i f f e r e n c e b e t w e e n the a m o u n t p a i d b y the s c h o o l d i s t r i c t
an d the sal ary p a i d to e a c h intern.
In ea ch center,

a m e m b e r of the u n i v e r s i t y

f acu lty

serves as p r o g r a m d i r e c t o r a n d is p e r m a n e n t l y b a s e d in the
off-campus

center.

He acts as g e n e r a l p r o g r a m coo r d i n a t o r ,

te aches s o m e of the e l e m e n t a r y m e t h o d s

courses,

coordinates

the s t u d e n t t e a c h i n g e x p e r i e n c e s , a n d s u p e r v i s e s
of the i n t e r n c ons ult ants .
responsibility
local d i s t r i c t s

the w o r k

C a m p u s - b a s e d f ac u l t y s har e the

for the m e t h o d s

i n s t r uct ion .

Cooperating

f u r n i s h all n e c e s s a r y p h y s i c a l

fa cil iti es

such as o f f i c e a n d c l a s s r o o m s p a c e , i n c l u d i n g u t i l i t i e s .
Si n c e its incepti on,

M i c h i g a n State U n i v e r s i t y has

e n g a g e d in a c o n t i n u o u s e v a l u a t i o n o f its I n t e r n s h i p
P r o g r a m th r o u g h

the use of s y s t e m a t i c d e p t h i n t e r v i e w s of

the s t u d e n t s t h e m s e l v e s a n d o f those w o r k i n g m o s t c l o s e l y
w i t h them.

S ome of the a d v a n t a g e s w h i c h h a v e b e e n d i s ­

c o v e r e d in this a p p r o a c h to t e a c h e r p r e p a r a t i o n are as
follows:
1.

E d u c a t i o n a l t h e o r y and p r a c t i c e can be i n t e g r a t e d
m u c h m o r e easily.

Methods

cou r s e s

are t a u g h t w h i l e

69
s t u d e n t s are s p e n d i n g p a r t of t h e i r time in the
e n v i r o n m e n t of the p u b l i c s ch o o l a n d thus rich
opportunities

for the i m m e d i a t e t r a n s f e r of formal

i n s t r u c t i o n in p e d a g o g y

to w o r k in the c l a s s r o o m is

possible.
2.

This p r o g r a m m a k e s p o s s i b l e
the student,

frequent evaluations of

u s i n g as the b a s i s

for d e c i s i o n s

and

r e t e n t i o n in the p r o g r a m e v i d e n c e of g r o w t h in the
student's
3.

a b i l i t y to w o r k e f f e c t i v e l y w i t h children,

E I P 's m a j o r c o n t r i b u t i o n has b e e n its d e v e l o p m e n t
of a new d i m e n s i o n in t e a c h e r p r e p a r a t i o n , the
i n t e r n c o n s u l t a n t po sit ion .

Ins t e a d of e x p e c t i n g

a b e g i n n i n g t e a c h e r to p e r f o r m w e l l all of the tasks
u n d e r t a k e n by an e x p e r i e n c e d teacher,
student receives
w h e n he a ssu m e s

the EIP

continuing individualized guidance
responsibility

for a classroom.

His i n t r o d u c t i o n to t e a c h i n g thus,

is g r a d u a l and

c a r e f u l l y d irected.
The i n t e r n c o n s u l t a n t s ,

s e l e c t e d f rom a m o n g the

m o s t able t e a c h e r s in the c o o p e r a t i n g sc h o o l
d i s t r i c t s , hav e d e v e l o p e d i n - s e r v i c e e d u c a t i o n of
ne w tea c h e r s

far b e y o n d the i n i t i a l ex pec tat ion .

Th e low ra t i o of i n t e r n s

to i n t e r n c o n s u l t a n t s and

the c l o s e n e s s a n d c o n t i n u i t y of the r e l a t i o n s h i p
o v e r time has m a d e it p o s s i b l e

for v e r y s p e c i f i c

h e l p to be o f f e r e d an d accepted.
the c o n s u l t a n t s h a v e h e l p e d

M o s t imp o r t a n t l y ,

"b rid ge the gap"

70
b e t w e e n the c o l l e g e c our se w o r k and th e p u b l i c
s c h o o l c l a s s r o o m b y h e l p i n g the int ern to rel ate
th e o r y and p r a c t i c e .
T he a c t i v e p a r t i c i p a t i o n of p u b l i c s ch o o l st a f f
m e m b e r s in c e r t a i n p h a s e s of the p r o g r a m tends to g u a r a n t e e
a r e a l i s m an d p r a c t i c a l i t y w h i c h may s o m e t i m e s be lost as
c o l l e g e p r o f e s s o r s w o r k in isolation.
approximately

A t the p r e s e n t t i m e ,

40 p e r c e n t of M i c h i g a n S t a t e U n i v e r s i t y e l e ­

mentary education majors
Stiles

(1973)

are e n r o l l e d in the EIP Program .

r e p o r t e d t h a t the e n t i r e state of

W i s c o n s i n w i t h its State D e p a r t m e n t o f E d u c a t i o n ,
tions of h i g h e r

l e a r n i n g a n d p u b l i c schools,

p r a c t i c e the i n t e r n - i n - t e a m plan.

has p u t into

A ke y f eature is the

r e s p o n s i b i l i t y t a k e n by s choo l s ys t e m s

for o r g a n i z i n g and

s u p e r v i s i n g the c l i n i c a l e x p e r i e n c e of p r o s p e c t i v e
Th is m e t h o d a l l o w s

the sch o o l s

institu­

teachers.

(where t he a c t i o n is)

b e s t d e t e r m i n e w h a t kinds o f e x p e r i e n c e s

to

a student teacher

s h o u l d be a w a r e of in o r d e r to m a x i m i z e hi s p o t e n t i a l to
b e c o m e a teacher.

Also,

implement appropriate

"We h a v e

the int ern a c h a n c e

to

th eor y u n d e r a s u p e r v i s e d situation;

t h e r e f o r e , he m i n i m i z e s
field.

it g i v e s

the ri sk of b e i n g out in left

learned,

for ex amp le,

that experience

for

e x p e r i e n c e 's sake m a y n o t b e the m o s t e f f e c t i v e w a y to
prepare teachers,"
4.

observed Stiles

(1973).

Public School Experiences

for U n i v e r s i t y F a c u l t y

M e m b e r s .— In the last few years, M i c h i g a n S t a t e U n i v e r s i t y

71
has m a d e s er i o u s
E d u c a t i o n f ac u l t y

attempts

to i n v o l v e its C o l l e g e of

in s i g n i f i c a n t t e a c h i n g e x p e r i e n c e s

the p u b l i c schools.

in

It w a s p a r t i c u l a r l y c o n c e r n e d tha t

p r o f e s s i o n a l c o u r s e s a t the u n i v e r s i t y

level are o f t e n

s t a f f e d b y p e r s o n n e l w h o hav e n o t h a d r e c e n t e x p e r i e n c e s in
i n n e r - c i t y c lassrooms.
P a r t i c i p a n t s in O p e r a t i o n R E F U E L
ences

for u r b a n e d u c a t i o n a l

instructional
Michigan.

leaders)

se rve o n on e o f four

teams at the A l l e n S t r e e t Sch ool in L ans i n g ,

E a c h t e a m c o n s i s t s o f two L a n s i n g t e a c h e r s , one

Michigan State University professor,
interns,

(relevant e x p e r i ­

o n e or two g r a d u a t e

a n d two to four s t u d e n t teachers.

responsible
elementary

E a c h is

for the i n s t r u c t i o n of a p p r o x i m a t e l y
s tud ents.

fifty

T h e M i c h i g a n S t a t e U n i v e r s i t y st aff

m e m b e r is a t e a m m e m b e r h a l f - t i m e

for t w e l v e weeks.

His

role i n the c l a s s r o o m is in the a r e a of his s p e c i a l t y a n d
i n v o l v e s a c t i v e p a r t i c i p a t i o n w i t h children.
p r i m a r y f u n c t i o n is c l a s s r o o m ins t r u c t i o n ,

A l t h o u g h his

a Michigan State

U n i v e r s i t y p r o f e s s o r m a y b e a s k e d to c o n s u l t w i t h m e m b e r s
of o t h e r i n s t r u c t i o n a l teams in his s p e c i a l i t y area.
S i m i l a r s e c o n d a r y s c h o o l o p p o r t u n i t i e s ar e o f f e r e d
to M i c h i g a n S t a t e U n i v e r s i t y

f a c u l t y at P a t t e n g i l l J u n i o r

Hi gh a n d E a s t e r n H i g h S c h o o l in L a n s i n g as a p a r t o f o u r
T T T Project.
in the f i r s t t h r e e ye ars ,
versity

approximately sixty uni­

f a c u l t y m e m b e r s w e r e e n g a g e d in t he d i r e c t

i n s t r u c t i o n of p u p i l s

in the p u b l i c schools.

The

72
participating professors

i n d i c a t e d that t h e i r e x p e r i e n c e in

the s chools h e l p e d t hem to i m p r o v e t h e i r m e t h o d s
teaching.

course

-It h e l p e d ea ch to fre shen his m e m o r y r e g a r d i n g

the d a y - t o - d a y d i f f i c u l t i e s e n c o u n t e r e d in p u b l i c s cho ol
teaching.

It a ls o h e l p e d h i m g a i n c r e d i t a b i l i t y a mon g

c o l l e g e stu den ts by his w i l l i n g n e s s
the

to p u t his ideas

"on

line" in a r e a l classroom.
M o s t of the p r o g r a m s

r e p o r t e d , especially under

(2) , are b a s e d on t he a s s u m p t i o n th at l a b o r a t o r y - f i e l d experience supplemented methods
to t r a n s l a t e

t heo ry into p r a c t i c e , t h e r e b y o v e r c o m i n g the

i n a d e q u a c i e s of a sterile#
me t h o d s course.
methods

co ur s e s e n a b l e s a s t u d e n t

the o r y - b a s e d #

A strong mandate,

cou rse s and f rom s tud ents,

student teaching

non-participation

b o t h f r o m i n s t r u c t o r s of
in s u p p o r t of the pre -

fi eld e x p e r i e n c e seems apparent.

While a

p r o g r a m o f this t y p e adds

to the p l a n n i n g time of i n s t r u c ­

tors,

d e f i n i t e l y p a y - o f f the e x t r a

the o v e r a l l e ffe c t s

e f f o r t involved.

It a lso p r o m o t e s

a b o u t t heir p o t e n t i a l
However,

the students'

confidence

f u n c t i o n i n g in s t u d e n t t e a c h i n g .

t her e is a d a n g e r if s u f f i c i e n t time is no t

a l l o w e d for i n - c l a s s d i s c u s s i o n a nd d e m o n s t r a t i o n of m e t h o d s
and m a t e r i a l s b e f o r e s e n d i n g the s t u d e n t s

to c l a s s r o o m s .

Selected Research and Evaluation Literature
on R e l a t e d (Research) M e t h o d o l o g y
of the Study
Herriot

(1967)

r e p o r t e d an e x p l o r a t o r y s t u d y in

w h i c h b e l o w a v e r a g e 7 and 9 g r a d e s

s t u d e n t s d i d as w e l l as

73
(in som e cas es b e t t e r than)

the a b o v e a v e r a g e s t u d e n t s

a l g e b r a c o v e r e d b y the fi rst g r o u p
ye ars

(below average)

and by above a v e r a g e s t u d e n t s in one year.

s e e m e d to s u p p o r t C a r r o l l 1s h y p o t h e s i s

(1963).

in

in two
T he study

C a r r o l l 's

hypothesis:
All,

or a l m o s t all,

stu d e n t s

c o u l d b e b r o u g h t to

the sam e l evel of a c h i e v e m e n t in any p a r t i c u l a r s c h o l a s t i c
topic, b u t the a m o u n t of

instruction

that w o u l d be n e e d e d

to b r i n g a s t u d e n t to a p a r t i c u l a r level of a c h i e v e m e n t
would vary

f r o m s t u d e n t to student.

Begle

(1971)

r e p o r t e d a s i m i l a r study in w h i c h

fourth g r a d e cla sse s w e r e
with base

tau g h t the s a m e c o n t e n t d e a l i n g

five n u m e r a t i o n u n d e r three d i f f e r e n t pla ns

d e v i s e d to c o v e r the same m a t e r i a l in o n e c las s p eri od,
class p eri ods ,

two

a n d three cla ss p e r i o d s r e s p e c t i v e l y t h e r e b y

c o n t a i n i n g d i f f e r e n t i n c r e a s i n g a m o u n t of t e a c h i n g t e c h ­
niques,

review of related materials,

p r a c t i c e time a n d all

the th r e e t r e a t m e n t s g i v e n to e a c h g r o u p of s t u d e n t s
c l a s s i f i e d as low, mi ddle,
of reasoning.
Carroll's

already

and high ability using pretest

A g a i n the r esults are in a c c o r d w i t h

hyp oth esi s.

Bloom

(19 71)

emphasized domain/criterion-referenced

m e a s u r e d w i t h h i g h c o n t e n t - v a l i d i t y are n e e d e d for e v a l u ­
ation of i n s t r u c t i o n / c u r r i c u l u m .
procedures

Milliman

for e s t a b l i s h i n g s t a n d a r d s

n u m b e r of it ems n e e d e d in

(1973)

reviewed

and d e t e r m i n i n g the

" c r i t e r i o n - r e f e r e n c e d " m eas u r e s .

He o r g a n i z e d the d i s c u s s i o n a n d p r o c e d u r e of s e t t i n g a

74
p a s s i n g sco re a r o u n d five f a c t o r s - p e r f o r m a n c e of others,
i t e m content,

e d u c a t i o n a l con s e q u e n c e s ,

fi nan c i a l costs a nd err ors

p s y c h o l o g i c a l and

due to g u e s s i n g an d i t e m s a m p l i n g

and b e l i e v e d t ha t they r e q u i r e judgment.
t h e o r y , b i n o m i a l , and s e q u e n t i a l m o d e l s
le n g t h w e r e c ons ide red ,
and the last w a s

Classical

test

for d e t e r m i n i n g test

the first w a s no t v i e w e d as useful,

jud ged as m o s t fea sib le w h e n e x a m i n e s s

i n t e r a c t w i t h c o m p u t e r s d u r i n g testing.
W a l b e s s e r an d C a r t e r

(19 68)

d i s c u s s e d the i m p o r t a n c e

of d e f i n i n g c u r r i c u l u m b y d e v e l o p i n g a s e q u e n c e d set of
instructional objectives while Atkin

(19 68) w a r n e d that the

e d u c a t i o n a l r e l e v a n c e of c u r r i c u l u m m i g h t b e r e d u c e d b y
s t r i c t a d h e r e n c e to s p e c i f i c a t i o n of b e h a v i o r a l o u t c o m e s of
i n s t r u c t i o n a l a c t i v it ies .
and S c r i v e n

(1967)

(1971), T a t e

(1956),

St a n l e y

(1967),

all d i s c u s s e d th e m e t h o d o l o g y of and

distinctions between
Recent writers

Bloom

f o r m a t i v e a nd s u m m a t i v e ev al u a t i o n s .

like R e y n o l d s a nd L i g h t

(1971), W e s t b u r y

(1970),

(1971), A b e d o r
an d W e i s s

(1971)

n o t e d the a m b i g u i t y in the d e f i n i t i o n o f the term,
ev alu ati on,
procedures

have

formative

and the c o n s e q u e n t p a u c i t y of w e l l - d e f i n e d
and t e c h n i q u e s

for c o n d u c t i n g such ev a l u a t i o n .

T e c h n i c a l te rms in the s t u d y are us ed in Sa n d e r s
Cunningham's

sense.

Metfessel and Michael

(1969)

and
discussed

m e t h o d s us ed to c o l l e c t e x t e r n a l i n f o r m a t i o n for i n t e r i m
f o rma tiv e e v a l u a t i o n .
model

Borich

(19 71)

suggested a conceptual

for f o r m a t i v e p r o d u c t ev a l u a t i o n .

m a t i o n is of u t m o s t i m p o r t a n c e

Contextual infor­

for f o r m a t i v e p r o d u c t

75
e v a l u a t i o n e m p h a s i z e d S a n d e r s a nd C u n n i n g h a m
S t u f f l e b e a m e t al.

(1971)

(1973).

d i s c u s s e d in d e t a i l forces that

ac t u pon p r o g r a m s d e s i g n e d to m e e t the needs of s t u d e n t s

in

i l l u s t r a t i v e e x a m p l e of w h a t they c a l l e d " C o n t e x t u a l source
of I n f o r m a t i o n . "

S t u f f l e b e a m et al.

(1971)

also discussed

steps tak en to c o l l e c t i n f o r m a t i o n for d i f f e r e n t kinds of
ev alu ati on.

In B l o o m e t al.

(1971), W i l s o n i l l u s t r a t e d

w h a t Sa nd e r s

and C u n n i n g h a m t e r m e d I n t e r i m

(Process)

f o r m a t i v e e v a l u a t i o n u s i n g i n t e r n a l s o u r c e of i n f o r m a t i o n
(which I n t e r i m i n t r i n s i c ev a l u a t i o n ,
1967),

on pp.

690-92.

f o l l o w i n g Scriven,

C o l l e c t i o n of such i n f o r m a t i o n

all the u n i t s of a p r o g r a m p u t t o g e t h e r
w h a t S anders and C u n n i n g h a m c a l l e d
evaluation"

(pp.

646-47)

for
is

"f o r m a t i v e p r o d u c t

f r o m i n t e r n a l sou r c e if u s e d s t r i c t l y

for f e e d ­

b a c k of the d e v e l o p e r b u t it w i l l be a s u m m a t i v e e v a l u a t i o n
if use d as c o n s u m e r r e p o r t - t y p e of appraisal.
Hunsen

(19 67)

s h o w e d t h a t t h r o u g h o u t the w o r l d

s t u d e n t a c h i e v e m e n t in m a t h e m a t i c s
e d u c a t i o n s o c i o - e c o n o m i c status.

is r e l a t e d to parents'
He s u g g e s t e d f u r t h e r

m u l t i v a r i a t e s tu d i e s o f t e s t sc o r e s a nd b a c k g r o u n d

factors

a f f e c t i n g p e r f o r m a n c e in m a t h e m a t i c s .
O n th e r e l a t i o n s h i p b e t w e e n m a t h e m a t i c s a c h i e v e m e n t
and a t t i t u d e

t owa rd m a t h e m a t i c s N e a l e

ar t i c l e b a s e d on C a t t e l l an d B u t c h e r ’s
Ne a l e was o f the o p i n i o n

(19 69) w r o t e an
(1968)

findings.

that i m p r o v e m e n t of a t t i t u d e

to w a r d m a t h e m a t i c s m a y n o t i n c r e a s e m a t h e m a t i c s
Aiken

a chievement.

(19 70) w a s o f the o p i n i o n t hat N e a l e m a d e too

76
d e f i n i t i v e c o n c l u s i o n s m o r e than w a r r a n t e d f r o m c o r r e l a t i o n a l
studies.

Aiken

(1970)

s a i d the a p p a r e n t c o n t r a d i c t o r y /

n o n - c o n c l u s i v e r esults on r e l a t i o n s h i p b e t w e e n m a t h e m a t i c s
a c h i e v e m e n t and a t t i t u d e is due to m i s — use of a t t i t u d e
to war d math.

Aiken

(1972)

c l a i m e d that the L i k e r - l i k e

i n s t r u m e n t is m o r e r e l i a b l e in h i g h s c h o o l an d c o l l e g e
than o t h e r types of i n s t r u m e n t s and d e f e n d e d his p osi tio n.
He r e a f f i r m e d his b e l i e f th at i m p r o v i n g te ach ers a t t i t u d e
to war ds m a t h e m a t i c s can r e s u l t in m o r e p o s i t i v e a t t i t u d e
on the p a r t of the students.

H e then r e c o m m e n d e d t h a t

a t t i t u d e t owards d i f f e r e n t as p e c t s of m a t h e m a t i c s w i l l be
m o r e m e a n i n g f u l than just a s i n g l e m e a s u r e o f a t t i t u d e
toward mathematics.

Aiken

ou t on t hre e age g r o u p s
st udy shows

that

(a)

(1972)

reported a study carried

(also c l a s s i f i e d by s e x ) ,

there is a g e n e r a l v a r i a b l e of a t t i ­

tude t o w a r d m a t h e m a t i c s t h a t i n c l u d e s
routine computations,
(b)

terms,

attitude toward

symbols and world problems,

attitude toward mathematics

is d i r e c t l y r e l a t e d to

in t e r e s t in p r o b l e m - s o l v i n g tasks in general,
r e l a t e d to i n t e r e s t in l a n g u a g e arts,
other

His

"verbal" pur su i t s ,

but inversely

s o c i a l studies,

a nd

(c) p e o p l e w i t h m o r e p o s i t i v e

a t t i t u d e t o w a r d m a t h tend to like d e t a i l e d w o r k and see
t h e m s e l v e s m o r e p e r c e i v i n g or s e l f - c o n f i d e n t ;
tend to m a k e h i g h e r m a r k s
general,

and

in m a t h an d in s c h o o l w o r k in

(d) a l t h o u g h there are age and se x d i f f e r e n c e s

in this regard,
the f a t h e r

they als o

the r e p o r t e d a t t i t u d e a n d a c h i e v e m e n t of

( par t i c u l a r l y in the c ase of male)

a nd t h a t of

77
mother

(partic ula rly in the ca se of female)

a s s o c i a t e d w i t h students'

are also

a t t i t u d e t o w a r d math.

W h i l e an o f t e n s t a t e d o b j e c t i v e in the p r e p a r a t i o n
of e l e m e n t a r y teachers of m a t h e m a t i c s

is

"the d e v e l o p m e n t

in these p r o s p e c t i v e tea che rs of f a v o u r a b l e a t t i t u d e t o w a r d
m a t h e m a t i c s " w i l l i n f l u e n c e students'
m e n t in m a t h e m a t i c s ,

Peskin

(1964)

attitude and achieve­

found no significant

r e l a t i o n b e t w e e n t e a c h e r a t t i t u d e a n d s t u d e n t a t t i t u d e nor
b e t w e e n t e a c h e r a t t i t u d e and s t u d e n t a c h i e v e m e n t .
(1973)

c l a i m e d that P e s k i n ' s

hi s design.
students

Phillip

fi ndi ngs w e r e c o n s e q u e n t i a l to

He then r e p o r t e d a s t u d y w h i c h s h o w e d t hat

a c h i e v e b e t t e r in a r i t h m e t i c if the y h a d s e q u e n c e

o f three teachers,

all of w h o m h a d p o s i t i v e a t t i t u d e t o w a r d

a r i t h m e t i c than if they h a d a s e q u e n c e of t h r e e t e a c h e r s
h a v i n g u n f a v o u r a b l e a t t i t u d e t o w a r d a r i t hme tic .

His s tud y

f u r t h e r s h o w that type of t e a c h e r a t t i t u d e t o w a r d a r i t h ­
metic,

student attitude toward arithmetic,

and student

i n t e l l i g e n c e do not i n t e r a c t in any w a y s uch as to p r o d u c e
a s i g n i f i c a n t s t u d e n t a c h i e v e m e n t in a r i t h m e t i c .

Thi s

s t u d y s hows that te ach ers * a t t i t u d e t o w a r d a r i t h m e t i c does
n o t hav e s i g n i f i c a n t e f f e c t o n s t u d e n t ' s

a t t i t u d e and

a c h i e v e m e n t un l e s s s t u d e n t - t e a c h e r i n t e r a c t i o n lasts for a
sufficiently

long period,

t h r e e years,

say.

Knaupp

(1973)

al so r e v i e w e d l i t e r a t u r e on c a u s a l r e l a t i o n s h i p b e t w e e n
a t t i t u d e an d a c h i e v e m e n t in m a t h e m a t i c s , he o bse rved,
Aiken

(19 70),

like

that in m o s t of the se s t u d i e s i n s t r u m e n t s ,

78
d e s i g n s , m e t h o d of a n a l y s i s are d e f e c t i v e and these a c c o u n t
for a p p a r e n t l y c o n t r a d i c t o r y / i n c o n c l u s i v e findings.
Todd

(1966),

Reys a n d D e l o n

prospective elementary school

(19 68)

s h o w e d th at

teachers w h o completed basic

m a t h e m a t i c s c o m p a r a b l e to C U P M L e v e l

I recommendations

demonstrated significant

t h e i r s cor es on the D u t t o n

scale.

gains o n

H u n k e r a nd Q u e s t (19 72)

n o t e d t h a t in none of

the

two c i t e d s t u d i e s w a s a c o m p a r i s o n m a d e b e t w e e n those
st ude n t s w h o h a d c o m p l e t e d the course.
Norton
1.

(19 56)

found th at te ache rs

display

a strong interest

P o f f e n b e r g e r and

who,
in subject,

2.

indicate a desire
m a t e r i a l , an d

3.

d i s p l a y a goo d con t r o l of the class w i t h o u t b e i n g
o v e r l y s t r i c t ten d to a f f e c t students' a t t i t u d e s
an d a c h i e v e m e n t po s i t i v e l y .

N o t i n g this
Quast

(19 72)

to h a v e s t u d e n t s u n d e r s t a n d the

f i n d i n g of P o f f e n b e r g e r an d No rto n,

H u n k e r and

r e p o r t e d a st u d y w h i c h c o m p a r e d a t t i t u d e of

p r o s p e c t i v e e l e m e n t a r y sc h o o l teachers,
a.

w h o have taken neither mathematics content nor
m a t h e m a t i c s - m e t h o d course,

b.

w h o have taken mathematics
method c o u r s e ,

c.

w h o have taken both mathematics content and math—
me thod, the l a t t e r b e i n g t a u g h t b y an i n s t r u c t o r
who displayed Poffenberger-Norton characteristics.

content but not math-

The r e s u l t s of their s t u d y show,
1.

the m a t h - m e t h o d d e s i g n e d for t he p r o s p e c t i v e
te ach e r s d i d i m p r o v e t h e i r m a t h e m a t i c s att itu de,

2.

t he m a t h - c o n t e n t t o g e t h e r w i t h m a t h - m e t h o d courses ,
can p r o b a b l y be u s e d to i m p r o v e the m a t h e m a t i c s
attitude of prospective elementary school t e a c h e r s .

79
Alexander,

et al.

(19 71)

r e p o r t e d a s t u d y in w h i c h

the p a s t g r a d e p o i n t s w e r e u sed to cla s s i f y s t u d e n t s
ranking)

in Hig h GP A and Lo w GPA.

as s i g n e d to two treatm ent s.

They were

(by

then randomly

I, II,

in w h i c h Ss in I w e r e

p o l i t e l y t r e a t e d by the i n s t r u c t o r

(but w i t h no p e r s o n a l

interest)

w h i l e Ss in II w e r e r e f e r r e d to by n a m e and the

i n s t r u c t o r i n i t i a t e d d i s c u s s i o n w i t h them.
gators r e p o r t e d

The i n v e s t i ­

that the t e a c h e r - i n i t i a t e d t e a c h e r - s t u d e n t

personal inter-actions did significantly
a c h i e v e m e n t in f a v o u r o f Ss II.

i n f l u e n c e the

Th ey a l s o f oun d th at Ss

w i t h Hi gh G P A a c h i e v e d s i g n i f i c a n t l y h i g h e r sc o r e s than Ss
with

low G P A a n d c o n t r a r y to p r e v i o u s

(19 70)

study,

b y Means,

et al.

c o n c e r n e d w i t h G P A the i n t e r a c t i o n was n o t s i g n i f i ­

c ant a l t h o u g h in the s a m e d i r e c t i o n as in the p r e v i o u s
study.

Th e i n v e s t i g a t o r s o b s e r v e d that as the s e m e s t e r

p r o g r e s s e d Ss II i n c r e a s i n g l y i n i t i a t e d i n t e r a c t i o n w i t h
the instructor.
Begle

(1972)

r e p o r t e d a study w h i c h s o u g h t the

relationship between teachers understanding of modern
al g e b r a and t h e i r s t u d e n t a c h i e v e m e n t in n i n t h g r a d e
algebra.

T h e y found that:

1•

the p r e t e s t s g i v e n the s t u d e n t s (math i n v e n t o r y
f r o m N L S M A an d R e f e r e n c e T e s t for C o g n i t i v e
Factors) t urn ed o u t to b e g o o d p r e d i c t o r s of
success,

2.

there was substantial variations
ness o f teachers,

3.

t e a c h e r s e f f e c t i v e n e s s w i t h m a l e s t u d e n t s w a s not
s i g n i f i c a n t l y d i f f e r e n t f r o m tea cher s e f f e c t i v e n e s s
on fe mal e s t u d e n t s ,

in the e f f e c t i v e ­

80
4.

t e a c h e r s u n d e r s t a n d i n g of m o d e r n a l g e b r a has no
s i g n i f i c a n t c o r r e l a t i o n w i t h s t u d e n t a c h i e v e m e n t in
n i n t h gra de a l g e b r a ,

5.

te ach ers u n d e r s t a n d i n g o f the a l g e b r a of r eal
n u m b e r s ha s no s i g n i f i c a n t c o r r e l a t i o n w i t h ni nth
g r a d e a l g e b r a i c skills,

6.

t eac her s u n d e r s t a n d i n g o f a l g e b r a of real n u m b e r s
is s i g n i f i c a n t l y c o r r e l a t e d w i t h st ud e n t s a c h i e v e ­
m e n t in u n d e r s t a n d i n g a l g e b r a i c c o n c e p t s b u t the
c o r r e l a t i o n is so low t h a t it is e d u c a t i o n a l l y
insignificant.

CHAPTER III
DESCRIPTIVE FEATURES AND DESIGN
O F T H E ST UDY
Introduction
This chapter summarizes

the e v a l u a t i o n a c t i v i t i e s

c a r r i e d ou t in the p r e s e n t study.

It is p r e s e n t e d in two

parts.
The

first p a r t i n v o l v e s

an a p p r a i s a l of the m a t h e ­

m a t i c s e d u c a t i o n c o m p o n e n t o f the p r o g r a m w i t h r e f e r e n c e
to the content,

goals,

grading procedures,

a p p r o a c h has b e e n t e r m e d

"intrinsic"

an a r m c h a i r a f f a i r

re lie s h e a v i l y o n the
mation .
goals,

It p r o v i d e s

(program)

This

evaluation.

c r i t e r i a are u s u a l l y n o t o p e r a t i o n a l l y
to s o m e extent,

etc.

The

formulated,

(Scriven,

it is,

1967).

It

i n t e r n a l s o u r c e of i n f o r ­

i n f o r m a t i o n a b o u t the r a t i o n a l e ,

a n d o b j e c t i v e s of the p r o g r a m w h i c h c o n t r i b u t e to an

understanding of value positions
and o t h e r p e r s o n s

ta k e n b y the d e v e l o p e r s

i n v o l v e d in th e p r o g r a m
Th is

(Sanders and

Cunningham,

1973).

is u s e f u l b e c a u s e o f the use

"o bje cti ve"

d a t a a l o n e is i n s u f f i c i e n t in the e v a l u a t i o n o f

le a r n i n g u n d e r d i s s i m i l a r s y s t e m s of i n s t r u c t i o n

81

(Brownell,

82
1966) .

Evaluation activities

in the st u d y u n d e r this

a p p r o a c h are;
Th e g e n e r a l c o n t e x t a n d p r o g r a m d e s c r i p t i o n :

An

a n a l y s i s of the m a t h e m a t i c s c o n t e n t in the m a t h e m a t i c s
e d u c a t i o n c o m p o n e n t o f the progra m;
A d e s c r i p t i o n of the m a t h e m a t i c s m e t h o d i n t e g r a t e d w i t h
the m a t h e m a t i c s c o n t e n t an d c l i n i c a l e x p e r i e n c e ;
A
They

c r i t i c a l a p p r a i s a l of the i n s t r u c t i o n a l met h o d .
form what Wittrock

(196 8) d e s c r i b e d as e v a l u a t i o n of

e n v i r o n m e n t s of l e a r n i n g .
The second part describes

the p r o c e d u r e

for

e x a m i n i n g the e f f e c t s o f the m a t h e m a t i c s e d u c a t i o n c o m p o ­
n e n t o f the p r o g r a m on the i n t e r n s .
termed

"pay off" e v a l u a t i o n

the e v a l u a t i o n of l ear ning,

(Scriven,

pre-

includes

and p o s t - t e s t s ,

1967),

It i n v o l v e s

the e v a l u a t i o n o f l e a r n e r s

the e v a l u a t i o n of i n s t r u c t i o n
activities

This approach has been

(Wittrock,

1968).

The

a p p r a i s a l o f the d i f f e r e n c e s b e t w e e n
and between experimental and control

groups tests on a number of criterial parameters.
e v a l u a t i o n relies h e a v i l y o n e x t e r n a l
texual

(entry behavior)

r e s e a r c h des ign,
assumptions

and

sources.

hypotheses

and l i m i t a t i o n s

(samples)

T h e samples,

a n d conmeasures,

tested, m e t h o d o l o g i c a l
are d e s c r i b e d .

The

83
General Context and Program Description
Its O r i g i n

■

T h e T e a c h e r C o r p s P r o g r a m is a U n i t e d States
federally

i n i t i a t e d r e f o r m e f f o r t c r e a t e d d u r i n g the 1960s

to i m p r o v e the w e l f a r e o f

l o w - i n c o m e people.

The original

p u r p o s e of the p r o g r a m as p r o v i d e d in a u t h o r i z i n g l e g i s ­
lation , w e r e :
1.

To strengthen educational opportunities
i n areas w i t h c o n c e n t r a t i o n s

2.

of l o w - i n c o m e

To attract and prepare persons
in su ch areas

for c h i l d r e n

to b e c o m e

families.

teachers

through coordinated work-study

experiences.
3.

To encourage colleges

an d u n i v e r s i t i e s ,

sc hools,

a n d st a t e d e p a r t m e n t s

of e d u c a t i o n to w o r k t o g e t h e r

to b r o a d e n a n d i m p r o v e t e a c h e r - e d u c a t i o n p r o g r a m s
(Corwin,
Th e C o r p s

1973).

r e s u l t e d f r o m t h e p r e m i s e t h a t t h e r e are c r i t i c a l

d i f f e r e n c e s b e t w e e n the s k i l l s r e q u i r e d to t e a c h in lowi n c o m e s ch o o l s

a nd m i d d l e - c l a s s schools.

d o e s n o t imply t h a t u n i q u e p r i n c i p l e s

of l e a r n i n g are

i n v o l v e d in the two d i f f e r e n t s e t t i n g s .
the d i f f e r e n c e s
me n t s

in val ues,

B u t this p r e m i s e

It is b e l i e v e d th at

prior experiences,

among children from various

income,

and e nviron­

ethnic,

and

r a c i a l s u b g r o u p s are so g r e a t t h a t the t e a c h e r s n e e d
s p e c i a l t r a i n i n g in o r d e r to a p p l y the p r i n c i p l e s
fashion the procedures

for e a c h group.

and

84
The

typical Corps program involves

li be r a l a r t s g r a d u a t e s

(interns)

f r o m 30 to 40

a n d five p r o f e s s i o n a l

t e a c h e r s w h o a c t as t e a m leaders.

T h e g r o u p r e c e i v e s a bou t

e i g h t w e e k s of s p e c i a l p r e s e r v i c e t r a i n i n g a t a c o l l e g e or
university,

a f t e r w h i c h it is d i v i d e d i n t o five teams,

c o m p o s e d o f at l e a s t s i x i n t e r n s

each

and o n e t e a m leader.

Each

t e a m is a s s i g n e d to a s c h o o l t h a t s e r v e a p o v e r t y area,
u s u a l l y an e l e m e n t a r y sc h o o l , w h e r e the t e a m s p e n d s
le ast 60 p e r c e n t o f its w e e k l y

time.

at

In the b e g i n n i n g the

t e a m m a y w o r k w i t h s m a l l g r o u p s of s t u d e n t s o n s p e c i f i c
l e s s o n p l a n s but,
become more

as the t e a m g a i n s e x p e r i e n c e ,

complex.

It s p e n d s

its tasks

a b o u t 20 p e r c e n t o f its

t i m e in a c a d e m i c w o r k at the u n i v e r s i t y

(some of this w o r k

is i n t e r d i s c i p l i n a r y a n d leads to t e a c h e r c e r t i f i c a t i o n and
a m a s t e r 's d e g r e e in two y e a r s ) .

F i n a l l y , t he i n t e r n s

also

are e x p e c t e d to s p e n d 20 p e r c e n t of t h e i r time on c o m m u n i t y
activities,

l e a r n i n g as m u c h as th ey c an a b o u t the e n v i r o n ­

m e n t of t h e i r s t u d e n t s .
Several

thousand interns

and experienced teachers

h a v e g r a d u a t e d f r o m the p r o g r a m s i n c e t he f i r s t c y c l e b e g a n
in 1966.

Corps

at o n e time,

t e a m s s e r v e in f r o m 30 to 70 u n i v e r s i t i e s

b u t they h a v e s e r v e d in m o r e t h a n 100 u n i ­

v e r s i t i e s a n d 250 s c h o o l s y s t e m s
Ri co at v a r i o u s

times.

city s c h o o l s y s t e m s ,

in 37 s t a t e s

and Puerto

A b o u t h a l f of the p r o g r a m s

including seventeen

a b o u t h a l f ar e i n s m a l l t o w n s
b e e n p r o g r a m s in N e w York,

large c ities,

a n d r u r a l areas.

Chicago,

are in

Detroit,

and

There have

Philadelphia,

85
Los A n g e l e s , K a n s a s C i t y , M i a m i , A t l a n t a ,
Dallas;

in A p p a l a c h i a n towns,

ru r a l South;

S e a t t l e , and

in the Ozarks,

in m i g r a n t co m m u n i t i e s ,

in S p a n i s h - s p e a k i n g c o m m u n i t i e s

an d in the

in Ind ian schools;

in N e w York,

F lor ida ,

a nd

and

the Southwes t.
Th e e i g h t h c y c l e T e a c h e r C o r p s p r o g r a m w h o s e m a t h e ­
m a t i c s e d u c a t i o n c o m p o n e n t is b e i n g e v a l u a t e d is c o o p e r ­
a t i v e l y i m p l e m e n t e d by the L a n s i n g S c h o o l District,
Michigan State University,
munity .

the

the M o d e l C i t i e s an d the c o m ­

T h e T e a c h e r C o r p s A d v i s o r y B o a r d c o n s i s t s of

r e p r e s e n t a t i v e s of t h e s e bodies.
Rationale
In its e f f o r t to r e c r u i t t e a c h e r s
w i t h d i s a d v a n t a g e d ch ild r e n ,

ca p a b l e of w o r k i n g

the L a n s i n g S c h o o l D i s t r i c t

has e n d e a v o r e d to r e c r u i t t e a c h e r s

f r o m e t h n i c gr o u p s

r e p r e s e n t a t i v e of t he s t u d e n t p o p u l a t i o n .

However,

it has

b e e n r e a l i z e d t hat al l t e a c h e r s n e e d t r a i n i n g in w o r k i n g
w i t h d i s a d v a n t a g e d c h i l d r e n a n d in d e v e l o p i n g p r o g r a m s
w h i c h b e n e f i t t h e s e a n d the o t h e r p u p i l s o f the d i s t r i c t .
T r a i n i n g of th e s e t e a c h e r s s h o u l d be c o m p e t e n c y - b a s e d

(in

b o t h the a c a d e m i c areas of t h e c u r r i c u l u m a n d in s t u d e n t
at t i t u d e s ) , community-based,

bilingual and bicultural

(Lansing S c h o o l D i s t r i c t a n d M i c h i g a n S t a t e U n i v e r s i t y ,
1972).
A f u n d a m e n t a l p u r p o s e of the L a n s i n g S c h o o l s y s t e m
is to s u s t a i n and n o u r i s h free s o c i e t y

through transmission

86
of c u l t u r a l and p o l i t i c a l h e r i t a g e to c h i l d r e n and youth.
R e c o g n i z i n g that the free s o c i e t y is s t i l l the e x c e p t i o n a l
society,

the schools plan

to d e v e l o p in p u p i l s

c r e a t i v e ta l e n t and i n t e l l e c t u a l vigor.
teaching methods

the n e c e s s a r y

Traditional

an d p e r s o n n e l h ave d i f f i c u l t y r e a l i z i n g

th e s e e d u c a t i o n a l o b j e c t i v e s w i t h m a n y e l e m e n t s of the
d i s a d v a n t a g e d c omm un i t y ,

n o r are they p a r t i c u l a r l y su it a b l e

for t e a c h i n g all c h i l d r e n w h o w i l l

live a n d w o r k in the

y e a r 2000.
M i c h i g a n State U n i v e r s i t y has h a d a l o n g - s t a n d i n g
i n t e r e s t in d e v e l o p i n g c o m p e t e n c y - b a s e d t e a c h e r educat ion .
It has b e e n h e a v i l y e n g a g e d in the T r a i n i n g of T e a c h e r E d u ­
cator P r o j e c t s as w e l l as in o t h e r p r o g r a m s w h i c h are
c o m p e t e n c y - b a s e d o r b a s e d o n b e h a v i o r a l ob j e c t i v e s .
M i c h i g a n State U n i v e r s i t y and the L a n s i n g S c h o o l
District have

joi n t l y s p o n s o r e d a s h o r t t erm t e a c h e r e d u ­

cation program which was oriented toward developing
C l i n i c a l T e a c h i n g S t r a t e g i e s in r e a d i n g a n d m a t h e m a t i c s .
C o u r s e s w e r e o f f e r e d o n the u n d e r g r a d u a t e a n d g r a d u a t e
level in c o m p e t e n c y - b a s e d e d u c a t i o n .

T h e s e c ou r s e s w e r e a

j o int o f f e r i n g of the C o l l e g e o f U r b a n D e v e l o p m e n t a nd the
C o l l e g e of Ed u c a t i o n .

New staffing patterns have been

e x p l o r e d c o o p e r a t i v e l y b y the u n i v e r s i t y a n d the school
system utilizing workshops and seminars on differentiated
s t a f f i n g a n d the o p e n classr oom .
T h e C o l l e g e of E d u c a t i o n has e s t a b l i s h e d a c o u n c i l
with members

from all departments and programs

i n v o l v e d in

87
competency-based education.

T h e c o u n c i l ' s task w a s

to

c o o r d i n a t e eff o r t s

and d i s s e m i n a t e i n f o r m a t i o n a b o u t the

college's programs

in this area.

U r b a n D e v e l o p m e n t has,

Fi nal l y ,

the C o l l e g e of

t h r o u g h the P r o j e c t D e v e l o p m e n t

S p e c i a l i s t in the s i x t h c y c l e

(who w a s

a f ac u l t y m e m b e r w i t h

the C o l l e g e ) , m a d e a c o m m i t m e n t to the d e v e l o p m e n t of
c o m p e t e n c y - b a s e d p r o g r a m s as a n o t h e r v i a b l e i n s t r u c t i o n a l
mode.

M u c h of the c u r r i c u l u m b e i n g p l a n n e d for the U r b a n

E d u c a t i o n c o m p o n e n t o f the C o l l e g e w o u l d be c o m p e t e n c y based .
C o m m u n i t y b a s e d e d u c a t i o n and i n v o l v e m e n t ,

parti­

c u l a r l y t h a t of p a r e n t s o f s c h o o l chi ld r e n , w a s

a central

fe a t u r e o f th e T e a c h e r C o r p s s t r a t e g y w h i c h w a s

facilitated

t h r o u g h the c o l l a b o r a t i v e d e c i s i o n - m a k i n g p r o c e s s o u t l i n e d
in the p rop osa l.

Developmental Community involvement

p r o g r a m s of any kind,

to b e c o m p l e t e l y s u c c e s s f u l ,

require

c e r t a i n c h a n g e s in the e x p r e s s e d a t t i t u d e s

and b e h a v i o r of

the p e o p l e w i t h i n the c o m m u n i t y c o n c e r n e d ,

o t h e r w i s e no

s u c h p r o g r a m n e e d be c o n t e m p l a t e d .

realized that

It w a s

th ese c h a n g e s c o u l d n o t b e d i c t a t e d by a few le a d e r s w i t h i n
or outside
city,

the T e a c h e r C o r p s s t r u c t u r e o p e r a t i n g at the

county,

or r e g i o n a l

level.

T hey c o u l d b e s t be

a c h i e v e d w h e n they are the r e s u l t o f the d e l i b e r a t i o n s of
the p e o p l e w o r k i n g o u t t h e i r o w n p r o b l e m s

in p r i m a r y g r o u p s

at the local level.
T h e c o m m u n i t y e d u c a t i o n c o m p o n e n t a i m e d at
d e v e l o p i n g i n s t i t u t i o n a l u n d e r s t a n d i n g of the c o m m u n i t y

88
a n d c o m m u n i t y u n d e r s t a n d i n g of th e i n s t i t u t i o n s .
c o m p o n e n t f u n c t i o n e d in a d u a l p a t t e r n .

This

The individuals

p a r t i c i p a t i n g i n this p r o g r a m w e r e n o t o n l y i n s t r u c t o r s
b u t a l s o l ist ene rs.

Th e c o m m u n i t y h a d an o p p o r t u n i t y to

interact through seminars

so t h a t i n s t i t u t i o n s h a d a c l e a r e r

a n d b e t t e r p e r s p e c t i v e o f t he c o m m u n i t y it w a s

serving.

Objectives
Teacher Corps Training Objectives
Cooperating Teachers,
1.

T e a m Le aders,

for Interns,

and Community Volunteers:

To m a i n t a i n a n d d e v e l o p c o m p e t e n c y - b a s e d i n ­
s t r u c t i o n a l p r o g r a m s w h i c h r e s u l t in p o s i t i v e
a c h i e v e m e n t of p u p i l s

in the L a n s i n g S c h o o l D i s t r i c t

w i t h p a r t i c u l a r r e f e r e n c e to c h i l d r e n in g r a d e s
Kindergarten-six from low-income

families.

c o n c l u s i o n of th e p r o g r a m the p a r t i c i p a n t s
be abl e to m e e t the f o l l o w i n g o b j e c t i v e s
competency
a.

A t the
should

at a

l e v e l e s t a b l i s h e d b y t he s c h o o l d i s t r i c t .

to u s e a s s e s s m e n t
diagnose

a n d o b s e r v a t i o n a l s k i l l s to

learning strengths and weaknesses of

all p u p i l s

i n c l u d i n g th o s e w i t h

learning dis­

abilities .
b.

to u s e d i a g n o s t i c

d a t a in d e v e l o p i n g b e h a v i o r a l

o b j e c t i v e s w h i c h s p e a k to t h e ne e d s of e a c h
p u p i l a n d to d e s i g n e f f e c t i v e s t r a t e g i e s
attain those objectives.

to

89
c•

to e v a l u a t e p u p i l g r o w t h t o w a r d i m p o r t a n t e d u ­
cational objectives

and plan curriculum

revisions based on such evaluations.
d.

t o e q u i p the p a r t i c i p a n t s w i t h t h e sk i l l s
n e c e s s a r y to b e able to a s s e s s h i s / h e r i m p a c t
o n s t u d e n t s a n d to m o d i f y

t hat i m p a c t by

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

to i d e n t i f y l i n g u i s t i c p r o b l e m s a n d c o n d u c t a
l a n g u a g e or d i a l e c t i c a l l y a d j u s t e d r e a d i n g s k i l l
d e v e l o p m e n t a l p r o g r a m so t h a t c h i l d r e n c a n
i m p r o v e t h e i r r e a d i n g level.

f.

to c o n d u c t an i n q u i r y - o r i e n t e d m u l t i - d i s c i p l i n a r y
pr o g r a m which,

organized cross-culturally, will

e q u i p c h i l d r e n to live in a p l u r a l i s t i c society.
g.

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

visual conceptualization of mathematical

constructs without
putation,

l o s i n g p r o f i c i e n c y in c o m ­

a nd r e l a t e s m a t h e m a t i c s

lessons

to

the e x p e r i e n c e s o f l o w - i n c o m e c hil dren.
h.

to e x a m i n e

interpersonal relationships between

staff members w h i c h impede the teachinglearning process.
i.

to e f f e c t i v e l y i n t e g r a t e i n s t r u c t i o n a l m e d i a
w i t h i n s t r u c t i o n a l m o d u l e s d e v e l o p e d in the
p r e v i o u s cycle.

90
j.

to d e v e l o p a nd i n v o l v e c o m m u n i t y r e s o u r c e s as
i n t e g r a l p a r t s of t h e t e a c h e r - l e a r n i n g pr oce ss;
and

k.

to i n t e r p r e t the s c h o o l 's i n s t r u c t i o n a l p r o g r a m
to t he c o m m u n i t y a n d to e n c o u r a g e p a r e n t s and
p a t r o n s o f the d i s t r i c t to take p a r t in the
o n g o i n g e v a l u a t i o n s of g o a l s a n d o b j e c t i v e s
w i t h i n the f r a m e w o r k e s t a b l i s h e d b y the B o a r d
of Education.

2.

To continually modify instructional programs
m e e t the c h a n g i n g e d u c a t i o n a l n e e d s

to

of th e s t u d e n t s

a nd to take a d v a n t a g e o f n e w m a t e r i a l s ,

techniques,

an d t e c h n o l o g y .
a.

to d e s i g n c u r r i c u l u m m o d e l s w h i c h r e l a t e to
the m u l t i - e t h n i c p o p u l a t i o n of t h e L a n s i n g
School District and which provide ways

for the

d e v e l o p m e n t of p o s i t i v e s e l f - i m a g e s a m o n g

low-

income students,
b.

to d e v e l o p

the use o f v i d e o r e c o r d e r s a n d o t h e r

r e l a t e d e q u i p m e n t as p a r t o f the i n s t r u c t i o n a l
programs,
c.

to m o v e b e y o n d the t r a d i t i o n a l t e a m t e a c h i n g
model

to a d i f f e r e n t i a t e d s t a f f t h a t w o u l d

include

community resource p e r s o n s , professional

and paraprofessional educators.

91
d,

to use m o d u l a r de s i g n in d e v e l o p m e n t of
tr ai n i n g p roc edu res and c u r r i c u l u m ma ter ial s
w h i c h p e r m i t i n d i v i d u a l i z e d instruction.

3.

To u n d e r s t a n d a n d use the "tasks of tea ching model."
This m o d e l includes the following:
a.

T o assess the "givens" p r e s e n t in the unique
i n s t r u c t i o n a l situation.
gathering,

This involves data-

d a t a - a n a l y s i s , communi cat ion ,

and

d e c i s i o n - m a k i n g skills.
b.

To set the goal by s p e c i f y i n g the i nte nde d
changes in s t u d e n t b ehaviour.
g o a l - ide nti fica tio n,

This involves

objective-specification,

c o m m u nic ati ons an d / o r n e g o t i a t i o n skills.
c.

To select, prepare,

and i m p l e m e n t strategies

for p r o d u c i n g int e n d e d changes.
d ecision-making,

This involves

p r e p a r a t i o n and i m p l e m e n t a t i o n

skills.
d.

To design,

prepare,

and i m p l eme nt e v a l u a t i o n

i n s t r u m e n t a n d procedures.
de cis ion -ma kin g,
ca tion skills

This involves

dat a-analysis,

(Henderson,

and communi­

1973).

La n s i n g S c h o o l and C o m m u n i t y
Involvement:
To a ssess a n d ar tic u l a t e w i t h the com mu n i t y those
e x p l i c i t an d / o r imp l i c i t needs that have b e e n i d e n t i f i e d
and to c o o p e r a t e in i m p l e m e n t i n g c o m p e t e n c y - b a s e d

92
e d u c at ion al p rograms w h i c h m e e t these needs and e ven tua lly
e lim ina te .them.
a.

to actively recruit interns,

t e a m leaders,

and

c o o p e rat ing teachers p r i m ari ly from, but not
limited to,

the L a n s i n g area,

for the Ei g h t h - C y c l e

program.
b.

to ed uca te p ar ents and community residents to the
concepts of c o m p e t e n c y - b a s e d education.

c.

to dev elop the m e c h a n i s m t hrough wh i c h p a r t i c i p a t i n g
p a rents and com munity residents can int eract wit h
principals,

c o o p e rat ing teachers,

team leaders,

and

interns to facilitate g e n u i n e c o m m u n i t y - b a s e d e d u ­
cation and involvement.
d.

to id entify and r ec r u i t parents

and com mun ity

residents for the L an s i n g T e a c h e r Corps.
e.

to o rg a n i z e sp ecific p a r e n t and commun ity residents
at ea ch p a r t i c i p a t i n g school to address and arti­
culate their un iqu e p r o b l e m s .

f.

to or gan i z e p a r e n t and co mmunity re sidents at
p a r t i c i p a t i n g schools w h o s e childre n are bu s e d to
n o n - n e i g h b o r h o o d schools.

g.

to devise a n d st ruc t u r e an e d u c a t i o n a l v eh i c l e
w i t h i n the T e a c h e r Cor ps

fra mework that addresses

itself to the communit ies a n d w i l l upon t er m i n a t i o n
of T e a c h e r Corps i n v o l v e m e n t con tinue to function
in a viable manner.

93
h.

to p r o m o t e aw are nes s of e x i s t i n g e d u c a t i o n a l
p r o g r a m s a m o n g i n s t r u c t i o n a l aides a nd c o o p e r a t e
w i t h the L a n s i n g S c h o o l D i s t r i c t in p r o v i d i n g c a r e e r
mobility opportunities

for l o w - i n c o m e c o m m u n i t y

persons.
i.

to e s t a b l i s h a v i a b l e w o r k i n g r e l a t i o n s h i p w i t h
va ri o u s

c o m m u n i t y gr oup s w h i c h w i l l

facilitate

c o m m u n i t y p a r t i c i p a t i o n at all levels of d e c i s i o n ­
m a k i n g o f the p r o g r a m — i.e.,
p r o g r a m d e v e l op men t,

needs a s s e s s m e n t ,

ev a l u a t i o n .

University Objectives:
1.

T o c o n t i n u e d e v e l o p m e n t of i n n o v a t i v e c o m p e t e n c y based teacher training programs which will result
in e f f e c t i v e t e a c h i n g an d l e a r n i n g a m o n g all
pupils.
a.

to s y s t e m a t i c a l l y t e s t a n d d e m o n s t r a t e c o m p o ­
n e n t s o f the T e a c h e r C o r p s t r a i n i n g p r o g r a m
w h i c h can b e c o m e the b a s i c units of an i n n o ­
vative competency-based teacher training
pr ogram.

b.

to p r o v i d e t i m e a n d s u p p o r t for u n i v e r s i t y
fa c u l t y to d e v e l o p t r a i n i n g m o d u l e s

for i n n o ­

vative competency-based teacher preparation
programs.

94
c.

to dev elop e m p i r i c a l r e s e a r c h a n d e v a l u a t i o n
- components w h i c h s u p p o r t i n n o v a t i v e co mpe ten cyb a s e d tea chi ng t r a i n i n g pr ograms.

d.

to pr o v i d e an i m p r o v e d c o m p e t e n c y —b a s e d tea che r
i n s t r uct ion al p r o g r a m w i t h e m p h a s i s on a p p l i ­
cation in the field.

e.

to es tab l i s h i n t e g r a t e d c o m p e t e n c y - b a s e d t eac her
tr aining s equ e n c e s w h i c h

i n v o l v e o t h e r colleges

in the University.
f.

to de v e l o p a c o m p e t e n c y - b a s e d b i - l i n gua l,

bi-

cu ltural course w h i c h w i l l b e c o m e a p a r t o f the
re qu i r e d e x p e r i e n c e s in the C o l l e g e of Education.
g.

to e s t a b l i s h a p r o c e s s by w h i c h te a c h e r tra ining
may be c o n s t a n t l y m o n i t o r e d t h r o u g h c o m m u n i t y
input so th at p r o g r a m s r e f l e c t the c h a n g i n g
needs of the c ommunity.

h.

to pro vid e

field s e t t i n g s w h e r e p r e - s e r v i c e

teachers can e x p e r i e n c e a v a r i e t y of i n ­
structional,

o r g a n i z a t i o n a l patter ns,

team teaching,

e.g.,

d i f f e r e n t i a t e d staffing,

indi­

v i d u a l i z e d in str uct ion .
i.

to con t i n u e the d e v e l o p m e n t of c o m p e t e n c y b a s e d t e a c h e r e d u c a t i o n t o w a r d te a c h e r c e r t i f i ­
cation b a s e d on f i e l d d e m o n s t r a t e d com petencies.

j.

to d ev e l o p a p r o c e s s w h i c h ins u r e s that te ach er
tr aining p r o g r a m s

i n v o l v e s c h o o l p e r s o n n e l in

de s i g n i n g o b j e c t i v e s a n d t r a i n i n g st rategies.

95
Teacher Education Program
Philosophy
T h e L a n s i n g T e a c h e r C o r p s p r o j e c t in c o n j u n c t i o n
w i t h the C o l l e g e of E d u c a t i o n a t M i c h i g a n St a t e U n i v e r s i t y
has s p e c i f i c a l l y

f o c u s e d its a t t e n t i o n on the p r e p a r a t i o n

of tea che rs w h o w i l l h a v e the c o m p e t e n c i e s

and s e n s i t i v i t y

to m e e t the needs o f l o w - i n c o m e a rea ch ildren,

including

the needs of c h i l d r e n w h o are c o n s i d e r e d c u l t u r a l l y d i f f e r ­
ent .
It is n o l o n g e r a c c e p t a b l e to p e r m i t p r o s p e c t i v e
t e ach ers

to a c c u m u l a t e c r e d i t o n the b a s i s of a t t e n dan ce.

It is i m p e r a t i v e

t hat s t u d e n t p e r f o r m a n c e be t i e d in

d i r e c t l y to the c o m p e t e n c i e s
instruct or;

therefore,

a n d p e r f o r m a n c e s o f the

a p r i m e o b j e c t i v e of this p r o g r a m

w i l l be to s t r e n g t h e n the type o f t e a c h e r t r a i n i n g p r o g r a m s
c u r r e n t l y b e i n g o f f e r e d a t the uni ver sit y.

R e a l i z i n g that

s u c h a p r o g r a m c a n n o t be a d m i n i s t e r e d in iso lat i o n ,

the

L a n s i n g S c h o o l Dis tri ct,

Model

Ci t i e s ,

Michigan State University,

L a n s i n g C o m m u n i t y Co lle ge,

a n d the S t a t e of

M i c h i g a n D e p a r t m e n t of E d u c a t i o n w i l l p u l l t o g e t h e r to
develop guidelines

that w i l l p r o v i d e for m o r e f l e x i b l e

teacher training p r o g r a m s .

T h e p r i m a r y c o n c e r n s at the

c o n s o r t i u m w i l l be p r o g r a m f o r m u l a t i o n ,

implementation,

and

e v a l u a t i o n o f t e a c h e r t r a i n i n g as it s p e c i f i c a l l y r e l a t e s
to the e d u c a t i o n of c h i l d r e n in l o w - i n c o m e areas.
A g r e a l d e a l w as

learned

fr om the s i x t h c ycl e

p r o j e c t an d a c o n t i n u e d e f f o r t to d e v e l o p a nd i n t e g r a t e int o

96
the sy s t e m those aspects of the p r o g r a m that have p r o v i d e d
for individu al d e v e l o pmen t in the areas of self-discipline,
critical thinking, e f f e c tiv e communication,
will be made.
room,

and c rea tiv ity

Past experiences w i t h the home,

the university,

the c l a s s ­

and the com mu n i t y indicate that a

viable tea cher tra ining p r o g r a m m u s t address itself to each
of these factors.
O u r society,
changes;

therefore,

as a whole,

is u n d e r g o i n g d r a m a t i c

it is our r e s p o n s i b i l i t y as educators

to e xplore n e w ways of re spo ndi ng to these changes as they
af fect o u r c h i l d r e n ’s lives as we ll as o u r own.
specifically,

M ore

e d u c a t i o n a l i nstitutions m u s t take a l ead er­

ship role in gi v i n g d i r e c t i o n to these social changes.
de aling w i t h these changes,

flexibility,

In

not stability,

wi ll be the m o s t im por t a n t c ata l y s t in o u r tea c h e r training
program,
-B ased Edu cat ion
In the com p e t e n c y - b a s e d mo d e l of education,
is de f i n e d in terms of pl an n e d b e h a v i o r change.

learning

If a

te acher is to p o s t u l a t e that le arning has taken place,

it

must be o b j e c t i v e l y d e m o n s t r a t e d that b e h a v i o r a l change,
the des i r e d direction,
subjectivity,

has b e e n manifested.

expectations,

in

The role of

and v a l u e - b a s e d i nte rpr eta tio ns

is m i n i m i z e d in d e t e r m i n i n g e x i s t i n g learning, p l a n n i n g and
i m p l e m ent in g pro grams d esi g n e d to teach n e w l ear nin g and
e v a l u a t i n g ac h i e v e m e n t of l ear n i n g objectives.

97
Further,

the t eac her is p e r c e i v e d as the m a n a g e r of

the learning environment,

r e s p o nsi ble and a c c o unt abl e for

its co ndition and events ar isi ng therein.

It is the teacher

wh o m u s t structure the m i l i e u such that a pr e d i c t a b l e
r e l a t i o n s h i p bet w e e n the student's c l a s s r o o m p e r f o r m a n c e
and the c l a s s r o o m e n v i r o n m e n t is established.

The t ra­

d i t i o n a l no tio n p l a g u i n g u rba n schools, w h i c h b a s e s
te aching me tho ds and learning p rog r a m s on the as sum pti on
that students,

given an "average enviro nme nt" w i l l learn

w h e n they are

"ready"

is c o u n t e r p r o d u c t i v e to e d u c a t i o n a l

achievement.

To o o f t e n in the urban setting,

made to u n d e f i n e d m e c h a n i s m s cal l e d

re fer e n c e is

"intelligence" or

"genetic endowment" if st udents fail to e x h i b i t w h a t is
n e b u l o u s l y d e f i n e d as learning, w h e n in fact no adequate,
e x p l i c i t p r o v i s i o n has b e e n m a d e for learn ing to take
place.
In the c o m p e t e n c y - b a s e d model,

faulty

learning is

p e r c e i v e d to be a p r o d u c t of the c l a s s r o o m e n v i r o n m e n t
ra the r than a p r o d u c t of p o s t u l a t e d i n c o m p e t e n c i e s and
i nca pac iti es of a faulty student.
pa rad oxi cal ,

This c o n t ra sts w i t h the

latent a s s u m p t i o n im pli cit and p r e v a i l i n g in

e d u c a t i o n that the schools c an n o t b a s i c a l l y teach due to
the fixed ef fects of ge n e t i c or early family experiences.
This r e s t r i c t i v e a n d u n f o u n d e d no t i o n is the c o n v e r s e of
a b a s i c tenet in c o m p e t e n c y - b a s e d e d u c a t i o n h o l d i n g that
b e h a v i o u r n o t only can be, b u t is m o d i f i e d — for b e t t e r or
w o r s e — in the school room,

a n d that such b e h a v i o u r change

98
(learning)

may be positively accomplished through exposure

to e f f i c i e n t and e f f e c t i v e

learning environments.

severe physiological impairment,

Barring

no s t u d e n t c a n l e g i t i m a t e l y

be d e e m e d to be l i m i t e d in w h a t a n d h o w m u c h he can

learn.

T h e c o m p e t e n c y - b a s e d m o d e l then does n o t a s s u m e t h a t the
e d u c a t i o n a l a c h i e v e m e n t of s t u d e n t s

is d e t e r m i n e d b y some

r e l a t i v e l y c o n s t a n t l e v e l of ab i l i t i e s ,

aptitudes,

characteristics

a n d l a b e l e d by

identified,

classified,

t e s t s or p e r c e p t i o n s of s u b j e c t i v e
R a t h e r it p e r c e i v e s

and

j u d g m e n t b y the evalu a t o r .

t h a t the m a j o r i t y of o v e r t b e h a v i o u r is

e n v i r o n m e n t a l l y d e t e r m i n e d and s u b j e c t to change.

T o say

t h a t b e h a v i o u r is d e t e r m i n e d and f i x e d by forces b e y o n d the
school's

i n f l u e n c e is c o n t r a r y

to the p r i n c i p l e t h e s i s

d e l i n e a t i n g the c o m p e t e n c y - b a s e d m o d e l of e d u c a t i o n .
C o m p e t e n c y - B a s e d M a n a g e m e n t of
the L e a r n i n g E n v i r o n m e n t
The competency-based trained teacher used behaviour
management techniques

to s t r u c t u r e an a p p r o p r i a t e

e n v i r o n m e n t and c o n s t r u c t r e l e v a n t

learning

learning strategies

d e s i g n e d to p r o m o t e p l a n n e d b e h a v i o u r change.

U s u a l l y the

c h a n g e p r o g r a m is i n t e n d e d to p r o m o t e the a c q u i s i t i o n a n d
m a i n t e n a n c e of b e h a v i o u r s
achievement.
however,

compatible with educational

Learning activities may

to p r e v e n t ,

incompatible with

decelerate,

also be d e s i g n e d

or eliminate behaviours

l e a r n i n g o b j e c t i v e s of the c l a s sr o o m .

In e i t h e r insta n c e ,

the object of attention and planned

99
m a n i p u l a t i o n is a c l e a r l y d e f i n e d c a t e g o r y of o b s e r v a b l e
a n d m e a s u r a b l e behavi o u r .
The assumptions underlying behavioural programs
f r o m w h i c h c o m p e t e n c y - b a s e d m o d e l issues are b a s e d u p o n
e m p i r i c a l l y v a l i d a t e d tenets o f s o c i a l
d e s c r i b e d b y C l a r k et al.,
1.
2.
3.
4.
5.

(1972)

learning.

As

these tenets h o l d thati

I n d i v i d u a l b e h a v i o r o c c u r s in the c o n t e x t of a
s o c i a l e n v i r o n m e n t a n d in i n t e r a c t i o n w i t h the
environment
S o c i a l b e h a v i o r is l e a r n e d in i n t e r a c t i o n w i t h the
environment
B e h a v i o r is t a u g h t a n d m a i n t a i n e d by the s o c i a l
environment
S o c i a l l e a r n i n g is a p r o c e s s o f r e c i p r o c a l i n f l u ­
ence.
P a r t i c i p a n t s i n t e r a c t i n g in a s o c i a l s y s t e m
mutually affect each others behaviors
The r e c i p r o c a l i n f l u e n c i n g p r o c e s s m a y b e e x p l i c i t
o r implicit, p l a n n e d o r u n p l anned, b u t m u s t b e
c o n s i d e r e d a f a c t o r i n s o c i a l systems.

In a c c o r d a n c e w i t h these e m p i r i c a l l y d e r i v e d p r i n c i p l e s of
l e a r n i n g theory,

the c o m p e t e n c y - b a s e d m o d e l p e r c e i v e s

i n d i v i d u a l s t u d e n t b e h a v i o u r as being;
1.
2.
3.
4.
5.

e x h i b i t w i t h i n the c o n t e x t o f the s o c i a l e n v i r o n m e n t
a f f o r d e d b y the c l a s s r o o m
m a l l e a b l e a n d a m e n a b l e to c h a n g e in the c o n t e x t of
i n t e r a c t i o n w i t h the c l a s s r o o m e n v i r o n m e n t
taught, m a i n t a i n e d , r e d u c e d o r e l i m i n a t e d as a
f u n c t i o n of i n t e r a c t i o n w i t h t h e c l a s s r o o m e n v i r o n ­
ment
r e c i p r o c a l l y i n f l u e n c e d in f o r m and f r e q u e n c y b y
t h o s e w i t h w h o m the s t u d e n t i n t e r a c t s in th e
classroom--the teacher and student
c o n t i n u a l l y s u b j e c t to c o n s c i o u s or u n c o n s c i o u s
i n f l u e n c e b y i n t e r a c t a n t s in t h e classroom.

T he c o m p e t e n c y - b a s e d m o d e l

is c e n t r a l l y c o n c e r n e d w i t h

effectuating explicitly defined and carefully planned
a f f e c t s o n the b e h a v i o u r o f s t u d e n t s
environment.

in the l e a r n i n g

T o the e x t e n t that s t u d e n t b e h a v i o u r s are

100
unsystematically,
the classroom,
cation.

randomly,

a n d i n e x p l i c i t l y m o d i f i e d in

it is n o t a c o m p e t e n c y —b a s e d p r o g r a m o f e d u ­

L e a r n i n g rather,

is left to c h a n c e a n d accident.

F a c u l t y and S t a f f O r i e n t a t i o n
and T r a i n i n g
O n e of the m a i n r e s p o n s i b i l i t i e s in a t e a c h e r
p r e p a r a t i o n p r o g r a m s h o u l d be t h e o r i e n t a t i o n and t r a i n i n g
of f a c u l t y a n d s t a f f members.
c y c l e o f T e a c h e r Corps
has b e e n involved,

Si n c e this was

the s e c o n d

in w h i c h M i c h i g a n S t a t e U n i v e r s i t y

a n u m b e r of r e t u r n i n g u n i v e r s i t y s t a f f

m e m b e r s w e r e on h a n d to form the nuc l e u s o f the e i g h t h
c y c l e project.
In o r d e r to f a c i l i t a t e u n d e r s t a n d i n g o f the focus
of the p r o j e c t and to c o n t r i b u t e t o w a r d its growth,

the n e w

p e r s o n n e l w a s e x p o s e d to the p h i l o s o p h y of the pro g r a m .
P a r t i c i p a n t s w e r e i n v o l v e d in b o t h the p r e - s e r v i c e a n d ins e r v i c e phases.

As a wh o l e ,

the C o l l e g e o f E d u c a t i o n w a s

v e r y i n v o l v e d w i t h the s i x t h cycle program.
the a reas of reading,

children's

Faculty from

literature, m a t h e m a t i c s ,

a n d s o c i a l s t u d i e s and i n t e r p e r s o n a l d e v e l o p m e n t w e r e
directly responsible
interns.

for c l a s s e s

During pre-service,

groups were

for the T e a c h e r C o r p s

S u m m e r 1973,

f o r m e d so t h a t s t a f f m e m b e r s

interaction

took a n a c t i v e

p a r t in the d e v e l o p m e n t a l p h a s e of the project.
Involvement with
m e m b e r s,

the p r o j e c t for f a c u l t y a n d staff

as w e l l as a d m i n i s t r a t o r s ,

took p l a c e d u r i n g b o t h

t he p r e - s e r v i c e a n d i n - s e r v i c e p h a s e s o f the p r o g r a m .

The

101
p e r i o d o f g r e a t e s t i n t e n s i t y t o w a r d o r i e n t a t i o n and t r a i n i n g
was

the p r e - s e r v i c e period;

however,

tion w a s an o n - g o i n g p r o c e s s

t r a i n i n g and o r i e n t a ­

t h r o u g h o u t the program.

Some of the o b j e c t i v e s of the o r i e n t a t i o n t r a i n i n g
p h a s e w ere:
1.

T o o r i e n t those p e o p l e d i r e c t l y or i n d i r e c t l y
i n v o l v e d w i t h t h e p r o j e c t w i t h the T e a c h e r C o r p s
philosophy.

2.

T o e x p l o r e a v e n u e s of i n t e g r a t i n g T e a c h e r C o r p s
o b j e c t i v e s i n t o the r e g u l a r U n i v e r s i t y t e a c h e r
t r a i n i n g program.

3.

T o seek w a y s to m a k e T e a c h e r Corps' o b j e c t i v e s
a p p l i c a b l e to local e d u c a t i o n a l c o n d i t i o n s .

4.

T o d e v e l o p n e w c o u r s e s t h a t w i l l be j o i n t l y o f f e r e d
in T e a c h e r C o r p s as w e l l as in the r e g u l a r t e a c h e r
t r a i n i n g p r o g r a m o f the U n i v e r s i t y ,

5.

T o i n v o l v e as c l o s e l y as p o s s i b l e U n i v e r s i t y s t a f f
in the d e v e l o p m e n t of the T e a c h e r C o r p s c u r r i c u l u m .
T h e o b j e c t i v e s o f the faculty and s t a f f o r i e n t a t i o n

was

to share a n d i n s t a l l s o u n d e d u c a t i o n a l o b j e c t i v e s

the t e a c h e r t r a i n i n g u n i t of the U n i v e r s i t y .
served a dual purpose
Corps

Orientation

in t h a t it a l s o e n a b l e d T e a c h e r

to i n c o r p o r a t e g o o d p r o g r a m s

a l r e a d y at M i c h i g a n State

U n i v e r s i t y into its o v e r a l l o b j e c t i v e .
ar e a s of c o m p e t e n c y - b a s e d educat i o n ,
education,

into

mathematics education,

education were emphasized.

Workshops

in the

bilingual/bicultural

and c o m m u n i t y - b a s e d

T h o s e i n c l u d e d are:

1.

Community-based Education.

2.

I n t r o d u c t i o n to C o m p e t e n c y - b a s e d E d u c a t i o n
Differentiated Staffing and Bilingual-Bicultural
Education.

3.

F o u r day I n t e r n Retreat.

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O r i e n t a t i o n p a r t i cipants i n c l u d e d Interns,
Teachers, T e a m Leaders,

Cooperating

and s e l e c t e d C o m m u n i t y Leaders.

Analysis of the M a t h e m a t i c s C o n t e n t in the
Mathematics Education Component
of the P r o g r a m
M i c h i g a n State U n i v e r s i t y u n d e r g r a d u a t e e l e m e n t a r y
e d u c a t i o n majors are requ i r e d to c o m p l e t e a s e q u e n c e of two
courses

in the m a t h e m a t i c s e d u c a t i o n c o m p o n e n t of their

training.
matics,

The first,

off e r e d b y the D e p a r t m e n t of M a t h e ­

is a four-quarter h o u r c o n t e n t co u r s e e n t i t l e d

F o u n d a t i o n s of A r i t h m e t i c
course,

(Mathematics 201).

Dur i n g this

p r o s p e c t i v e e l e m e n t a r y teac h e r s s p e n d three hours

a w e e k in lecture rooms and two hours in a m a t h e m a t i c s
laboratory.

The second,

m e n t a r y Education,

o f f e r e d by the D e p a r t m e n t of E l e ­

is a t h r e e - q u a r t e r h o u r m e t h o d s c ourse

en t itles T e a c h i n g of M a t h e m a t i c s in E l e m e n t a r y Grade s
(Education 3 2 5 E ) .

Thus students in r e g u l a r e l e m e n t a r y

e d u c a t i o n p r o g r a m earn a total o f s e v e n - q u a r t e r c r e d i t
hours.
The m a t h e m a t i c s e d u c a t i o n c o m p o n e n t of the T e a c h e r
Corps p r o g r a m b e i n g e v a l u a t e d w a s an i n t e g r a t e d content,
me t h o d s and p r a c t i c e experience,

s e v e n - q u a r t e r h o u r credits

be i n g e a r n e d for the con t e n t and m e t h o d s p a r t s

(the credits

for p r a c t i c e experi e n c e asp e c t b e i n g e a r n e d u n d e r i n t e r ­
ship) .
of 1973.

The interns s p e n t six hours p e r w e e k in F a l l t e r m
However,

an a d m i n i s t r a t i v e p r o b l e m a r o s e in

W i n t e r and Sp r i n g terms w h i c h r e d u c e d the c l a s s - m e e t i n g to

103
f o u r - h o u r s p e r week.

T h u s t h e i n t e r n s r e c e i v e d se v e n -

quarter hour credits

for f o u r t e e n h o u r c l a s s - m e e t i n g .

will be explained later

This

(under c r i t i c a l a p p r a i s a l of

instructional m e t h o d ) .
Data Gathering M e t h o d
Educational reform can b e

c o n s i d e r e d to i n c l u d e two

c o m p o n e n t s — e i t h e r or b o t h o f w h i c h m a y b e p r e s e n t in n e w
teaching p r a c t i c e s :
teaching m e t h o d s .

(1)

n e w c u r r i c u l a r content;

In e v a l u a t i n g

(2) n e w

the e f f e c t o f a c h a n g e in

e i t h e r of t h e s e c o m p o n e n t s m a n y p e o p l e b e l i e v e

that t h e task

is g e n e r a l l y e a s i e r if i t is c o n f i n e d to e v a l u a t i o n of
alone.

This

is so b e c a u s e

compared wi t h

curriculum

the t e a c h i n g m e t h o d m i g h t b e

an a l t e r n a t i v e m e t h o d in t e a c h i n g the s a m e

curricular contents
both m e t h o d s .

and h e n c e u s e c r i t e r i a t h a t are f a i r to

However,

(content)

w h e r e w e are c o m p a r i n g a n e w

w i t h a n o t h e r we m a y w e l l be t r y i n g to

c o m p a r e two t e a c h i n g - c o n d i t i o n s
criteria of success

to e a c h o f w h i c h d i f f e r e n t

are a p p r o p r i a t e .

The

teaching-aims

th a t a c c o m p a n y t h e u s e o f one c u r r i c u l u m are
differ greatly
other

(2)

liable

to

f r o m t h o s e t h a t a c c o m p a n y the u s e of the

(NCTM C o m m i t t e e o n A n a l y s i s of E x p e r i m e n t a l M a t h e ­

matics Programs,

1963, W i l l i a m s

(1967),

and B r o w n e l l

(1966)) .
Hicks

and Perrodin

(1967)

s e l e c t i o n o f topics a p p r o p r i a t e

p r o v i d e d a b a s e f o r the

for the p r e - s e r v i c e e d u ­

c a t i o n in m a t h e m a t i c s o f e l e m e n t a r y s c h o o l t e a c h e r s .

Four

104
types of sources w e r e i n t e nsively revi e w e d by them to
p r o v i d e the n e c e s s a r y data.
1.

T h e y were:

Re v i e w of forty-six s e l e c t e d research studies
p o i n t i n g out the m a t h e m a t i c a l competencies or
w e a k n e s s e s of el e m e n t a r y school teachers.

2.

Re v i e w of t h i r t y - t w o sets of recomm e n d a t i o n s of
m a t h e m a t i c s educa t o r s and n a t i o n a l l y - r e c o g n i z e d
advis o r y groups o r organizations.

3.

P a g e - b y - p a g e a nalysis of sixteen recent textbooks
d e s i g n e d for college courses in m a t h e m a t i c s for
e l e m e n t a r y school teachers.

4.

A n a l y s i s of e l e v e n a r i t h m e t i c series or teacher's
guides for grades K-7 p u b l i s h e d since 1962.
A compo s i t e list of m a t h e m a t i c a l topics

from the

ab o v e sources was then c o m p i l e d by Hicks and P e r r o d i n
(1967)

a n d a s y s t e m of r a t i n g these topics was devised.

T o p i c s w h i c h appea r e d at least once in the compo s i t e list
w e r e c a t e g o r i z e d as Level I.

To be c a t e g o r i z e d as Le v e l II,

topics had to m e e t one of the following conditions:
—

a p p e a r in at least three

of the r e s e a r c h studies;

—

a p p e a r in at least five o f the r e c o m m e n d a t i o n s of
the m a t h e m a t i c s educa t o r s o r advi s o r y groups;

—

a p p e a r in at least e i g h t
textbooks in m a t h e m a t i c s

of the sixt e e n college
for e l e m e n t a r y school

teachers;
—

a p p e a r in at least six of the eleven arithmet i c
s eries or teach e r ' s guides

for grades K-7,

105
Finally,

to be c l a s s i f i e d as L e v e l III,

a t o p i c h a d to m e e t

at l e a s t two of the f o u r c r i t e r i a l i s t e d ab o v e

for L e v e l

II

topics.
A total of 9 8 topics w e r e l o c a t e d in the
(19 in s o u r c e one,
a n d 79 in s o u r c e

54 in s o u r c e two,

four).

4 sources

84 in s o u r c e three,

Of t h e s e topics,

fifty-one were

c a t e g o r i z e d as L e v e l II and t h i r t y - f i v e w e r e c a t e g o r i z e d
as L e v e l

III.
Table

the sources
this t a b l e

1 shows the topics

in level t h r e e a l o n g w i t h

in w h i c h they appeared.

It is o b v i o u s f r o m

that the last t h r e e s o u r c e s

are in c l o s e a g r e e ­

m e n t o n w h a t s h o u l d be i n c l u d e d in s o m e m a n n e r in the m a t h e ­
m a t i c s c u r r i c u l u m of the e l e m e n t a r y s c h o o l teacher.
relatively

low p e r c e n t a g e in the f i r s t s o u r c e does

i n d i c a t e d i s a g r e e m e n t w i t h the o t h e r sources;
indicates

not

it o n l y

the lack of e x p e r i m e n t a l r e s e a r c h d o n e on the

s e l e c t i o n of m a t h e m a t i c a l topics
elementary school teachers.

for the p r e p a r a t i o n of

T o t e s t the v a l i d i t y of this

w e r e v i e w e d p u b l i c a t i o n s of s i m i l a r sour c e s
1 9 6 8 - 1 9 73.

The

for the y e a r s

T h e t o p i c s s u g g e s t e d in these s o u r c e s

are very

c o n s i s t e n t w i t h the list d e s c r i b e d a b o v e e x c e p t in the
f i e l d o f G e o m e t r y and in the f i e l d o f Logic.
A n a l y s i s o f the c o n t e n t of five t e x t b o o k s
mentary mathematics

for t e a c h e r s

for e l e ­

revealed that coordinate

g e o m e t r y a n d m a t h e m a t i c a l l o g i c w e r e n o t i n c l u d e d in the
list developed by Hicks and Per r o d i n
m e t r i c Teac h e r ,

annually publishes

(1967).

The Arith-

a s u m m a r y of r e s e a r c h

106
Table 1.— Suggested Topics for the Mathematical Preparation of
Elementary school Teachers.
Topic
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.

Set Terminology
Set Operations
Relations & Functions
Whole Number Operations
Counting and One-to-One
Correspondence
Order and Cardinality
Field Operations
Different Numberation
Systems & Place Value
Ancient Numeration
Systems
Roman Numeration
Primes and Composite
Factors and Multiples
Exponents & Exponential
Notations
Divisibility Rules
The Number Line
Common Fractions
Decimal Fractions
Percentages
Ratio & Proportions
Real Numbers
Square Root
Measurement
Precision and Error
Formulae & Substitution
Basic Concepts of Geometry
Geometric Figures
Metric System & Conversion
Equations and Symbols
Inequations
Central Tendency
Statistical Graphs
Probability
Problem Solving
Making Estimations
Rationalizing Algorithm

Source 1

Source 2

Source 3

Source 4

X

X

X

X

X

X

X

X

X

X

x
X

X

X

X

X

X

X
X

X

X

X

X

X

X

X

x

X

X

X

X

x

X

X

X

X

x
x
x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x
X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X
X

X
X

X

X

X

X

X

Total

13

28

31

32

% of Total No. of Topics

37

80

89

91

107
and articles on m a t h e matics e d u c ation conducted in the
U n i t e d States during the prece d i n g year.

Review of these

summaries for the years

and 1971 again

1968,

1969,

1970,

p o i n t e d out that m o s t research done on t h e content was in
topics noted in the Level III list as def i n e d by Hicks and
Perrodin.
(1969)

However,

two pieces of research, one by Shah

on the applicability of teaching geom e t r y to e l e ­

m e n t a r y school children,

the ot h e r by O'B r i e n and S h apiro

(1968)

c o n f i r m e d children's ability to learn m a t h e m a t i c a l

logic.

Rese a r c h conducted by Suppes

(1969)

at Stanf o r d

Unive r sity in teaching logic to elemen t a r y school children
has not as y e t p r o v i d e d conclusive evidence to the children's
ability to learn and c o m p rehend m a t h e m a t i c a l
on this review of recent literature,

logic.

Based

the i n v e s t i g a t o r c o n ­

c luded that only the topic "Coordinate Geometry" m e t the
q u a l i f i c a t i o n s of the Level III p r e s c r i b e d by Hicks and
Perrodin,

and therefore decided to include it as the

t h i r t y -sixth topic in the criteria list.

It sh o u l d be

m e n t i o n e d that the list contains thirty-six topics w h i c h is
much less than the latest requirements in the "Guidelines
for the P r e p a r a t i o n of Teachers of M a t h ematics" p u b l i s h e d
by N C T M C o m m i s s i o n on P r e s ervice E d u c a t i o n of Teachers
(1973) .
The criteria list is then u s e d to compare the
m a t h e m atics contents
o f the T e a c h e r Corps,

in the m a t h e m a t i c s e d u c a t i o n programs
the regular e l e m e n t a r y e d u c a t i o n and

a n o t h e r e x p e r i m e n t a l class of elementary e d u c a t i o n students

108
jointly taught by the T e a c h e r Corps' m a t h e matics ins t r u c t o r
and a n other m a t h e matics educator.
Bl o o m et al.

(1971)

noted that most fundamen t a l to

the use of formative evalua t i o n is the selec t i o n of a unit
of learning.

Each course or e d u c a tional p r o g r a m can be

c o n s i d ered to have separable parts or divisions
purposes.

for a n a l y t i c

It is still poss i b l e to consider the parts

separately,

though these parts may be i n t e r related in v a r i ­

ous w a y s so that the learning
one p a r t has consequences

(or level of learning)

of

for the learning of others.

The m a t h e matics e d u c a t i o n component of the T e a c h e r
Corps p r o g r a m was origin a l l y pla n n e d to cover seven units.
They are Measurement, Numeration,

A d d i t i o n and Su b t r a c t i o n

of W h o l e Numbers, M u l t i p l i c a t i o n and Division of Who l e
Numbers,

Fractions,

Geome t r y and P r o b a bility and Statistics.

T h r e e of these Measurement,

Fractions,

P r o b ability and

St a t i s t i c s - - w e r e to be w r i t t e n by the instructor whi l e the
r e m a ining units are M a t h e m a t i c s Met h o d s Program,

developed

by the M a t h e m a t i c s E d u c a t i o n D e v e l o p m e n t C e n t e r of Indiana
University.

The first five of these seven units w e r e w e l l

co v e r e d and m a s t e r y tests w e r e taken on them.

There was

no time to look at the sixth and seventh units due to
(administrative)
instructor.
interns.

circumstances bey o n d the control of the

However,

the sixth unit was g i v e n to the

There is no u n a n imous agree m e n t among the m a t h e ­

m atics educators about stat i n g i n s t r u c t i o n a l objecti v es
s p e c i f i c a l l y in b e h a v i o r a l terms

(A l l e n d o e r f e r , 1971,

109
Forbes,

1971).

The i n s t r u c t i o n a l objectives,

s p e c i f ically st a t e d by the instructor,
to those of the
(Shakrani,

are very similar

"Trainers of T e a c h e r Trainers"

1973).

though not

(TTT) p r o j e c t

F i n d i n g s w i l l be d i s c u s s e d in ch apter IV.

D e s c r i p t i o n of the M a t h e m a t i c s Methods
In t e g r a t e d w i t h M a t h e m a t i c s C o n t e n t
and C l i n i c a l E x p e r i e n c e
F i v e units w e r e s t u d i e d in the m a t h e m a t i c s e d u ­
ca t i o n c o m p o n e n t of the p r o g r a m .
M e asurement,

These w e r e units o f

Numeration, A d d i t i o n and S u b t r a c t i o n of Whole

Numbers, M u l t i p l i c a t i o n and D i v i s i o n of W h o l e Numbers,
Fractions.

and

T h e first and the last units. M e a s u r e m e n t and

Fractions, w e r e p r e p a r e d by the i n s t r u c t o r w h i l e the
r e m a i n i n g three w e r e p a r t of M a t h e m a t i c s Met h o d s Program,

a

p r o j e c t of the M a t h e m a t i c s E d u c a t i o n D e v e l o p m e n t Ce n t e r
s p o n s o r e d jointly by the M a t h e m a t i c s D e p a r t m e n t and the
S c hool of E d u c a t i o n of Ind i a n a U n i v e r s i t y and funded
t h r o u g h the U P S T E P p r o g r a m of the N a t i o n a l Science
F oundation.

A l l the five units are c o n t e n t - m e t h o d s

inte­

grated.
Measurement
The m e t h o d p a r t o f this u n i t d i s c u s s e d m a j o r topics
as m e a s u r e m e n t as a comparison,
m e a s u r i n g units,
process,

the a r b i t r a r y na t u r e of

the a p p r o x i m a t e nature of m e a s u r i n g

p r e c i s i o n and accuracy,

and skills in measuring,

d e v e l o p i n g concepts of

d e v e l o p i n g concepts of new units.

A l l these ideas sta r t e d w i t h li n e a r m e a s u r e m e n t and e x t e n d e d

110
to area, volume, wei g h t ,

c a p a c i t y and time.

The use of

m a n i p u l a t i v e s and e x p e r i m e n t a t i o n w a s e m p h a s i z e d at the
b e g i n n i n g and d e v e l o p e d to the d e r i v a t i o n and use of
formula.
T h e i m p l i c a t i o n s of P i a g e t i a n rese a r c h on m e a s u r e ­
m e n t w as e x t e n s i v e l y discussed.

The interns came to realize

that the n e c e s s a r y concepts w i l l d e v e l o p
is o l d enough
P i a g e t ) , and

(1) w h e n the c h i l d

(eight to e i g h t and one-half,

a c c o r d i n g to

(2) w h e n he is a l l o w e d to ope r a t e on

m e n t with, manipulation)

(experi­

objects u s e d in m e a s u r e m e n t and

that b o t h c o n d i t i o n s are n e c e s s a r y for the o p e r a t i o n a l
t h o u g h t n e c e s s a r y to p e r f o r m meas u r e m e n t .

Further impli­

c a t i o n s of P i a g e t i a n r e s e a r c h w e r e d e m o n s t r a t e d a n d d i s ­
cussed:
child

that b e f o r e a t t e m p t i n g s y s t e m a t i c m e a s u r e m e n t the

(a) m u s t be able to cons e r v e the idea of len g t h of an

object,

(b) m u s t u n d e r s t a n d the conc e p t of s u b d i v i s i o n s

s ince the o b j e c t to be m e a s u r e d m u s t be s u b d i v i d e d into
s u b - u n i t s of t h e same
ruler,

and

length as a m e a s u r i n g i n s t r u m e n t or

(c) m u s t real i z e that a d i s t a n c e b e t w e e n two

o b j e c t s is c o n s e r v e d w h e n o t h e r objects are p l a c e d b e t w e e n
them.

Interns finally b e c a m e aware of the fact t h a t w h i l e

c h i l d r e n can u n d e r s t a n d the c o n c e p t of area u s i n g i n t u i t i v e
methods

(of super position)

(by building)

and conserve interior volume

a r o u n d the age o f eight,

the m e t h o d of

d e t e r m i n i n g a r e a a n d v o l u m e by formulas s h o u l d n o t b e
e x p e c t e d to d e v e l o p u n t i l e l e v e n to twelve ye a r s of age
(Copeland,

1973).

Ill
Numeration '
The m a i n ideas p r e s e n t e d in the unit w e r e those o f
sets,

number,

and numeral,

grouping,

place value,

and the

use of these ideas.

Informal learning w a s e m p h a s i z e d for

k i n d e r g a r t e n level.

T he d e v e lopment again p r e c e d e d from

the use of p h y s i c a l objects to pictures,
tions,

to m e r e r e p r e s e n t a ­

a n d finally to the use of symbols,

the enactive,

thus embracing

iconic a nd symb o l i c levels of r e p r e s e n t a ­

tional thinking i d e n t i f i e d by B r u n e r

(19 66).

A c t i v i t i e s that led to r e c o g nition of impor t a n t
c h a r a c t eristics of a g o o d n u m e r a t i o n systems w e r e provided.
More a c t i v i t i e s that led to d i s t i n c t i o n and rela t i o n s h i p
b e t w e e n group i n g and p l a c e - v a l u e w e r e provided,

and i m p o r ­

tance of the latter in o p e r a t i o n w i t h numbers w a s e m p h a ­
sized.

Exerc i s e s w e r e p r o v i d e d in s e q u e n c i n g of n u m e r a t i o n

activi t ies in e l e m e n t a r y schools.
lesson plans on counting,
numbers,

r o u n d i n g numbers,

to decimal,

D e v e l o p m e n t of N u m e r a t i o n

numeral reading,

o r d e r i n g of

exten s i o n of n u m e r a t i o n s ystem

and e x p o n e n t i a l n o t a t i o n w a s encouraged.

The

diagnosis of co m m o n errors that chil d r e n o f t e n m a k e in
e l e m e n t ary schools t o g e t h e r w i t h their r e m e d i a t i o n p r o c e s s
w e r e discussed.
F i n a l l y the p s y c h o l o g i c a l j u s t i f i c a t i o n of each of
the a c t ivi t i e s at d i f f e r e n t stages w a s established.

1X2
A d d i t i o n a n d S u b t r a c t i o n of
Whole Numbers
In this u n i t the p e d o g o g i c a l
mathematics
was

and real w o r l d was

that mathematical

relationship between

stressed.

T h e c e n t r a l theme

l e a r n i n g of y o u n g c h i l d r e n s h o u l d

flow f r o m real w o r l d of e x p e r i e n c e s

to symbols.

th a t c o u l d d e v e l o p n u m b e r r e a d i n e s s

in c h i l d r e n w e r e

provided,

viz.

activities

d e v e l o p m e n t of n u m b e r s ,
approaches
line),

that p r e s e n t p r e — n u m b e r c o n c e p t s ,

a n d p r e - a d d i t i o n concepts.

to t e a c h i n g o f a d d i t i o n — sets, m e a s u r e

and function

occurrence and methods

were provided.

Three
(number

(func t i o n m a c h i n e ) — and d i f f i c u l t i e s

e x p e r i e n c e d by c h i l d r e n i n s u b t r a c t i o n s ,

w e r e d iscus s e d .

Activities

reasons

for t h e i r

of a v o i d a n c e o f t h e i r o c c u r r e n c e

A i d s u s e d in e a r l y a d d i t i o n a c t i v i t i e s
S e q u e n c i n g of a d d i t i o n a n d s u b t r a c t i o n

activities was p r a c t i c e d .

Three models

t a k e - a way,

and c o mparison— and w r iting

lessons

m i s s i n g addend,

for s u b t r a c t i o n —

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

extensively d i scussed.
T h e d i s c u s s i o n o f the r e v e r s i b i l i t y o f t h o u g h t and
the i n c l u s i o n r e l a t i o n at P i a g e t ' s s t a g e s
r e l a t i o n to a d d i t i o n a n d s u b t r a c t i o n w a s

1 a n d 2 in
u s e d to a p p r e c i a t e

the n e c e s s i t y for m a n i p u l a t i o n o f c o n c r e t e m a t e r i a l s
th o s e s t a g e s

and see t h a t c h i l d r e n are n o t r e a d y for

systematic addition
are i n s t a g e

in

3.

emphasized while

"facts"

in a b s t r a c t

form until they

T h e p r i m a r y i m p l i c a t i o n s of P i a g e t ' s w o r k
teaching add i t i o n and subtraction of

113
w h o l e numbers w e r e on kinds of activities that should
p r e c e d e and be p r e - r e quisite for such work.
M u l t i p l i c a t i o n and Divi s i o n of
W h o l e Numbers
P e d a g o g i c a l a s p e c t of the unit b e g a n by devel o p i n g
an u n d e r standing of the models that could be used to i n t e r ­
p r e t m u l t i p l i c a t i o n and division situations since these
operat ions arise quite n a t u rally from the child's real
world.

T h r o u g h o u t the unit it was d e m o n strated that m a t h e ­

m a t i c a l properties can be used to help children in early
m u l t i p l i c a t i o n and division.

The use of number line and

o t h e r p i c t o r a l models before symbolizing was highly
r e c o m m ended at initial stage.

The role of using thinking

p a t terns in help i n g chil d r e n learn ba s i c number facts was
e x t e n s ively discussed.

This was followed by sequenc i n g

i n itial w o r k in m u l t i p l i c a t i o n and division.
N u m e r o u s activities that could lead children from
i n t r o d uctory concepts to m e m o r i z a t i o n of facts was d e v e l o p e d
and these w e r e followed b y the use of properties and
n umber patte r n s in learning the number facts.

Practice was

p r o v i d e d on w r i t i n g an activity and outli n i n g a lesson to
achieve an objec t i v e in m u l t i p l i c a t i o n or division.
S e q u e n c i n g of o b j e ctives for d e v e l o p i n g m u l t i p l i c a t i o n and
division

(both stan d a r d and non-standard)

algorithms was

practiced.

On the wh o l e there was a long series of

activities,

me n t a l and sensory,

that led from the initial

114
ideas of m u l t i p l i c a t i o n to a ma t u r e concept and an e f f i c i e n t
algorism.
P r o m developmental point of view children are able
to learn m u l t i p l i c a t i o n at the same time that they are able
to learn addition,

approximately seven years of age, yet

m u l t i p l i c a t i o n is delayed.

In fact children can mu l t i p l y

s m aller numbers as readily as they add them, this m a y be
due to close relationship of addition and m u l t i p l i c a t i o n —
processes of put t i n g together.
emphasized.

This relationship w a s w e l l

Piaget's w o r k implies that m u l t i p l i c a t i o n at

abstract or symbolic should be introduced at approximately
the same time as addition.*1 Paradoxically,

the natural

situations for application of the concepts of addition and
m u l t i p l i c a t i o n do not arise as often in y o u n g c h i l d r e n ’s
social environment as do d i v i s i o n and subtraction situations.
In reality the child probably begins w i t h partition
division bef o r e other operations, but the algorithms of the
processes of divi s i o n and subtraction are diffi c u l t and
should be left until the child has used the concepts on a
p r e - n u m b e r basis for a long time and has devel o p e d a deep
u n d e r standing of their m e a n i n g

(Crowder and Wheeler,

1972).

The s ymbo l i s m and p a p e r w o r k should come only after the
i n verse r e l a t ionship b e t w e e n m u l t i p l i c a t i o n and division
is understood using concrete material.

Such problems

should be done at the concrete material levels combining

^"Copeland, op,

cit. , p.

146.

115
and separating sets of objects*

M u l t i p l i c a t i o n facts

should involve the c o r r e s p o n d i n g d i v i s i o n f a c t s — as sets of
objects are manipulated.
It was e m p h a s i z e d that the studies of Pi a g e t i n d i ­
cate that pa p e r work should n o t b e g i n
and division)

(on m u l t i p l i c a t i o n

until a c h i l d has the r e v e r s i b i l i t y of thought

c h a r a c t e r i s t i c of the third or "operational"

thought level,

and that for many youngsters the symbol or abst r a c t w o r k
m i g h t best be done in s y s t e m a t i z e d fashion toward the later
part of the first grade or in the se c o n d grade since the
n e c e s s a r y o p e r a t i o n a l thought level does not oc c u r in m a n y
ch i ldren until around the age of seven.
level,

A t this abstract

it is r e c o m mended that the c h i l d should o r g a n i z e his

m u l t i p l i c a t i o n facts into a table by u s i n g m a n i p u l a t i v e
materials.

Moreover,

these b a s i c m u l t i p l i c a t i o n facts

(up

to 9 x 9) should be c o m m i t t e d to memory w i t h some kinds of
r e i n f orcement activities in form of e n j o y a b l e games w h i c h
p r ovide prac t i c e in the recall of these b a s i c facts.

The

u n d e r standing of nu m b e r p r o p e r t i e s and p l a c e value w i l l then
be n e c e s s a r y as they

(children)

m o v e to w a r d the con v e n t i o n a l

p r o c e dures invol v i n g m u l t i p l i c a t i o n and d i v i s i o n problems.
Since the learning involves d i s c o v e r y a n d r e l e a r n i n g n e w
facts,

recalling all p r e v i o u s learning,

a t t e ntion span,

ha v i n g grea t e r

and b e i n g able to h a n d l e a more c o m p l e x

o p e r a t i o n i n v o lving storage and recall of numbers d u r i n g
operation,

it is r e c o m m e n d e d that the w o r k be sp r e a d over

a p e r i o d of years

(Fehr and Phillips,

19720).

Fractions
The unit sta r t e d w i t h thor o u g h e x a m i n a t i o n of the
m e a n i n g of 1/a w h e r e a is a n o n - z e r o w h o l e number.

Following

this was a set of activities that p r o v i d e d students w i t h
learning experiences that could help them grasp the
fraction concept.

Each e x p e r i e n c e was desi g n e d to b r i n g

the student into a p e r s o n a l e n c o u n t e r w i t h fractions as
they are r e p r e s e n t e d by p h y s i c a l m o d e l s or referents.

The

interns w e r e exp o s e d to e x p e r i e n c e s w h i c h they could later
use p e r i o d i c a l l y w i t h ei t h e r an e n t i r e class
pupils)

(of their

in c o n j u n c t i o n w i t h a m o r e compr e h e n s i v e unit on

fractions or w i t h sele c t e d groups of pupils w h o ap p e a r to
need additional prac t i c e w i t h concrete re p r e s e n t a t i o n s
order to bec o m e familiar w i t h b a s i c concept.

in

The interns

w e r e given the o p p o r tunity to w o r k independent,

to explore,

to guess and to see if h i s / h e r gues s e s w e r e correct.

The

ac t i v ities r e q u i r e d e q u i p m e n t and m a t e r i a l s that w e r e
relatively easy to obtain and safe to use w i t h a m i n i m u m of
c l a s s r o o m supervision.
Tho u g h many m a n i p u l a t i v e and vi s u a l aids w e r e
used,
viz.

three of them w e r e p r e d o m i n a n t l y used as models,
cuisen a i r e rods,

nu m b e r line and r e c t a n g u l a r arrays.

The interns at the end of the unit b e l i e v e d that the use of
c u i s e naire rods o f f e r e d them a w e a l t h of m a t h e m a t i c a l l y
co rrect e x p e r i e n c e in fractions w h i c h c o u l d n o t be a c q u i r e d
through no o t h e r m e t h o d
by a f r a c t i o n ) .

(especially d i v i s i o n of a fraction

They b e l i e v e d it was b o t h abst r a c t a n d

117
concrete.

The power o f these colou r e d rods lies, m a t h e ­

matically,

in their p e n e t r a t i o n to the core of r e l a t i o n ­

ships and structures,

and, psychologically,

stimulus to intui t i o n and enquiry.

in their

They o b s e r v e d that

fractions are m e n t a l structures e x t r a c t e d from str a i g h t
forward,

simple situations involving the rods,

and m a d e

e v i d e n t both through the colors and l e n g t h s , m o r e i m p o r t a n t
is the feet that ins t e a d of considering one fraction alone,
they c o u l d cons i d e r those e q u i v a l e n t to it as well;
r e c t a n g l e - m o d e l is very helpful in this case.
Clinical Experience
T he interns sta r t e d w h a t was called the inservice
p a r t of their p r o g r a m in Fall 19 73.
morning

(four hours)

Each intern spent full

daily in Fall and Spring terms

in

e l e m e n t a r y school a n d s p e n t the after n o o n on the un i v e r s i t y
campus.

In Winter,

19 74, the interns spent m o r n i n g hours

on u n iver s i t y campus and after n o o n hours in e l e m ent a r y
school.

T h e interns w e r e un d e r the s u p e rvision of e x p e r i ­

e nced i n s e r v i c e teachers,
e l e m e n t a r y schools.

c a l l e d team leaders,

in the

These t e a m leaders also attend e d the

interns m a t h e m a t i c s classes at the u n i v e r s i t y throu g h o u t
the a cad e m i c year.

The purpose of this was to solve some

of the prob l e m s of

(though p r o f e s s i o n a l l y e x p e r i e n c e d but)

m a t h e m a t i c a l l y i n c o m p e t e n t s u p e r v i s i n g teachers ra i s e d b y
Hatfield

(1972).

118
The interns after d e v e l o p i n g their lesson p l a n s in
c o o p e ration w i t h their t e a m - l e a d e r s , taught these lessons
to their pupils.

It should be m e n t i o n e d again that these

interns w e r e not fully res p o n s i b l e for their classes they
w e r e supp o s e d to be invo l v e d in t e a m - t e a c h i n g w i t h their
c o o p e r a t i n g teachers.
observation,
small groups,

Tho u g h they w e r e to start by

they p r o c e e d e d g r a d u a l l y to tutoring,

teaching

and eventu a l l y teac h i n g the w h o l e class.

The clin i c a l e x p e r i e n c e p r o v i d e d the interns with:
1.

The opp o r t u n i t y to relate theory to practice,

by

appl y i n g the k n o w l e d g e g a i n e d at the u n i v e r s i t y to
actual te aching s i t u a t i o n s at the e l e m e n t a r y
school.
2.

The o p p o r t u n i t y to obs e r v e d i f f e r e n t classes,
teachers,

3.

and t e a c h i n g methods.

The o p p o r t u n i t y to init i a t e their t e a c h i n g e x p e r i ­
ence star t i n g by w o r k i n g w i t h a small group of
children,

thus b e n e f i t i n g from cl o s e r indivi d u a l

relations

and m i n i m i z e d p r o b l e m s of d i s c i p l i n e and

control,
4.

a n d e v e n t u a l l y h a n d l i n g the en t i r e class.

The o p p o r t u n i t y to rec e i v e i m m e d i a t e feedba c k on
the met h o d s of teaching u t i l i z e d from e x p e r i e n c e d
in-ser v i c e teachers and fac u l t y members.
An A p p r a i s a l o f the I n s t r u c t i o n a l M e t h o d
The i n s t r u c t i o n a l stra t e g y u s e d by the i n s t r u c t o r

w as B loom's model of M a s t e r y Learning.

The p r o c e d u r e was

119
very sim i l a r to that d e s c r i b e d in chapter II u n d e r p r a c t i c e
of m a s t e r y learning.
A look at the prev i o u s grades of the interns
r e f l ected that many of them w e r e short of good b a c k g r o u n d
in m a t h e m a t i c s o r that they could not learn ma t h e m a t i c s
effectively in the t r a d i tional setting.

A review of

literature showed that laboratory approach in c o n j u n c t i o n
with mas t e r y learning strategy w o u l d be very e f f e c t i v e for
these interns.

More so this sh o u l d be a us e f u l me t h o d for

them to teach their pupils

(taking into c o n s iderati o n

results of studies and r e c o m mendations of p r o f e s s i o n a l
bodies on teaching of m a t h e m a t i c s in the inner-city
schools).
During the Fall 1973 two doct o r a l students
cluding the investigator)

(in­

w o r k e d w i t h the i n s t r u c t o r to see

that the needs of the i n d i v i d u a l intern was met.

T h e team

leaders also he l p e d in this attempt.
On the basis of the formative e v a l uations c a r r i e d
out in Fall term of 1973,

the interns w e r e d i v i d e d into two

groups in W i n t e r 19 74,

for the p u r p o s e of p r o v i d i n g

learning c o r r e c t i v e s .

The g r o u p containing the ave r a g e

and b e l o w average interns w e r e p r o v i d e d with reme d i a l w o r k
w h i l e the above average was p r o v i d e d w i t h enrichment.
instructor together w i t h three d o c t o r a l students
the investigator)

The

(including

were cha r g e d w i t h this responsibility.

An two-hour c l a s s - m e e t i n g per w e e k w a s s p e c i a l l y set aside
for this though interns c o n s u l t e d w i t h the d o c t o r a l students

120
and the instru c t o r outside the class as well.

Two doctoral

students w e r e w o r k i n g with interns w h o ne e d e d remedial w o r k
w h i l e the third doctoral student was w o r k i n g w i t h interns
w h o needed enrichment.
groups.

The instructor w o r k e d w i t h both

This a r r a n g ement gave the interns an oppor t u n i t y

not only to remove their deficiencies b u t also to explore.
The activities

included

p r o b l e m sessions,

(for remediation)

individual tutoring,

small group

and use of alternative

learning materials w h i l e the e n r i c h m e n t group was i ntroduced
to N u m b e r Theory.

In Spring term this special class-

m e e t i n g could not hold due to some a d m i nistrative problems
in the schedule of the interns,

however, help and learning

correctives w e r e p r o v i d e d in and outside the regular classmeetings.
In order to unders t a n d the overall goal of the
m a t h e m a t i c a l i n s t r uction let us examine some of the common
goals of instruction.

Lewin's

(19 35)

theory of m o t i v a t i o n

p o s t u l a t e s that a state of tension w i t h i n an i n d i vi d u a l
m o t i v a t e s m o v e m e n t toward the a c c o m plishments of desired
goals.

Three goals emerged from Lewin's notions:

w h e r e there is cooperative goal interdependence,

one
one w h e r e

there is competitive goal i n t e r dependence and one w h e r e a
p e r s o n has indivi d u a l i s t i c goals u n r e l a t e d to anyone else's.
B u i l d ing a field theory of c o o p e ration and competition,
D e u tsch

(1949,

1962)

defined

(1) a soc i a l situa t i o n as one

w h e r e the goals of the separate individuals are so linked
t o g ether that there is a positive c o r r e lation bet w e e n their

goal attainments,

(2) a c o m p e t i t i v e s o c i a l s i t u a t i o n as o n e

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

are so l inked

t h a t there is a n e g a t i v e c o r r e l a t i o n b e t w e e n t h e i r goals
a t t a i n m ent,

and

(3)

an i n d i v i d u a l i s t i c s i t u a t i o n as one

w h e r e the goals of i n d i v i d u a l s are i n d e p e n d e n t of ea c h
other.

T o D e u t s c h un d e r the first an i n d i v i d u a l can o b t a i n

his g o a l if,

and o n l y if,

the o t h e r p e r s o n w i t h w h o m he is

l i n k e d can o b t a i n his goal;
can o b t a i n his goal if,

un d e r the s e c o n d an i n d i v i d u a l

and only

if,

the o t h e r s w i t h w h o m

he is l inked c a n n o t o b t a i n t h e i r goals;

u n d e r the t h i r d

w h e t h e r o r n o t an i n d i v i d u a l a c c o m p l i s h e d his g o a l ha s no
b e a r i n g u p o n w h e t h e r o t h e r i n d i v i d u a l s a c c o m p l i s h th e i r
goals,

in this s i t u a t i o n the i n d i v i d u a l seeks an o u t c o m e

t h a t is b e s t
others

for himself,

r e g a r d l e s s of w h e t h e r or n o t

a c h i e v e t h e i r goals.

upon l e a r n i n g theory,

In a c o n c e p t u a l i z a t i o n b a s e d

Kelly and Thibaut

(1969)

defined a

c o o p e r a t i v e s t r u c t u r e as one in w h i c h the i n d i v i d u a l s
r e w a r d s are d i r e c t l y p r o p o r t i o n a l to the q u a l i t y of the
g r o u p work;

a c o m p e t i t i v e s t r u c t u r e is one in w h i c h

v i d u a l s are r e w a r d e d so t h a t o n e r e c e i v e s
and the o t h e r r e c e i v e s a m i n i m u m reward;
s t r u c t u r e is o n e in w h i c h

indi­

a maxi m u m reward
an i n d i v i d u a l i s t i c

i n d i v i d u a l s are r e w a r d e d on the

b a s i s of the q u a l i t y of t h e i r w o r k i n d e p e n d e n t of the
q u a l i t y of w o r k of o t h e r s t u d e n t s .

Deutsch

(1962)

empha­

sizes t h at an i n d i v i d u a l w i l l t e n d to f a c i l i t a t e the
a c tions of others w h e n he p e r c e i v e s t h a t t h e i r a c t i o n s w i l l
p r o m o t e his cha n c e s of g o a l a t t a i n m e n t and w i l l tend to

122
o b s t r u ct th e i r actions w h e n he perce i v e s that they w i l l be
d etrim ental to his goal attainment.

F o r Kelley and T h i b a u t

the reward d i s t r i b u t i o n m o t i v a t e d individuals
cooperatively,

competitively,

upon the reward structure.

to beh a v e

individualistically depending

For Deu t s c h it is the drive for

goal a c c o m p l i s h m e n t that m o t i v a t e s cooperative,
or i n d i v i d u a l i s t i c behaviour.
e x t r i n s i c motivation,
helpful;

competitive,

W h e n one is focusing upon

Kelley and T h i b aut's d e f i n i t i o n is

w h e n focusing upon intri n s i c m o t i v a t i o n Deuts c h ' s

c o n c e p t u a l i z a t i o n is helpful.
The past success of p r o g r a m m e d learning m a t e r i a l s
and m a s t e r y programs

{Block,

1971)

indicate that i n d i ­

v i d u a l i s t i c goal s t r u ctures are a p p r o p r i a t e for the
learning of spec i f i c c o g n i t i v e mater i a l s and skills.

Due

to lack of i n t e r action a m o n g students and their inde p e n d e n c e
from each other,

feelings of loneliness a n d i s o l a t i o n m a y

b l o c k the de v e l o p m e n t of i n t e r p e r s o n a l and group skills
w h i c h m a y lead to the s u f f ering of affec t i v e outc o m e s and
pr o c e s s variables.
McKeachie

The w o r k of Deu t s c h

(19 67) , Hammond and Gol d m a n

(1957) , Kogan and W a l l a c h

{19 49a), Hain e s and

(1961) , Thomas

(1967) , J o h n s o n

(1971,

1974a),

show that c o o p e rative goal structures s h o u l d be u s e d w h e n
i n s t r u c tional object i v e s
affective outcomes as:
group productivity;
mation;

focus upon such c o g n i t i v e and
p r o b l e m solv i n g effectiveness ?

m e m o r i z a t i o n and retri e v a l of i n f o r ­

compet e n c e in co o p e r a t i v e situations,

cognitive

d e v e l o p ment and its related areas of social adjustment.

123
co m m u n i c a t i o n effectiveness,
and e m p a t h e t i c ability;
areas,

autonomous m o r a l judgment,

p o s i t i v e attitude toward su b j e c t

i n s t r u c t i o n a l activities,

teachers and students;

r e d u c t i o n of p r e j u d i c e and the app r e c i a t i o n of cultural
and indivi dual differences;
attitudes

d e v e l o p m e n t of posit i v e s e l f ­

and a bel i e f in one's b a s i c competence and worth;

d e v e l o p m e n t of a c h i e v e m e n t and motivation;
i n t e r personal skills;

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

upon intri n s i c motivation;
sizes m o d e r a t e

d e v e l o p m e n t of

learning processes w h i c h e m p h a ­

levels of anxiety,

posi t i v e inte r p e r s o n a l

r e l a t ionships and rel a t e d c o h e s i o n and p s y c h o l o g i c a l
s u p p o rt and safety;
a m o n g students;
students;

the r e d u c t i o n of h o s t i l i t y and c o n f l i c t

o p e n and effective c o m m u n i c a t i o n among

trusts; m u t u a l influence p r o m o t i n g a c h i e v e m e n t

a nd t ask-orientation;
m u t u a l helpfulness;
and tasks;

sha r i n g of ideas and m a t e r i a l s and

i n v o l v e m e n t in i n s t r u c t i o n a l activities

c o o r d i n a t i o n of efforts and d i v i s i o n of labour;

a n d d i v e r g e n t and r i s k - t a k i n g thinking.
Deutsch

(1949a,

and L e w i c k i

1962),

(1969)

Crombag

D e u t s c h and Krauss

(1966),

(1962), J o h n s o n

s h o w that c o m p e titive situa t i o n p r o d u c e s

the above results in negat i v e direction.
In the p u r e l y acad e m i c

(cognitive)

areas of the

m a t h e m a t i c s e d u c a t i o n c o m p onent of this p r o g r a m the
i n s t r u ctor's goal s t r u cture c o u l d be d e s c r i b e d as
c o o p e r a t i v e l y - i n d i v i d u a l i s t i c in the sense that p r e p a r a t i o n
for the p o s t - t e s t c r i t e r i o n - m e a s u r e s was co o p e r a t i v e in
n ature w h i l e the m a s t e r y appr o a c h m a d e the o u t c o m e

124
individualistic.
of l esson plans,

The p r o f e s s i o n a l a s p e c t like p r e p a r a t i o n
units,

journals were c o o p e rative in some

o c c a sions and c o o p e r a t i v e l y - i n d i v i d u a l i s t i c in others.

The

reason for using this appr o a c h is evident from the above
review,

b a c k g r o u n d of the interns,

and the na t u r e of the

job for w h i c h they are be i n g prepared.
i n d i v i d u a l i s t i c appr o a c h removed

The c o o p e r a t i v e l y -

lack of interaction,

feelings of isolation and loneliness,

and

b l o c k i n g of i n t e r ­

p e r sonal and group skills that could re s u l t under p u r e l y
i n d i v i d u a l i s t i c goal struct u r e s and still p r o d u c e d the
learning of speci f i c c o g n i t i v e m a t e r i a l s and skills.

The

interns also e x h i b i t e d m o s t o f the p o s i t i v e o u t c o m e s of
the p u r e l y c o m p e t i t i v e goal s t r u c t u r e approach.

Advantages

of m a s t e r y appr o a c h have already b e e n d i s c u s s e d in
c h apter II.
Samples
Three groups of students w e r e

involved in the

study.
The first and pri m a r y sample o f inte r e s t c o m p r i s e d
of thirty interns

(forming g r o u p G 1 ) sele c t e d for the

e i g h t h cycle T e a c h e r Co r p s p r o g r a m at M i c h i g a n State U n i ­
versity.

The s e l e c t i o n o f the interns was b a s e d on the

f o l l owing criteria:

1

^Lansing School D i s t r i c t E i g h t h C y c l e T e a c h e r
Corps Proposal, V I I — 8 and 9.

125
Educational Requirement
1.

Be a c i t i z e n / p e r m a n e n t resid e n t of the U n i t e d
S tates of America.

2.

Have no legal b a c k g r o u n d that will ha m p e r teacher
certification.

3.

Have a m i n i m u m of 60 semes t e r hours or 90 q u a r t e r
hours,

f r o m an a c c r e d i t e d institution,

that can be

a p p l i e d towards a Bachelors Degree.
Human C h a r a c t e r i s t i c s
1.

K n o w the o b j e c t i v e s of T e a c h e r Corps as they apply
to the L a n s i n g project.

2.

A willingness

to w o r k on a teac h i n g team.

3.

Be sensi t i v e to the needs o f low-income area
children.

4.

W i l l i n g n e s s to deal w i t h school p e r s o n n e l and
ad m i n i s t r a t i o n s

5.

in an ef f o r t to i m p l e m e n t new ideas

U n d e r s t a n d the three components of T e a c h e r Corps
(school,

university,

and community)

and h o w they

w o r k together.
6.

U n d e r s t a n d a n d be w i l l i n g to deal w i t h the v i g o r o u s
schedule demands of the T e a c h e r Corps program.

Many of the interns

are mem b e r s of m i n o r i t y or low-income

g roups w i t h w e a k e d u c a t i o n a l background.
a l a b o r at o r y - o r i e n t e d ,

They received

c o n t e n t - m e t h o d s integrated,

126
m a t h e m a t i c s e d u c at ion p r o g r a m w i t h m a s t e r y - l e a r n i n g approach
in fourteen q u a r t e r - h o u r s spr ead over three terms.
The sec o n d g r o u p ,

, com pr i s e d of twenty -tw o

students w h o w e r e ran domly sel e c t e d out of v o l u nte ers

from

about 150 students w h o reg ist e r e d for m a t h e ma tic s 201 in
the fall q u a r t e r of 1973 as students in the r egular teacher
e d u c a t i o n program.

The group w a s given an in s t r u c t i o n very

si milar to that of G-^ and jointly t aught by the i nst ru c t o r
w h o t aught G^ and a not her i n s t r u c t o r in six q u a r t e r - h o u r s
in fall.
week.
to
type.

T h e y also h a d cli nical exp eri e n c e of one h our per

The m a i n di ffe r e n c e b e t w e e n the instruc tio ns given
and G 2 was that

r ece i v e d t i m e - m a s t e r y - l e a r n i n g

M o r e o v e r mem b e r s of G 2 w e r e not n e c e s s a r i l y from a

p a r t i c u l a r social-background.
Th e t hir d group,

G^ r c o m p r i s e d of students in the

re gular p r o g r a m w h o w e r e e x p e c t e d to have h a d m a t h e m a t i c s
201 and w h o r e g i s t e r e d for e duc a t i o n
in m a t h e mat ics ,

in fall,

1973.

32 5E, a me th o d s course

These students di ffe r

su b s t a n t i a l l y from G^ and G 2 in that m o s t of t h e m w e r e
juniors.

This Group

(G3) was u s e d to compare the m a t h e ­

m a t i c a l u n d e r s t a n d i n g and attitud e toward a r i t h m e t i c after
the c o m p le tio n of the me thods course w h i c h was s e p a r a t e d
from m a t h e m a t i c s content.
Me as u r e s and I nst ru m e n t a t i o n
T h e follo win g i n s t r u men ts we re used in g a t h e r i n g
data for the pa y - o f f p a r t of the study;

127
1.

F i v e c r i t e r i o n - r e f e r e n c e d a c h i e v e m e n t m e a s u r e s to
as ses s the m a t h e m a t i c a l a n d p e d a g o g i c a l
on p r e s c r i b e d o b j e c t i v e s

2.

Dossett's

(two p a r a l l e l

competencies
forms).

test of B a s i c M a t h e m a t i c a l U n d e r s t a n d i n g

of P r o s p e c t i v e E l e m e n t a r y S c h o o l T e a c h e r s
parallel
3.

forms)

(Dossett,

(two

1964).

R e v i s e d fo rm of D u t t o n A t t i t u d e I n v e n t o r y F o r m C
( D u t t o n , 19 62) .

4.

Attitudes

S cal es

t o w a r d d i f f e r e n t a spe cts of m a t h e ­

m a t i c s d e v e l o p e d by t h e I n t e r n a t i o n a l Stu dy of
A c h i e v e m e n t in M a t h e m a t i c s
5.

(Husen,

1967).

A i k e n ' s E n j o y m e n t and V a l u e of M a t h e m a t i c s
(Aiken,

Scales

1974).

D e v e l o p m e n t and E v a l u a t i o n of
Criterion-Referenced
Achievement Measures
In o r d e r to as ses s the e f f e c t i v e n e s s o f the m a t h e ­
m a t i c s e d u c a t i o n c o m p o n e n t of the T e a c h e r C o r p s P r o g r a m on
the p r o s p e c t i v e e l e m e n t a r y tea c h e r s p a r t i c i p a t i n g in t he
program ,

it w a s n e c e s s a r y to d e v e l o p a s e r i e s o f c r i t e r i o n -

r e f e r e n c e d tests d e s i g n e d s p e c i f i c a l l y to t e s t w h e t h e r the
prospective

t e a c h e r c o u l d or c o u l d n o t e x h i b i t

te ncy i m p l i e d b y the p r e s c r i b e d o b j e c t i v e s
unit

(Glaser,

also essential

1963,

1971;

P o p h a m a n d Husek,

in e a c h

lea r n i n g

1969).

It was

to d e v e l o p two e q u i v a l e n t forms

test in o r d e r to a sse ss

the c o m p e ­

the e n t e r i n g b e h a v i o u r s

for e a c h
and the

t e r m i n a l b e h a v i o u r s of th e p r e - s e r v i c e t e a c h e r t o w a r d the

128
p r e s c r i b e d o b j e c t i v e s w i t h i n e a c h l e a r n i n g unit.
tages for the c ho i c e o f the

Advan­

"unit" as the c o n v e n i e n t

c u r r i c u l u m s e g m e n t for a n a l y s i s hav e b e e n d i s c u s s e d by
Hively,

et al.

(1973).

A r e v i e w of the l i t e r a t u r e h e l p e d g a i n d e e p e r
i n s i g h t on the m e t h o d o l o g y o f c o n s t r u c t i n g g o o d t e s t s .
M u c h of the t h e o r y of a c h i e v e m e n t t e s t i n g was
the E d u c a t i o n a l M e a s u r e m e n t
which Lindquist recommends

(ed. by L i n d q u i s t ,

1951) , in

the f o l l o w i n g steps in the

p r e p a r a t i o n of an e d u c a t i o n a l
n i n g the test,

o u t l i n e d in

achievement t e s t :

(2) w r i t i n g the test items,

(3)

(1) p l a n ­

try ing out

the test form an d a s s e m b l i n g the f i n i s h e d t e s t a f t e r t r y ­
out,

(4) p r e p a r i n g the d i r e c t i o n s

s c o r i n g the t e s t , and

(5)

for a d m i n i s t e r i n g and

r e p r o d u c i n g the t e s t .

T h o u g h the

m o s t c o m m o n a p p r o a c h to c o n s t r u c t i o n of c r i t e r i o n - r e f e r e n c e d
m e a s u r e s has b een to c o n s t r u c t p r o t o t y p i c a l t e s t i tem s that
are

"keyed"

to m o r e g e n e r a l l y s t a t e d o r i m p l i e d d e s c r i p t i o n s

of the d e s i r e d b e h a v i o r

(Mager,

1969;

1969;

M e r w i n and Womer,

Hively,

e t al.

(1973)

1962;

Gagne,

196 7; Bloom,

L i n d v a l l a nd Cox,

1970),

d i s c u s s e d the p r o b l e m w i t h this

approach.
In this s t u d y th e i n v e s t i g a t o r , w i t h a s s i s t a n c e
from mathematics educators

at the M i c h i g a n S t a t e U n i v e r s i t y ,

d e v e l o p e d the items in e a c h of the c r i t e r i o n - r e f e r e n c e d
m e a s u r e s b e a r i n g in m i n d th e r e c o m m e n d a t i o n s o f L i n d q u i s t
(1951), P o p h a m a n d H u s e k
and N o v i c k

(19 73).

(1969),

Simmon

(1969), H a m b l e t o n

T h e p a s s i n g scores a nd lengths w e r e

determi ned by Mil lim a n ' s
tained ten items.

(1973)

criteria.

Each test con ­

Two e q u i v a l e n t tests were p rep are d for

each of the five units,
other as post-test.

one s erved as pre - t e s t and the

A s e c o n d form of p o s t - t e s t for each

unit was p r e p a r e d and a d m i n i s t e r e d as s u g g e s t e d b y Block
(1972).

Many of the items w e r e not m u l t i p l e - c h o i c e because

of the nature of the content b e i n g tested.
test c o n s tru cto rs a dm i n i s t e r e d the test,

it w as not

necessa ry to p rep are the d e t a i l e d di rec tio ns
examiner.

There was

for testing,
two hours.

Since two of the

for the test

a great flexibility on time a llo cat ed

it v a r i e d be t w e e n one and one -h a l f hours

to

Partial credits w e r e al lowed in various parts

w h i c h w e r e non-multiple-choice.
The p r e - t e s t was a d m i n i s t e r e d pr i o r to the i n­
struction on the c o r r e s p o n d i n g unit,

and the po st- t e s t was

a d min ist ere d b e t w e e n four to s e v e n days after i nst ruc tio n
to assess the e f f e c t of eac h unit.

Th e i nv e s t i g a t o r was

pr es e n t at all testing.

W h e n a st u d e n t was absent during

the the p r e - t e s t period,

he was asked to take the test

before st ar t i n g on the ac tiv i t i e s

for that unit.

Tho ugh

long-range or s p i r a l i n g effects m i g h t also have bee n
as sessed after i n s t r u c t i o n in s pec i f i e d units,
end of a cademic year,

or later,

or at the

the e v a l u a t i o n focused

p r i m a r i l y on the im med iat e eff ects of i n s t r uct ion in each
unit.

130
V a l i d i t y .— C r i t e r i o n - r e f e r e n c e d mea sures are
v a l i da ted pr imarily in terms of the adequacy w i t h whi ch
they r epr esent the criterion;

therefore,

approaches are suited to such tests
1971).

content v ali dit y

(Popham and Husek,

The inherent m e t h o d by wh i c h the set of tests wer e

d e v e l o p e d as sured co ntent validity,

since the test items,

in the judgment of the team of m athematics e duc ators who
d e vel o p e d and des igned the learning units,

did in fact

reflect the s pecific objectives wi t h i n the m a t h e m a t i c a l
co ntent of that unit.
R e l i a b i l i t y .— Since each test was c o n s t ruc ted to
assess the instr uct ion al object ive s w i t h i n a specified
topic,

it was n e c e s sar y to estima te the reliabi lit y of each

test i n d e p end ent ly

(Popham and Husek,

1971).

Students in five sections of the regular methods
course

(Education 32 5E)

(three in W i n t e r and two in Spring)

were made available to test the r e l i a b ili ty of the pre- and
p o s t - c r i t e r i o n measure a c h i e veme nt tests.

There wer e about

thirty students in each of these classes and the i n v e s t i ­
ga t o r was allowed a p p r o x i m a t e l y one and o ne- h a l f hours

for

te sting purposes.
F o l l o w i n g Cook and S t u f f l e b e a m

(1967)

(who d e m o n ­

st rated e m p i r ica lly that group p e r f o r m a n c e is more e f f i c i ­
ently m e a s u r e d using small subsets of items d i s t r i b u t e d
among large numbers of students
et al.

than vice versa)

and Hively,

(19 7 3), it was dec i d e d to rando mly select a 5-i tem

131
s a m p l e f r o m e a c h o f the five p r e — a n d p o s t - t e s t s

(50%).

W h e n the s e l e c t i o n of the se p a r t i c u l a r items w as

comple ted ,

five t e n - i t e m tests w e r e assembled:
1.

T e s t I c o n t a i n e d five items
items

2.

from post-tests

T e s t II c o n t a i n e d
items

3.

f r o m p r e — a n d five

on M e a s u r e m e n t .

five items

f r o m p r e - and five

f r o m p o s t - t e s t s on N u m e r a t i o n .

T e s t III c o n t a i n e d five items
i t ems

f r o m p r e - and five

f r o m p o s t - t e s t s on A d d i t i o n an d S u b t r a c t i o n

of W h o l e Num bers.
4.

T e s t IV c o n t a i n e d
i t ems

five items

f r o m p r e - a n d five

f r o m p o s t - t e s t s on M u l t i p l i c a t i o n and D i v i s i o n

of W h o l e N umbers.
5.

T e s t V c o n t a i n e d five items
items

from post-tests

f r o m pre - and five

on Fr act ion s.

T w e n t y co pie s of eac h of the f i r s t t hre e tests w e r e
r a n d o m l y d i s t r i b u t e d to t he s i x t y s t u d e n t s
s e c t i o n s of E d u c a t i o n
E l e m e n t a r y Grades)

3 2 5E

in the three

(Teaching of M a t h e m a t i c s

in

in W i n t e r t e r m w h i l e t w e n t y c opi es of

e a c h of the las t two t est s w e r e r a n d o m l y d i s t r i b u t e d to
forty s t u d e n t s

in two s e c t i o n s of the class in S p r i n g term.

B a s e d o n the s t a t i s t i c a l r e s u l t s o f t hes e t e s t s , r e l i a b i l i t y
estimate

for e a c h t est w a s o bta ined.
E s t i m a t e o f the r e l i a b i l i t y o f e a c h o f the item-

sampled test was
Coefficients
technique.

calculated using

( H o y t , 1941)
Tables

the Hoy t R e l i a b i l i t y

t h r o u g h an a n a l y s i s of v a r i a n c e

c o n t a i n the s t a t i s t i c s

for the a n a l y s i s

132
of v a r i a n c e for e a c h test.

The Spearman-Brown

a p p l i e d to the H o y t R e l i a b i l i t y C o e f f i c i e n t s
total t e s t r e l i a b i l i t y .

Table

formula was

to o b t a i n the

2 shows the r e s u l t s o b t a i n e d

for e a c h test f r o m the s t a t i s t i c a l p r o c e d u r e s

described

above.
The reliability coefficients
f r o m a low of

for the tests v a r i e d

.79 for the p r e - t e s t on f r a c t i o n to a h i g h of

.94 for the p o s t - t e s t o n M u l t i p l i c a t i o n and D i v i s i o n of
W h o l e Numbers.
T h e s e c o e f f i c i e n t s are c o n s i d e r e d to be a c c e p t a b l e
for a c r i t e r i o n - r e f e r e n c e d test
Table

(Gagne,

1967).

2.— R e l i a b i l i t y C o e f f i c i e n t s for Pre- an d Pos tCriterion- Referenced Achievement Measures.
Measures

Pre- T e s t
<2)b
(l)a

P o s t -Test
(2)
(1)

Measurement

.7581

. 8623

.8352

.9100

N u m e r a t i o n Sys tem s

. 8240

.9033

. 8276

.9054

A d d i t i o n a n d S u b t r a c t i o n of
Whole Numbers

.7692

.8693

.8713

.9316

M u l t i p l i c a t i o n an d D i v i s i o n
of W h o l e N u m b e r s

. 8475

.9172

. 8922

.9432

Fractions

.6550

. 7910

.7924

.8844

aHoyt Reliability coefficients obtained
c e n t i t e m - s a m p l e d test.

from 50 p e r ­

L.

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 otal test a f t e r
a p p l y i n g th e S p e a r m a n - B r o w n fo r m u l a to H o y t R e l i a b i l i t y
coefficients.

133

R
=
R tt

2R .
St
1 + R ,
St

= R e l i a bi li ty of total test.
Rs ^_ = R e l i a bi lit y of sampled test.
Thorndi ke' s guide lin e for p r e p a r a t i o n of eq uiv ale nt
pre- and p ost -te sts was

followed

(Lindquist,

1951).

The

equivale nce of the p a i r e d tests was further che c k e d by
c o mputing P e a r s o n - m o m e n t correla tio n coefficients on test
scores of these 100 students.

The c o r r e lat ion coefficients

between pre- a nd p o s t - t e s t scores v a r i e d from a low of
for the test on F r a c t i o n s to a high of

.72

.93 for the test on

M u l t i p l i c a t i o n and D ivi s i o n of W h o l e Numbers.
Table 3.— C o r r e l a t i o n C o e f f i c i e n t s Be t w e e n Pre- and PostT e s t Scores of the Students in R egu lar Me thods
C ourse (Education 325E) on I t e m - Sa mpl ed CriterionRe f e r e n c e d Achievement.

Tests

N

Correlat ion C o e f fi cie nts

M e a s ur eme nt

20

.8857

Numeration

20

.7402

Ad d i t i o n and S u b t r act ion of
Wh o l e Nu mbers

20

.8576

Mu l t i p l i c a t i o n and D i v i s i o n of
Whole Numbers

20

.9324

Fr act ion s

20

.7213

134
Se lection of a Test of
M a t h e m a t i c a l Unders tan din g
This phase of the study b e g a n by sea rching for a
w e l l - d o c u m e n t e d instrument for m e a s u r i n g m a t h em ati cal u n d e r ­
standing.

A f t e r a careful search of the literature,

the

i n v e st iga tor de cided to use an i n s t r u m e n t d esi g n e d by
M i l d r e d J. Dossett
appropr iat e

(1964).

The test was deemed m ost

for the pu rpose of this invest iga tio n since the

test items co vered mat h e m a t i c a l topics r e c o mme nde d by p ro­
fessional and advisory groups in mathema tic s education.
P e r m i s s i o n was g r a n t e d by the author to use the test for
the pr e s e n t study.
Do sse t t * s instrume nt e n t i t l e d "Test of Basic M a t h e ­
ma ti c a l Unders tan din g" had a reliabil ity c o e f f i cie nt of
0.87 o b t a i n e d by correl ati ng the scores made by fifty
college students on the two e q u i v a l e n t forms of the test.
E q u i va len cy of the two forms was determi ned by using a
t-test s u g g e s t e d by McNemar.

The t-value o bta i n e d i n d i ­

cated no s i g n i f i c a n t differences be tw e e n the scores o n the
two forms of the test w h e n adm i n i s t e r e d to the fifty college
students.
F o r m A of Dossett*s test was
groups at the beg in n i n g of Fall term,

admin ist ere d to all
19 73 and p ost -te st

was a d m i n i s t e r e d to each group at the end of instruction.
Se lection of A t t i t u d e Inventory
The

"Arithmetic Att i t u d e

Inventory,"

an attitude

scale d e v e lop ed by W i l b u r D ut t o n at the U n i v e r s i t y of

135
California, was used in this study
this scale,

(Dutton,

1962).

For

D utton ut il i z e d a technique p e r f e c t e d by

Th urs ton e and C h a v e .
wr i t t e n statements

He first se le c t e d a large n umber of

re garding attitudes toward arithmetic

o bta i n e d from papers of six hu nd r e d university students
over a p e r i o d of five years.

The statements w e r e sorted

by judges using a scale of one to e leven
favorable to extre mel y f a v o r a b l e ) .

(extremely u n ­

The p r o p o r t i o n of

judges w h o pl a c e d each s t a t e m e n t in the d iff ere nt categories
c o n s t itu ted the b a s i c data for c omp uting the scale values
of the statements.
students,

The in str u m e n t was used w i t h over 2 89

A r e l i a b ili ty of

retest p roc ed u r e s

(Dutton,

.94 was o bta i n e d through test1965).

On the a ttitude instrument,

the fifteen items have

values that range from 1.0 to 10.5 r e p r e s ent ing e xtr emely
ne ga t i v e to e xt rem ely pos itive attitudes.

The individual

score is the average scale value of the statements which
the i ndi vi d u a l checked.
The i nst rum ent was a d m i n i s t e r e d to each group the
same day as

(but preceding)

Dosset t's tests.

Se lec t i o n of A t t i t u d e s Scales
T oward D i f f e r e n t As pects
"*
of M a t h e m a t i c s
The attitud e scale d e v e l o p e d by the I n t e rn ati ona l
Pr oject for the E v a l u a t i o n of E d u c ati ona l A c h i e v e m e n t in
Mathematics,

Husen

(1967), was used to measure subjects'

attitude toward some aspects of m a t h e ma tic s in r ela t i o n to

136
me a s u r e subjects'

attitude toward some aspects of m a t h e ­

ma tic s in re la t i o n to school and life in general.

This

test was a d m i n i ste red to ward the end of i n s t r u cti on p e r i o d
for various groups.

The coe ffi cie nt of re pro duc tiv ity of

the items ob ta i n e d from Gut tma n Scale were genera lly above
the

.80 to .85 wh i c h are considered acceptable

19 54)

through slightly b e l o w the

(Stouffer,

et al.,

1950).

scales are in Hunsen

(Guilford,

.90 recommen ded by G ut t m a n

Details for construc tio n of the

(1967, Vol.

I).

Se lection of Enj oy m e n t and Value
of: Ma the mat ics Scales
Aiken

(19 74)

de signed two scales w h i c h were to

m e a s u r e b o t h parts A and B of O b j e c t i v e IV,

"Appre cia tio n

and use of ma thematics," of the m a t h e mat ics objecti ves of
N a tional A s s e s s m e n t of E d u c a t ion al Pro gress

(1970).

These

su b-c ate gor ies are:
a.

R e c o gni zin g the import anc e and relevance of m a t h e ­
m a tic s to the indivi dua l and to society.

b.

Enj oy m e n t of mathematics.

Several attitudes scales
Scale by Aiken,

(including The M a t h em ati cs A t t i t u d e

19 72) m e a s u r e ob jec tiv e V I - B fairly, w h i l e

little a tte n t i o n has b e e n given to VI-A.
c o n s t r u c t e d 12 items,

initially,

Aiken

on E-S cale

initially,

(1974)

(Enjoyment of

Mathematics)

and 11 items,

on V - S c a l e

(Value of

Mathematics)

and random ly ar ranged these in a format of

the L ike rt type to get h e r w i t h 17 other items con cer ned w i t h
interests,

achievement,

and other b i o g r a p h i c a l information.

137
The res ulting 40 -item o p i n i o n n a i r e wa s a d m i ni ste red to 100
wo men and 9 0 m e n of a freshman class at a s out h-e ast ern
college.

Completed,

usable o p i n i o nna ire s w e r e re tu r n e d by

9 8 w o m e n and 8 7 men.
An analysis of these s h o w that the final 11-item
E- Scale used in the p r e s e n t study has a high internalconsist enc y reliabi lit y of

.95 and the final 10-item V-

Scale used in the p r e s e n t study has a m o d e r a t e l y high
in ter na l - c o n s i s t e n c y re l i a b i l i t y of

.85.

Inf orm ati on on

the val idity of the two scales w e r e o b t a i n e d by co rre lat ing
the total scores on E and V w i t h verbal and mat h e m a t i c a l
scores on the Sc hol ast ic A p t i t u d e Tests

and w i t h T -score

equivale nts of rank in h i g h school g rad ua t i n g class.

The

analyses of the co r r e l a t i o n analysis in dicate that the EScale is m o r e h ighly rel a t e d to m easures of m a t h e m a t i c a l
ab ility and interest, w h e r e a s the V- S c a l e is m o r e highly
c o r r el ate d w i t h measure s of verbal and g e n e r a l - s c h o l a s t i c
ability.
The E and V scales w e r e a d m i n i s t e r e d to
the end of instruction.

toward

T h e i r scores on these together

wi th other scores w e r e used,

via st epw i s e r e g r e s s i o n method,

to pre d i c t their scores on Dos set t's post-test.
De s i g n of Study
Wittrock

(1969)

m a i n t a i n e d that to eva lua te

instruc tio n one m u s t first m e a s u r e at least three components
of instruction:

(1) the e n v i r o n m e n t s of learning,

(2)

the

138
i n t e l lec tua l and s ocial pro cesses of learners,
learning.

He w e n t further s tating that:

and

(3) the

(4) the r e l a t i o n ­

ship b et w e e n these three parts of i n s t r uct ion m u s t then be
q u a n t i t a t i v e l y estimated.

W i t t r o c k di scu s s e d the e v a l u a t i v e

ac tiv i t i e s that c o u l d be ca rried out under each of these.
The non qua n t i t a t i v e e val ua t i o n of the e n v i r o n m e n t of
learning has b e e n d i s c u s s e d in various sections under the
" i n t r i n s i c ” p a r t of this evaluation.

The r e m a i n i n g three

activiti es come under our "pay-off" evaluation.
A.

The E v a l u a t i o n of L e a r n i n g
This c omp o n e n t makes e x p l i c i t the changes in

students'

b e h a v i o u r to try to d ete r m i n e w h a t had been

learned d u r i n g instruction.

This e val uates relativ ely

p e r m a n e n t change in b e h a v i o u r o c c u r r i n g as a r esult of the
experience.

This in s t r u c t i o n c o n s i s t e d of five units;

Measurement;

Numeration, A d d i t i o n and S u b t r a c t i o n of Who le

Numbers,

M u l t i p l i c a t i o n and D i v i s i o n of W h o l e Numbers,

Fractions.
The r ese a r c h d e s i g n o v e r time,

u s i n g C a m p b e l l and

Stanley 's notation,

was a one g r o u p p r e - t e s t - p o s t - t e s t design.

w a s a set

of five p re- t e s t s c o r r e s p o n d i n g the five units, 0 2 w a s a
se t of p o s t -te sts c o r r e s p o n d i n g to the units and X was the
set of five units.

The p r e - t e s t s w e r e u s e d to m o d i f y the

139
d e p e n d e n t var ia b l e ,

post-test,

i n s t r u c t i o n a l pr og r a m .
of the p r o c e s s
ham,
B.

to a sse ss

the e f f e c t o f the

It p r o v i d e d a f o r m a t i v e e v a l u a t i o n

u s i n g e x t e r n a l s ou r c e s

(Sanders and C u n n i n g ­

1973),
The E v a l u a t i o n of L e a r n e r s
T h e s e c o n d way,

s u g g e s t e d by W i t t r o c k

(1969),

to

e v a l u a t e i n s t r u c t i o n is to m a k e e x p l i c i t the students'
abilities,

interests,

an d a c h i e v e m e n t s

to d e t e r m i n e s t u d e n t

p e r f o r m a n c e at the e n d of i n s t r u cti on.
the i n t e l l e c t u a l a n d s o c i a l p r o c e s s e s

This,

thus,

of learners.

w a s don e in this s t u d y b y a s s e s s i n g c h a n g e s
basic mathematical knowledge and attitude

discuss
Thi s

in the interns'

toward m a t h e ­

matics .
Th e r e s e a r c h d e s i g n o v e r time,

°11
wa s

X

°12

°21

a o n e - g r o u p p r e - t e s t - p o s t - t e s t design,

X

°22

0 ^ 1 ant* ° 1 2 w e r e

pr e a nd p o s t of D o s s e t t ’s tes t of B a s i c M a t h e m a t i c a l
U n d e r s t a n d i n g of p r o s p e c t i v e e l e m e n t a r y s c h o o l

teachers,

w h i l e ^ 2 1 ant^ °22 w e r e P re and. p o s t of D u t t o n ' s A r i t h m e t i c
In ven tor y,

T h i s p r o v i d e d a f o r m a t i v e e v a l u a t i o n of the

product using external source

(Sanders a n d C u n n i n g h a m ,

1973).
C.

Th e E v a l u a t i o n of I n s t r u c t i o n
T h e p u r p o s e of this is to d e t e r m i n e

of m e t h o d of i n s t r u c t i o n

(three

levels)

(1)

the e f f e c t

an d e n t r y a t t i t u d e

140
to w a r d m a t h e m a t i c s

(three levels)

u p o n the m a t h e m a t i c s

a c h i e v e m e n t at the e n d o f i n s t r ucti on,
m e t h o d of i n s t r u c t i o n
aptitude

(three-levels)

(three-levels)

(2)

the e f f e c t of

and mathematical

up on the a t t i t u d e t o w a r d m a t h e ­

m a t i c s at the end of in s t r u c t i o n .

This provides

formative

e v a l u a t i o n u s i n g c o n t e x t u a l and e x t e r n a l s o u r c e s
and C u n n i n g h a m ,

1973,

p.

(Sanders

229).

Th e des igns o v e r t i m e ,

G2 :
G3!

°2

X1

°3

°i

°2

X2

°3

°i

°2

X3

°3

(2)

G lS
G2 :
G3:

H
O

Gj^s

H
O

(1)

°2

X1

°4

°1

°2

X2

°4

°1

°2

X3

°4

were non-randomized contro1-group pre-test— post-test
designs.

The treatments

( X 1s)

are

X ^ — T e a c h e r C o r p s M a t h e m a t i c s E d u c a t i o n P rog r a m ,
X£— Experimental Mathematics Content-Method-one weekly
F i e l d E x p e r i e n c e I n t e g r a t e d Pr ogram,
X ^ — R e g u l a r Program,
a n d G^,

G 2 , G^ w e r e

tr eat men ts.

the three g r o u p s

that received these

an d O 2 w e r e p r e - t e s t s of D o s s e t t ' s

tes t of

B a s i c M a t h e m a t i c a l U n d e r s t a n d i n g and D u t t o n ' s A t t i t u d e
Inventory.

an d

were post-tests

of D o s s e t t ' s

t e s t of

B a s i c M a t h e m a t i c a l U n d e r s t a n d i n g and D u t t o n ' s A t t i t u d e
Inventory.

In

(1)

was

us ed as a c o v a r i a t e

to m o d i f y

dependent variable

(criterion)

assigning students

to le vel s of an i n d e p e n d e n t var iab le,

attitude.

In

the

O^, w h i l e O 2 w a s u s e d in

£2) C>2 w a s u s e d as a c o v a r i a t e to m o d i f y the

141
dependent variable

(criterion)

0 4 , while

w a s u s e d in

assigning students

to levels o f a n i n d e p e n d e n t var iable,

m a t h e m a t i c a l aptitude.
Th e de si g n s o v e r v a r i a b l e s w e r e ;
Treatment Group

(1 )

_

.

ti2

G3

<D

1 High

!

G1

•

+J T t
0)
E
S
O
i-q

Treatment Group
0

(2 )

H

C 2

Low

Med.

High

....

In

(2)

the i n d e p e n d e n t v a r i a b l e s w e r e a t t i t u d e t o w a r d

m a t h e m a t i c s and t r e a t m e n t group,

the 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 a c h i e v e m e n t on the p o s t - t e s t ,

for m B, D o s s e t t ' s

test w h i l e the a c h i e v e m e n t on p r e - t e s t of D o s s e t t ' s
wa s u sed as covaria te.

In

(2)

te st

the i n d e p e n d e n t v a r i a b l e s

w e r e m a t h e m a t i c a l a p t i t u d e and t r e a t m e n t group,

the

142
de pendent v ariable was s tud e n t a c h i e vem ent in Dutton's
At ti t u d e Inventory used as a p o s t - t e s t w h i l e the a c h i e v e ­
m e n t on the same test used as p r e - t e s t was used as c ovari­
ate.

In each case the h ypotheses w e r e tested via a two-way

Analysi s of C o v a r i a n c e at the a =

.05 level.

D.
The Q u a n t ita tiv e Eva lua t i o n
of Environ men ts o F L e a r n i n g
Wh i l e no n - q u a n t i t a t i v e evalua tio n of en vironments
of learning has been done in various sections under
" i n t r i n s i c ” evaluation,

an attempt is made here to quantify

the students'

p erc ep t i o n of their learning environments.

Specifically,

the interns,

G^, and other two groups we re

co mpared on their percepti ons of M a t h e m a t i c s as a process
(B^) , diffic ult ies in learning m athematics
of m a t h em ati cs in the society
and school learnin g
e n v i r onm ent

(B 2^ * t*ie P ^ ace

(Bc ) , at titude toward school

(B^), and attitude toward man and his

(Bg).

Gl :

°1

X1

°2

G2 :

°1

X2

°2

G3:

°1

X3

O
to

The de s i g n over time,

was a two-fa cto r by o ne - w a y rep e a t e d m e a s u r e design.
treatments

The

(X's) w e r e g roup treatments as d i s c u s s e d above,

the p r e - t e s t O^, was Dutton's At ti t u d e
as signing students to levels o n e - f a c t o r
v a r i a b l e ) , attitude.

Inventory u s e d in
(an in d e p e n d e n t

O 2 was the d e p e n den t var iable wi th

143
five c o m p o n e n t s , B^,

B c * B 4 * B 5 b e i n g scores on five

me asures in the p r e c e d i n g paragraph.
The design over var i a b l e was,
Two-way Multivariate Design
ATTITUDE
HIGH
B1

B2

B3

MEDIUM
B4

B5

B1

B2

B3

LOW
B4

B5

B1

B2

B3

B4

B5

G1

&
§
O
Eh
Z

£

s
ah
E

°2

G3

The two factors w e r e gro ups

(three levels)

and attitude

(three l e v e l s ) .
Ana l y s i s of Data
The i n v e s t i g a t o r s ele c t e d several st ati sti cal
procedu res to analyze the data col le c t e d during the study.
A on e - w a y m u l t i v a r i a t e analysis of va ri a n c e t e c h ­
nique as d e s c r i b e d by W i n e r

(19 71) was s ele c t e d for testing

the s i g n i fi can ce of the g a i n b etw een the p o s t - t e s t scores

144
and the p r e - t e s t scores o f the interns
c r i t e r i o n - r e f e r e n c e d m easu res
p r o g r a m ve rsion 4

(1968)

(G^)

on the five

(Hypothesis A).

Finn's

on analysis of gain scores was

utilized.
To test the hy pot hes es related to the ef f e c t of the
mathemat ics c u r r i c u l u m on the b a s i c m a t h e m a t i c a l u n d e r ­
st anding

(Hypothesis Bl)

(Hypothesis B2)

and attitudes toward arithmetic

one - w a y mult i v a r i a t e analysis

(and the corresp ond ing univar iat e techniques)
was used.

Finn's

(1968)

technique
on gain scores

p r o g r a m on analysis of gain scores

was utilized.
To assess the rel ati ve pe r f o r m a n c e of the interns
( G ) as c o m p a r e d w i t h two o t h e r groups
basic mat h e m a t i c a l u n d e r s t a n d i n g
attitude toward a r i t h m e t i c

(G2 * G 3 ) (1) on the

(Hypothesis C l ) f (2) on

(Hypotheses C 2 ) , a 3 x 3

An alysis of C o v a r i a n c e was used in each case.
To compare the interns,
on

(their p er c e p t i o n s of m a t h em ati cs

ent aspects of attitudes
analysis of va riance

and the two other groups
learning)

(Hypothesis D) a three-f act or

(one-factor w i t h r epeated measures)

technique was used as s u g g e s t e d by W i n e r
Finn's

five d i f f e r ­

(1971,

559-69).

(196 8) p r o g r a m on on e-way r epe a t e d me as u r e s was

utilized.
To de ter min e h o w w e l l the scores on p ost -t e s t s of
the c r i t e r i o n - r e f e r e n c e d mea s u r e s and p r e - t e s t on Dos sett's
test of m a t h e m a t i c a l u n d e r s t a n d i n g p r e d i c t interns'

(G^)

scores on p o s t - t e s t of Dos sett's test of mat h e m a t i c a l

145
u n d e r s t a n d i n g , to es ti m a t e the p r o p o r t i o n of v ari anc e in
po st- t e s t score of Dossett's

test a cco unted for by each of

learning unit, and to assess

the r elative order of im por ­

tance of the predictors, m u l t i p l e regress ion and stepwise
re gre ssi on techniques w e r e employed.

Finn's

(196 8) p rogram

was again utilized.
Fi na l l y to d ete r m i n e h o w w e l l the scores o n Dutton's
pre-test,

five aspects of attitude and A iken's E and

V-Scales p r e d i c t interns
At ti t u d e Inventory,

(G-^) score on p o s t - t e s t of Dutton's

to estim ate the p r o p o r t i o n of variance

in p o s t - t e s t score of Dutton's A t t i t u d e Inventory accounted
for by each of the eigh t

(measures)

pr edi cto rs and to

assess the r ela tiv e order of importance of the predictors,
mu lt i p l e r eg res sio n and stepwis e r e g r e s s i o n techniques we re
employed.
Si gni fic anc e L evel Chosen
The

.05 of si gni fic anc e

for the r eje ction of sta­

tistical h y p o t h e s e s be i n g i n v e s t i g a t e d w as se lected as
b e i n g s u f f i c ien tly rigorous for the c onditions of this
study.

Thus,

if the p r o b a b i l i t y was at or less than five

times in one hu n d r e d that the o bse r v e d dif fer e n c e cou ld be
a t tri but ed to chance,

the re se a r c h h ypo the sis was accepted;

if the o b s e r v e d di ffe r e n c e was of such m a g n i t u d e that it
m i g h t arise m o r e than five times in one hun d r e d through the
o p e r a t i o n of chance factor,
rejected.

the res earch hy pot hes is was

146
M e t h o d o l o g i c a l A s s u m p t i o n s a nd L i m i t a t i o n s
of the St udy
One Group Pre-Test—
P o s t - T e s t D esi gns
In a s s e s s i n g the e f f e c t of the i n s t r u c t i o n a l p r o g r a m
o n the a c h i e v e m e n t of int e r n s o n the p r e s c r i b e d m a t h e ­
matical competencies,

w e w i s h e d to t e s t the null h y p o t h e s i s

of e q u a l i t y o f p e r f o r m a n c e w i t h r e s p e c t to al l d e p e n d e n t
variables
tests)

(five u n i t s ) a cro ss

(Hypothesis A ) .

p a r a m e t r i c test
hypothesis

the m e a s u r e s

In s t a n d a r d m u l t i v a r i a t e an al y s i s

(employed in the study)

to te st this

it is u s u a l l y a s s u m e d that

X ij ~ w + a j + ei + e ij

j = If
i = 1,

w h e r e y,

(pre- an d p o s t ­

2.

(K)

2,...,

24.

(n)

a j , a nd 0^ are c o n s t a n t v ec t o r s o f p o p u l a t i o n m e a n s

(elements of this v e c t o r are o v e r a l l m e a n s on e a c h d e p e n d e n t
variable),

treatment

(subjects)

effects.

(measurement)

e ffe cts ,

T h e v e c t o r s e^j are

48

and block
(nk)

i n d e p en den t,

normally distributed vectors wit h a common variancec o v a r i a n c e m atrix.

F r e q u e n t l y some o r all of th ese

a s s u m p t i o n s c a n n o t be j u s t i f i e d e s p e c i a l l y b l o c k a d d i t i v i t y
an d normal ity .

In this s t u d y a s s u m p t i o n of n o r m a l i t y

is

i n c o n s i s t e n t w i t h the u n d e r l y i n g the ory of m a s t e r y - l e a r n i n g
(chapter II).

It is,

th ere for e,

p a r a m e t r i c m u l t i v a r i a t e approac h.

i l l o g i c a l to use
A

the

logical tes t w o u l d

have b e e n G e r i g ' s m u l t i v a r i a t e e x t e n s i o n of the F r i e d m a n

147
test

(Gerig,

1969),

a n o n p a r a m e t r i c test w h i c h r e l a x e d the

a s s u m p t i o n s of b l o c k a d d i t i v i t y a n d n o r m a l i t y .
ga t o r attemp ted ,

b u t failed,

The

investi­

to s e c u r e a c o m p u t e r p r o g r a m

that c o u l d do this fro m the o f f i c e of r e s e a r c h c o n s u l t a t i o n
at the M i c h i g a n S t a t e U n i v e r s i t y .

T h e u n i v a r i a t e p a r t s of

h y p o t h e s i s A w o u l d h a v e b e e n t e s t e d b y F r i e d m a n or W i l c o x o n ’s
tests

(Conover,

1971).

T h e s e rem a r k s h o l d for t e s t i n g

h y p o t h e s e s B as w e l l if the g a i n s cor es w e r e due to the
e f f e c t of i n s t r u c t i o n

(mastery l e a r n i n g ) .

A w o r d is in o r d e r a b o u t the w e a k n e s s e s

of the one-

g r o u p p r e - t e s t — p o s t - t e s t d e s i g n u s e d in a s s e s s i n g the
e f f e c t of the i n s t r u c t i o n a l program.
c o n t r o l l e d for "selecti on"
(since the same s u b j e c t s

and

W h i l e this d e s i g n

"mortality" v a r i a b l e s

took tests

a s s u r a n c e t h a t the i n s t r u c t i o n

a n d O^)

(X) w a s

m a j o r f a c t o r in C>2— 0^ d i f f e ren ce.

there

is no

the o nly or ev en

P l a u s i b l e rival causes

c o u l d h a v e b e e n h i s t o r y , m a t u r a t i o n , t es t i n g e f f e c t s ,
c h a n g i n g e f f e c t s of i n s t r u m e n t a t i o n , s t a t i s t i c a l r e g r e s s i o n .
A n o b v i o u s w a y to c o n t r o l this is the use of c o n t r o l
groups.

Hively,

et al.

(19 73, p.

35)

a r g u e d th at w h e n on e

is p r i m a r i l y i n t e r e s t e d in f i n d i n g o u t w h a t c u r r i c u l u m can
do a nd w h e t h e r it s a t i s f i e s its o w n o b j e c t i v e s
g r o u p s are n o t useful.

then c o n t r o l

Of these five p l a u s i b l e r iva l

causes, m a t u r a t i o n a n d t e s t i n g e f f e c t s s e e m to t h r e a t e n the
i n t e r n a l v a l i d i t y of the d esi gns .
of the a t t i t u d e i nventory,

Since, w i t h the e x c e p t i o n

r e g u l a r c l a s s r o o m tests w e r e

u s e d as 0 ^ ’s i n t e r a t i o n o f t e s t i n g a n d t r e a t m e n t

(X) w h i c h

148
usually threatens ex ternal va lidity of this d esign is not
a threat in this study.

The same remark holds

for in ter ­

action of selection and treatment.
Tw o-way A N C O V A Designs
In determin ing
st ruction
m atics
levels)

Cl) the effect of me t h o d of i n­

(three levels)

(three levels)

and entry attitude toward m a t h e ­

and math e m a t i c a l aptitude

(three

o n the attitude toward ma the mat ics at the end of

instruction,
gator was

it should be m e n t i o n e d that w h i l e the i n v e s t i ­

aware of the differe nce s in time spent by these

groups on instruction,

Kirk

(1968, p.

457)

showed that it

is inc orrect to ad just the dep en d e n t var iable for the
co n c o m i t a n t variable,

time.

Differ enc es in learning

ab ility o r m a t h e mat ica l b a c k g r o u n d or attitude toward
m a t h em ati cs m i g h t exist bet w e e n these G^, G 2 , and
to the i n t r o duc tio n of the i n s t r u cti ona l methods,
extraneous variables w o u l d bias the evaluation.

pr ior
these
Since

previou s studies have in dic a t e d that these variables have
some effects on the a c h i e v eme nt of an ins tru cti ona l method,
it was,

therefore,

n e c e s sar y to co ntrol them statistically.

Two me t h o d s have been s u g g est ed for this, p r o v i d e d that the
co n c o m ita nt variables are measurable.
First,
covariables,

measure the c o n c o m i t a n t variables,

ca lled

in ad dition to the va riate of pri m a r y i n t e r ­

est, termed the criterion,

and in this case use analysis

of covariance s i n g l e - f a c t o r design.

Winer

(1971)

sug gested

149
the second m e t h o d
menter,

(p. 780).

In this method,

an e x p e r i ­

rather than using analysis of c ova ria nce

(ANCOVA)

m i g h t att empt to use covariate as a c l a s s i f i c a t i o n or
s t r a t i f i c a t i o n factor.

If this w e r e done in an ex p e r i m e n t

i n volving only one covariate,

the e x p e ri men t w o u l d be

a n alyzed as a two-factor ANOVA.

This m i g h t r es u l t in quite

small cell frequencies and po ss i b l y no entries
cells.

in some

If eac h of the resulting cell f requencies is

r elatively large,
Co c h r a n

(1957),

say five or more, Wi n e r

(1971),

following

is of the opinion that this type of

s t rat i f i c a t i o n on the covariate is g e n e r a l l y to be p r e f e r r e d
to ANCOVA.

Moreover,

Cox

(1957)

found that r a n d o m i z e d

block design is b e t t e r than A N C O V A if the c o r r e lat ion
b e t w e e n c r i t e rio n an d c o n c o m i t a n t variable,

p, is less than

0,6 w h i l e A N C O V A is appreci abl y b e t t e r than randomiz edb l o c k design if p is g re a t e r than 0.8.

A c c o r d i n g to C ox

(195 7) no p ref er e n c e b et w e e n the two w h e n
(1957)

.6 <_ p <_ .8.

Cox

p o i n t e d out that w her e tre atment eff e c t is not

s u s p e c t e d to be ent i r e l y i n d e p end ent of the co n c o m i t a n t
v a ria ble

(that is w h e r e there m i g h t be tre atm ent by c o n ­

c o mitant v a r i a b l e i n t e r a c t i o n ) , such an i n t e r a c t i o n m igh t
give u seful i nsight into the m e c h a n i s m u n d e r l y i n g the
t r eatment effects and mi g h t also change any p r a c t i c a l
r e c o m m e n d a t i o n s to be m a d e in the study.
In this study,

two covariates are of importance,

m a t h e m a t i c a l aptitude and entry p o i n t attitude,

and this

wi ll res ult in t h r e e -f act or e x p e r i m e n t — each of the

150
c o v a r i a t e s h a v i n g at l east two levels

(High, L o w ) .

Previous

st u d i e s h a v e i n d i c a t e d that it is b e t t e r to c o n s i d e r three
levels
Low.

for e a c h of t hes e c o v a r i ate s,
This w o u l d

i n t o nine cells.

force e a c h g r o u p ,
Since

viz;

High, Ave rag e,

, to b e d i s t r i b u t e d

had 24 m e m b e r s a n d G 2 h a d o n l y

21 m e m b e r s , this w o u l d n ot s a t i s f y the c o n d i t i o n s
which Winer prefers

under

this s t r a t i f i c a t i o n m e t h o d to ANCOVA.

Consequently,

the i n v e s t i g a t o r d e c i d e d to use one

of the c o v a r i a t e s — p r e - t e s t sco re on a t t i t u d e t o w a r d
a r i t h m e t i c as a s e c o n d factor, w h e n the c r i t e r i o n was p o s t ­
test s c o r e o n D o s s e t t 's test w h i l e us ing p r e - t e s t sc ore on
the sa me t e s t as c o v a r i a t e .

The same r e a s o n i n g was

carried

o u t in d e t e r m i n i n g the e f f e c t of i n s t r u c t i o n o n the
af f e c t i v e b e h a v i o u r of G^,

G 2 , G^.

Th us

a 3 x 3 A N C O V A was

u s e d in e a c h case of h y p o t h e s e s Cl and C2.

M o r e o v e r the

c l a s s i c a l p a r a m e t r i c A N C O V A test has b e e n used.
assumes

test

l i n e a r i t y of r e g r e s s i o n of the c r i t e r i o n on

co var i a t e ,

no rma l i t y ,

a n d others.

n o r m a l i t y m i g h t h o l d for G 2 and G^,

W h i l e the a s s u m p t i o n
it is q u e s t i o n a b l e

G^ if the a c h i e v e m e n t in the c r i t e r i o n w as d ue
struction
and Sen

This

(mastery-learning).

for

to in ­

Rank A N C O V A p r o p o s e d by Puri

(1969) w h i c h does n o t a s s u m e n o r m a l i t y and l i n e a r i t y

w o u l d h ave b e e n u s e d .

T h e i n v e s t i g a t o r c o u l d n o t do this

for r e a s o n g i v e n e a r l i e r .
It s h o u l d b e m e n t i o n e d t h a t w h i l e i n t e r a c t i o n of
s e l e c t i o n an d m a t u r a t i o n , and i n t e r a c t i o n o f t e s t i n g and
t r e a t m e n t w h i c h are n o r m a l l y s o u r c e s o f i n t e r n a l and

151
e xternal invalidity to n on- equ iva len t control-group,
q u a s i - e x p e r i m e n t a l design,
study,

a

are not threaten ing in this

it is n o t c ertain that reg res s i o n is n o t a source of

internal invalidity.
No n - S t a t i s t i c a l Unc o n t r o l l e d
Sources
A m o n g n o n - s t a t i s t i c a l factors that could affect the
d i ffe ren ces at the end of instructions

(apart from the

instructions)

i nstructors e x p e r i ­

ence,

are instructors ability,

class constancy,

u npl anned changes in instructions,

the p r o b l e m of m e a s u r i n g understanding,

the p r o b l e m of

c o n s t r uct ing or o bta i n i n g testing i nst ru m e n t that w o u l d
favor no group a nd e xp e r i m e n t a l s t i m u l ati on
effect).

(Hawthorne's

Most of these could not be c o n t r o l l e d in the

compara tiv e parts of the study and should,

therefore,

be

noted as wea kne s s e s of the study.
Summary
This cha p t e r d e s c ri bed the m a t h e m ati cs

compon ent

of the ei ghth cycle T eachers Corps p r o g r a m at M i c h i g a n State
Un ive r s i t y and the p roc edu res followed for its assessment.
The formative eva lua tio n of the m a t h e mat ics c o m p o ­
nent of the eighth cycle T e a c h e r Corps p r o g r a m at M i c h i g a n
State Uni ver s i t y took p lac e d uring the a c a d e m i c y e a r 197319 74.

T w e n t y - f o u r out of thirty interns w h o w e r e ori gin all y

a d mi t t e d into the p r o g r a m w e r e utilize d for this evaluation.
Two of the interns withdrew,

two fell ill, and two w e r e

152
randomly dro p p e d to o b t a i n an o r t h o g o n a l design.
addition,

In

samples of o t h e r st udent groups were used for

compari son purposes.
The following steps were

followed for the e v a l u ­

ation of the program:
1.

G e n e r a l con tex t and p r o g r a m description.

2.

An al y s i s of the m a t h e m a t i c s con t e n t in the p r o g r a m
by means of a c r i t e r i o n - r e f e r e n c e d list d e v e lop ed
a c c o r d i n g to topics s ugg e s t e d by specialists in the
p r e p a r a t i o n of e l e m e n t a r y school teachers,

3.

A p p r a i s a l of the ma t h e m a t i c s met h o d s i nte gr a t e d w i t h
m a t h e m a t i c s con t e n t and cli nical experience.

4.

A p p r a i s a l of the in st r u c t i o n a l m e t h o d used in the
program.

5.

A s s e s s m e n t of the content of the i n t e g r a t e d contentm e t h o d s course by means of the following i n s t r u ­
ments :
a.

Five c r i t e r i o n - r e f e r e n c e d a c h i e v e m e n t mea s u r e s
to assess m a t h e m a t i c a l c o m p e ten cie s on
p r e s c r i b e d objectives.

b.

Test of Bas ic M a t h e m a t i c a l U nde rstanding.

c.

A t t i t u d e In ven t o r y an d A t t i t u d e Scales.

Th e d e v e l o p m e n t and use of the test instrum ent s as w e l l as
the s t a t i st ica l p r o c e dur es used for the a nalysis of data
w e r e d e s c r i b e d in the last sec t i o n of this chapter.
R e sults o b t a i n e d f r o m the d i f f e r e n t ana lyses and
their i n t e r p r e t a t i o n are d i s c u s s e d in the following chapter.

CHAPTER IV
PRESENTATION AND ANALYSIS OF DATA

This c h a p t e r p r e s e n t s a sum m a r y of the d a t a
c o l l e c t e d d u r i n g this i n v e s t i g a t i o n ,
a n d findings b a s e d o n this analysis.

the a n a l y s i s of data
It c o n s i s t s of e i g h t

sections:
1.

A n a l y s i s of the m a t h e m a t i c s

c o n t e n t and m e t h o d s of

the l e a r n i n g u n i t s ;
2.

the e v a l u a t i o n of

l e a r n i n g o b t a i n e d by a s s e s s i n g

the in te r n s p e r f o r m a n c e o n the c r i t e r i o n - r e f e r e n c e d
a c h i e v e m e n t m e a s ur es?
3.

the e v a l u a t i o n of lea rne rs w h i c h p r o v i d e s

the e f f e c t

of the t e a c h e r c orps p r o g r a m on the b a s i c m a t h e ­
matical understandings

and attitude

toward m a t h e ­

m a t i c s of the i n t e r n s ;
4.

the e f f e c t of i n s t r u c t i o n and a t t i t u d e on m a t h e ­
matical understandings;

5.

the e f f e c t of i n s t r u c t i o n and m a t h e m a t i c a l a p t i t u d e
on attitude toward mathematics;

6.

the e f f e c t of i n s t r u c t i o n and a t t i t u d e on s tud e n t s
p e r c e p t i o n of m a t h e m a t i c s

153

l e a r n i n g a n d en vi r o n m e n t ;

154
7.

the cont r i b u t i o n of the learning units to basic
m a t h e m a t i c a l understandings of the interns;

8.

the r e l a t ions hip between the attitude toward m a t h e ­
matics and the interns percep t i o n of mathematics
learning,

enjoyment,

value and environment.

The c h apter ends w i t h a summary of results.
Anal y s i s of the M a t h e matics Content
and Met h o d s of"the L e a r n i n g Units
Like other c u r r i c u l u m projects,

the most logical and

conve n ient seg m e n t for analysis in the mathematics e d u ­
cation component of the T e a c h e r Corps p r o g r a m was the
"unit" since this was treated by the

(competency-based)

p r o j e c t as the a d m i n i s t r a t i v e and the theoretical bu i l d i n g
block.

The use of the unit is therefore,

fundamental to

formative e v a l u a t i o n in general and helpful in provi d i n g
learn i ng correctives in p a r t i c u l a r

(Keller,

196 8, Bloom,

et a l . , 1971).
There w e r e

five learning units in this study,

each

conta i ned one m a t h e m a t i c a l topic and methods of teaching.
Findings
Item by item c o m p a r i s o n bet w e e n Hicks and Pe r r o d i n
c r i t e r i o n - r e f e r e n c e d list

(1967)

and the m a t h e m a t i c a l

contents of the five learning units covered shows that the
following topics are not included, or covered to be precise,
in the m a t h e m a t i c s c u r r i c u l u m of the T e a c h e r Corps program:

155
1 . Cardinality
2.

Divisibility Rules

3.

Percentages

4.

Ratio and Proportions

5.

Square Root

6.

Formulae and Substitution

7.

Basic Concepts of Geometry

8.

Equation and Symbols

9.

Inequations

10.

Central Tendency

11.

Statistical Graphs

12.

Probability

13.

Coordinate Geometry

One s hould not rush to a c o n c l u s i o n that the mathema t i c s
content of this p r o g r a m is shallow because of the absence
of thirteen out of the thirty-six topics in the criterionlist.
topic

It should be m e n t i o n e d that every m a t h e m a t i c a l
(not methods)

in the five covered units is already

c o n t a i n e d in the criterion-list.

The units contain both

c ontents and methods of teaching them.
Discussion
The Tea c h e r Corps Trai n i n g Objectives as vie w d by
L a nsing School Distr i c t includes,
To conduct a mo d e r n m a t h p r o g r a m wh i c h emphasizes
structure w i t h o u t losing p r o f i c i e n c y in computation,
a nd w h i c h can be a p p l icable to low-income children's
experiences with numer i c a l variables. . . .

156
The succ e s s or failure in a c h i e v e m e n t of this
o b j e c t i v e c a n n o t be
the m a t h e m a t i c s

fully a n a l y z e d h e r e just by a n a l y z i n g

content.

can be e x a m i n e d is ".
computation."

The p a r t of the o b j e c t i v e

that

. . w i t h o u t losing p r o f i c i e n c y in

T h e q u e s t i o n that m i g h t be r a i s e d h e r e is

"how h i g h can the c o m p u t a t i o n a l p r o f i c i e n c y of the s e i n t e r n s
be w i t h o u t e x p o s i n g t h e m to those m i s s i n g m a t h e m a t i c a l
topics."
terms.

T h i s q u e s t i o n c a n n o t be a n s w e r e d in abso l u t e
T h i s is w h e r e A s t i n and P a n o s ' (1970)

v i e w that,

the n a t u r e of e v a l u a t i v e r e s e a r c h is that the i m p a c t
of any e d u c a t i o n a l p r a c t i c e or p r o g r a m can be a s s e s s e d
o n l y by c o m p a r i s o n w i t h some a l t e r n a t i v e p r a c t i c e s or
programs. . . .
The p u r p o s e of the c o n t e n t a n a l y s i s
m i s s i n g eleme n t s ,

is to p o i n t to those

the i n f o r m a t i o n m i g h t be useful in

explaining differences

(success or failure)

of the i m p a c t of the p r o g r a m on the interns,
w h i c h w i l l be c a r r i e d out b e l o w .
of the m a t h e m a t i c s

in m a g n i t u d e
an e x e r c i s e

At present a comparison

c o n t e n t o f the p r o g r a m w i t h two or

t h r e e o t h e r p r o g r a m s e x i s t i n g or that have e x i s t e d on the
c a m p u s m i g h t h e l p in m a k i n g o n e
the T e a c h e r Co r p s p r o g r a m
suffices

(a bit)

comfortable with

(see the table b e l o w ) .

It

to say that e m p h a s i s on m a s t e r y o f m a t h e m a t i c a l

c o n t e n t as o p p o s e d to c o v e r a g e is s t i l l an issue in m a t h e ­
m a t i c s ed u c a t i o n .
four m a j o r issues.
examine

Ward

(1970)

d i s c u s s e d this as o n e of

A t this stage it is n e c e s s a r y

to

f u r t h e r the m a t h e m a t i c s e d u c a t i o n o b j e c t i v e of the

T e a c h e r C o r p s p r o g r a m p r o p o s a l and ask:

157
Table 4.---A Comparison of Mathematical Topics Covered by the Teacher
Corps, the Regular Elementary Education Program, and the TTT
Experimental Program.

Topic
1 . Set Terminology
2. Set Operations
3. Relations £ Functions
4. Whole Number Operations
5. Counting and One-to-One
Correspondence
6. Order and Cardinality
7. Field Operations
e. Different Numeration
Systems £ Place Value
Ancient
Numeration Systems
9.
10. Roman Numeration
11. Primes and Composite
12. Factors and Multiples
13. Exponents £ Exponential
Notations
14. Divisibility Rules
15. The Number Line
16. Common Fractions
17. Decimal Fractions
18. Percentages
19. Ratio £ Proportions
20. Real Numbers
21. Square Root
22. Measurement
23. Precision and Error
24. Formulae £ Substitution
25. Basic Concepts of Geometry
26. Geometric Figures
27. Metric system £ Conversion
2B. Equations and Symbols
29. Inequations
30. Central Tendency
31. Statistical Graphs
32. Probability
33. Problem Solving
34. Making Estimations
35. Rationalizing Algorithm
36. Coordinate Geometry

Teacher
Corps*

Regular
Elem. Ed.
Program

Another Elem.
E d . Expl.
Class Taught
TTT
by T.C.
Project
Instruction
(1972)

X

X

X

X

X

X

X

X

X

X

X

X

X

X
X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X
X

X

X

X

X

X

X

X

X
X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

*x means the topic was taught.

158
H a ve t h e s e interns b e e n t a u g h t all m a t h e m a t i c a l topics
" a p p l i c a b l e to l o w - i n c o m e c h i l d r e n e x p e r i e n c e s ? "
This s h o u l d be p r e c e d e d by,
W h a t m a t h e m a t i c a l topics are n o t a p p l i c a b l e to lowi n come c h i l d r e n ' s e x p e r i e n c e s ?
as the last p a r t o f the p r o j e c t o b j e c t i v e p r e s u p p o s e s .

This

p a r t of the o b j e c t i v e i s , in f a c t , c o n t r a r y to p o p u l a r v i e w
like those of A d l e r
Bloom

(1957),

(1969) , S i l b e r m a n

Bruner

(1970) , A d l e r

specific recommendations made
Mathematics Education

(1962),

Carroll

(1972)

(1963),

and on e of the

to SMSG by a C o n f e r e n c e on

for B e l o w A v e r a g e A c h i e v e r s

in 196 4.

T h e r e are r e c o m m e n d a t i o n s , in the l i t e r a t u r e , on m e t h o d s
of t e a c h i n g m a t h e m a t i c s

to the d i s a d v a n t a g e d child r e n ,

none

on w h a t to t e a c h th e s e children.
E v a l u a t i o n of L e a r n i n g
In this p a r t of the study,

the r e s u l t s of p r e - and

p o s t - t e s t scores w e r e u s e d to assess

the e f f e c t o f the

i n s t r u c t i o n a l p r o g r a m o n the a c h i e v e m e n t of i n t e r n s
prescribed mathematical competencies.

in the

The evaluation was

c a r r i e d o u t in two parts:
1.

T o d e t e r m i n e the s i g n i f i c a n c e of g a i n in a c h i e v e m e n t
on the p r e s c r i b e d m a t h e m a t i c a l c o m p e t e n c i e s b e t w e e n
pre-

2.

and f i r s t p o s t - t e s t scores.

T o d e t e r m i n e w h e t h e r a s p e c i f i e d d e g r e e of m a s t e r y
o v e r these c o m p e t e n c i e s has b e e n a c h i e v e d b y the
inte r n s

at the f i r s t p o s t - t e s t o r s e c o n d p o s t - t e s t .

159
Hypotheses T e s t e d
The following m u l t i v a r i a t e hypothesis and a s s o c i ­
ated univariate hypotheses w e r e tested.
A.

There w i l l be no s i g n i ficant differences b e t w e e n
the p o s t - t e s t means and p r e - t e s t means of the
interns on the c r i t e r i o n - r e f e r e n c e d measures.
Symbolically:
Y - X = O
w h e r e Y, a 5 x 1 vector,

is the p o s t - t e s t me a n

scores on the five meas u r e s and X, a 5 x 1 vector,
is the p r e - t e s t m e a n scores on the five measures.
The a s sociated univar i a t e hypoth e s e s also tested were:
The p o s t - t e s t mean of the e x p e r imental group w i l l not
s i gnificantly di f f e r from their p r e - t e s t m e a n on the
c r i t e r i o n - r e f e r e n c e d mea s u r e in:
1.

Measurement

2.

Numeration

3.

A d d i t i o n and S u b t r a c t i o n of W h o l e Numbers

4.

M u l t i p l i c a t i o n a n d Divi s i o n of Wh o l e Numbers

5.

Fractions

Data A n a l y s i s
Data colle c t e d through the a d m i n i s t r a t i o n of prea nd p o s t - t e s t forms o f the c r i t e r i o n - r e f e r e n c e d m e a s u r e s
d e v e l o p e d as d e s c r i b e d in cha p t e r III w e r e used to test
Hypoth esis A ,

160
The I n t e r n s 1 scores on these meas u r e s are p r e s c r i b e d
in appendix F.

D a t a included in the tables in this section

w e r e drawn from appe n d i x F.
Pre- and p o s t - t e s t means,

standard deviations,

and

m e a n differences for the c r i t e r i o n - r e f e r e n c e d meas u r e s are
s hown in Table 5.
Univar i a t e and m u l t i v a r i a t e analysis of var i a n c e
t e c h n ique w e r e used in the analysis of data rel a t e d to
H ypothesis A.
F i n d i ngs
H ypoth e s i s A .— The data in Table 5 s h o w gains made
by interns on all c r i t e r i o n - r e f e r e n c e d measures.
i n c rease ranged from 8.42 to 21.54 points.

The

W h e n the v ector

of m e a n d i f f e rences was tested aga i n s t zero vector,

the

r e s u l t i n g m u l t i v a r i a t e F value w a s 29.38 w h i c h was highly
significant

tp < 0.0001).

B a s e d on this result,

the m u l t i ­

va r i a t e H y p o thesis A w h i c h st a t e d that there w i l l be no
s i g n i f i c a n t differ e n c e b e t w e e n the p o s t - t e s t m e a n s and
p r e - t e s t means of the interns on the c r i t e r i o n - r e f e r e n c e d
m e a s u r es was r e j e c t e d at .05 level of significance.
To e x a m i n e the students resp o n s e to e a c h m e a s u r e
separately,
level.

the u n i v a r i a t e h y p o t h e s e s w e r e tested at .01

Table 6 summarizes the findings for each un i v a r i a t e

h y p o t h eses t h a t w a s tested.
indicated:

Results of the analysis

161
Table 5.—-Means and Standard Deviations of Pre-test, Post-test, and
Gain Scores on the Five Criterion Measures for the Interns.
Pre-test

Post- test

Gains

Variable
Mean

S.D.

Mean

S.D.

Mean

S.D.

Measurement

28.91

11.29

37.63

9.57

8.42

11.06

2.

Numeration

20.33

13.41

41.88

6.17

21.54

12.39

3.

Addition and
Subtraction of
Whole Numbers

24.04

10.51

39.25

8.43

15.21

9.81

Multiplication and
Division of Whole
Numbers

19.83

9.23

33.67

10.90

13.83

9.07

Fractions

13.67

9.29

31.04

9.36

17.38

9.18

1

.

4.

5.

Table 6.— Multivariate Analysis of Interns on Differences Between Preand Post-test Scores on the Five Criterion Measures.

Variable

1

.

Measurement

Between
Mean Square

Univariate
F

Significance
Probability

1700.1667

13.8871

.0012

11137.0417

72.5241

.0001

2.

Numeration

3.

Addition and
Subtraction of
Whole Numbers

5551.0417

57.6677

.0001

Multiplication and
Division of Whole
Numbers

4592.6667

55.7912

.0001

Fractions

7245.3750

86.0041

.0001

4.

5.

162
The instructional treatment of the integrated
c ontent-methods course had a positive ef f e c t on the interns
pe r f o r m a n c e on the five c r i t e r i o n - r e f e r e n c e d measures.

The

univariate tests wh i c h stated that the p o s t - t e s t m e a n of the
interns w i l l not be significantly higher than their p r e ­
test means on each of the c r i t e r i o n - r e f e r e n c e d tested were
re j ected at

.01 level for each of the univariate hypotheses.

The d e t e r m i n a t i o n of a c h i e vement of the spec i f i e d
m a s t e r y level was not strictly statistical.
conseq uences discussed by Mil l m a n
the p a s s i ng-score
measure.

(mastery)

The edu c a t i o n a l

(1973) was used in setting

at 80 per c e n t for each criterion

Each intern was awarded mastery on each criterion-

r e f e r e n c e d m e a s u r e if his raw score was not less than
95 p e r c e n t of the pas s i n g score.^
interns that reac h e d mastery

Table 7 shows nu m b e r of

level out of the twenty-four

interns used in this analysis.
Second post-tests w e r e a d m i n istered to interns w h o
coult not reach mas t e r y

level in the first post-tests.

Students w h o could not reach mas t e r y at the second p o s t ­
test were only given individual help outside c l a s s r o o m until
they w e r e competent in areas in w h i c h they w e r e deficient.
They w e r e n o t given a third c o m p r ehensive post-test.

^Passing score was 40 out of 50 points.
was a w arded if a student

Mas t e r y

163
Table 7.— Num b e r and Percentage of Interns that Rea c h e d
Mas t e r y Level on L e a r n i n g Units.

CriterionR e f e r enced
Measure

N u m b e r of Interns that A t t a i n e d
Mas t e r y Level
—
— — ■ ■—
—
1st P o s t - t e s t
2nd P o s t - t e s t

Total
% of
Interns
(N=24)

Measurement

14

8

91.67

N u m e r ation

20

3

95.83

A d d i t i o n and
S u b t r action of
Whole Numbers

16

5

87. 50

Multiplication
and D i v i s i o n of
W h o l e Numbers

10

10

83.33

6

10

66.67

Fractions

An a l y sis of Results
Table 7 shows the p e r c e n t a g e all interns w h o attains
m a s t e r y on each c r i t e r i o n - r e f e r e n c e d measu r e s by the end of
s e c o n d post-test.

While about 9 3 perc e n t reached mastery

level on M e a s u r e m e n t only 83 and 67 percents reache d
m a s t e r y level on the last two topics.
e x p l a i n e d in two ways.

This m i g h t be

F i r s t the H a w t h orne's eff e c t of

laboratory approach turned them on at the b e g i n n i n g of the
program.

Secondly,

the entire T e a c h e r Corps p r o g r a m s e e m e d

to be w e l l o r g a n i z e d at that time.

However,

in Spr i n g term

there w e r e administrative d i f f i culties in schedulin g the
m a t h e m a t i c s class.

It was not certain,

that m a t h e m a t i c s class w o u l d be held,
in the o r i g i n a l proposal.

until very late,

though this was

By the time this was

164
settled,
matics

a n o t h e r class has b e e n s c h e d u l e d to use the m a t h e ­

l a b o r a t o r y at the h o u r s a v a i l a b l e for the interns.

W h i l e e v e r y e f f o r t w a s m a d e to b r i n g the n e c e s s a r y m a t e r i a l s
to the c l a s s r o o m used this term, m o s t of the interns
t h a t this w a s n o t truly a l a b o r a t o r y setting.

This

felt
feeling

m i g h t h a v e some s o r t o f n e g a t i o n of H a w t h o r n e ' s e f f e c t —
t h a t t h e y w o u l d n o t do w e l l in n o n - l a b o r a t o r y s e t t i n g — on
t h e interns.
instructor)

T h i s e x p e c t a t i o n of the int e r n s

(not the

m a n i f e s t e d into a m o d i f i e d form of J a c o b s o n —

R o s e n t h a l e f f e c t on the interns*

achievements

this

term.

A p a r t f r o m the p e r f o r m a n c e o f int e r n s on F r a c t i o n s
w h i c h m i g h t a l s o be a f f e c t e d by the e n d of t e r m 's p r e s s u r e
on students

f r o m o t h e r cou r s e s

in t h e i r s c h e d u l e , the

o v e r a l l p e r f o r m a n c e seems to follow

findings on p e r c e n t a g e

of s t u d e n t s t h a t a t t a i n m a s t e r y u n d e r B l o o m ' s m o d e l of
mastery

learning

(Peterson,

19 72).

T h e E v a l u a t i o n of L e a r n e r s
In this p a r t o f the study,

the e f f e c t the m a t h e ­

m a t i c s e d u c a t i o n c o m p o n e n t of T e a c h e r C o r p s p r o g r a m upon
the b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g s and a t t i t u d e t o w a r d
m a t h e m a t i c s of the i n t e r n s w e r e analyzed.
Hypothesis Tested
T h e f o l l o w i n g m u l t i v a r i a t e h y p o t h e s i s a n d the
a s s o c i a t e d u n i v a r i a t e h y p o t h e s e s w e r e tested:

165
Hypothesis B .

There w i l l be no d i f f e r e n c e b e t w e e n p r e -

a nd p o s t - t e s t m e a n scores of the interns on b a s i c m a t h e ­
m a t i c a l u n d e r standings and atti t u d e toward a r i t hm e t i c as
m e a s u r e d by Dossett's tests and Dutton's A t t i t u d e Inventory.
Symbolically,
y - x = o
w h e r e X, a 2 x 1 vector,

is the p r e - t e s t

m e a n scores of the interns on D o s s e t t a n d Dutton's tests
Y, a 2 x 1 vector,

is the p o s t - t e s t m e a n s scores of the

interns on D o s s e t t and Dutton's tests.
The a s s o c i a t e d u n i v a r i a t e h y p o t h e s e s also t ested
were:
The p o s t - t e s t means of the interns'

groups w i l l not

be d i f f e r e n t from their p r e - t e s t m e a n on
Bl.

b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g as m e a s u r e d by
the Dossett's tests,

B2.

attitude toward a r i t h m e t i c as m e a s u r e d by the
Dutton's A t t i t u d e inventory.

Findings
Hypothesis B .— The d a t a in table 8 show gains made
by the i n t e r n s ' group on b o t h m e a s u r e s .

The o b s e r v e d gain

on the m e a n score on b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g was
7.375 w h i l e that on atti t u d e to w a r d a r i t h m e t i c was

7.708.

W h e n the vec t o r of m e a n di f f e r e n c e s was t e s t e d a g a i n s t zero
vector,

the resul t i n g m u l t i v a r i a t e F was

h i g h ly s i g n i f i c a n t

(p < .0001).

15.9 824 w h i c h was

B a s e d on this result,

the

166
Table

8.— Means a n d S t a n d a r d De v i a t i o n s of Pre- and P o s t ­
test Scores of Interns on Dossett*s a n d D u t t o n 1s
Tests.
Pr e - T e s t
■
Mean S.D.

Variable

1.

D o s s e t t ' s Test

2.

D u t t o n 's
I n ventory

27.56

Post-Test
■ 1
— ■■
Mean
S.D.

Gains
■■ ■ ---Mean
S.D.

8.23

36.12

9.68

7.375

6.69

5.52 1.9 4

6.29

1.79

7.70 8

11.76

m u l t i v a r i a t e h y p o t h e s e s B w h i c h stated t h a t there w i l l be
no s i g n i f i c a n t differ e n c e b e t w e e n the p o s t - t e s t m e a n s and
the p r e - t e s t me a n s of the interns'

g r o u p on D o s s ett' s and

Du t t o n ' s i n s t r uments w a s reje c t e d at

.05 level of s i g n i f i ­

cance .
On the univariate tests,

the d i f f e r e n c e b e t w e e n

pre- a n d p o s t - t e s t means of the interns on Dossett's test
of b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g was s i g n i f i c a n t at
.025 level of confidence.

The univariate null h y p o t h e s i s

w h i c h s t a t e d that the p o s t - t e s t mean score of the interns
w i l l be n o t d i f f e r e n t from their p r e - t e s t mean score on
b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g as m e a s u r e d b y Dosse t t ' s
test w a s rejec t e d

(p <

.0001).

The d i f f e r e n c e b e t w e e n p r e -

and p o s t - t e s t scores o f the interns on Du t t o n A t t i t u d e
In v e n t o r y w a s also s i g n i f i c a n t at .025 level.
v a r i a t e n u l l hypoth e s i s

The u n i ­

that the p o s t - t e s t m e a n score of

the interns w i l l be not diffe r e n t from t h e i r p r e - t e s t m e a n
score o n atti t u d e t o w a r d a r i t h m e t i c as m e a s u r e d by D u t t o n ' s

167
Inven t ory was reje c t e d

(p < .0039).

Table 9 gives a summary

of the results.
Table 9. — M u l t i v a r i a t e and Univar i a t e Analysis of V a r i a n c e
of Interns on Gains D o s s e t t and D u t t o n ’s Tests.
M u l t i v a r i a t e F = 15.9824
D.F.

2 and 2 2

Bet w e e n
M e a n Square

Variable

p < .0001
Signifi c a n c e
Pr o b a b i l i t y
P<

Univar i a t e
F

Gain on
D o s s e t t ' s Tests

1305.3750

29.1598

.0001

G a i n on
D u t ton's Test

1426.0417

10.3046

.0039

R e l a t e d Questions
In the above analysis average value soore co m p u t e d
for e a ch intern on Dutton's test was used.
also a s k e d questions p e r t a i n i n g to
tudes w e r e developed,
disliked,
metic,

The instru m e n t

(1) grade w h e r e a t t i ­

(2) aspects of a r i t h m e t i c liked or

(3) estim a t e s of general feeling toward a r i t h ­

and

(4) average grade in arithmetic.

F i n dings on R e l a t e d Q u e s tions
1.

Grades w h e r e attitudes w e r e d e v e l o p e d :

t w e n t y - f o u r interns,

none indic a t e d on the p r e - t e s t that

he d e v e l o p e d his attitude in any of grades 2,
and 14.

Of the

O n the p r e - t e s t however,

d e v e l o p e d their attitudes

4, 5, 12,

13,

three i n d i c a t e d that they

in g r a d e 4, one d e v e l o p e d his

168
a t t i t u d e in g r a d e

5, one in grade 6 and a g a i n n o n e in

grades

A c o m p a r i s o n o f t h e i r r e s p o n s e s on p r e —

2 and 13.

a n d p o s t — tests r e v e a l e d some i n c o n s i s t e n c i e s w i t h a c o r r e ­
l a t i o n of 0.58 b e t w e e n pre-

and p o s t - r e s p o n s e s s h o w i n g that

m a n y s tude n t s c o u l d n o t c l e a r l y r e m e m b e r m u c h a b o u t t h e i r
attitudes

in early age w h i c h is c o n s i s t e n t w i t h P o f f e n -

berger's study

( P o f f e n b e r g e r , e t a l . , 1956,

the most cricual years

1959).

However,

for the inte r n s w e r e in the f o u r t h

t h r o u g h tenth g r a d e s , as r e p o r t e d in b o t h the pre- a n d p o s t ­
test

(see table 10).

findings

(Dutton,

2.

This period overlaps wit h Dutton's

1962).

General feeling toward a rithmetic;

Each intern was

a s k e d to c i r c l e a n u m b e r b e t w e e n 1 a n d 11 to show his
overall

fee l i n g t o w a r d a r i t h m e t i c

(1 r e p r e s e n t i n g e x t r e m e

d i s l i k e a n d 11 r e p r e s e n t i n g e x t r e m e l i k e n e s s ) .
p r e - t e st,

O n the

a c o m p a r i s o n t h e i r e s t i m a t i o n of t h e i r o v e r a l l

f e e l i n g w i t h t h e i r a v e r a g e scores y i e l d e d a c o r r e l a t i o n o f
0.73 w h i l e the s a m e c o m p a r i s o n on p o s t - t e s t y i e l d e d a c o r r e ­
l a t i o n o f 0.67.
i d e a of o v e r a l l

This

shows

that the int e r n s h a v e a g o o d

fee l i n g t o w a r d arithmetic.

between their overall
a v e r a g e s c o r e on items

feelings

The difference

and their corresponding

1 to 15 w a s a t t r i b u t e d to a v e r a g i n g

o f b o t h f a v o u r a b l e and u n f a v o u r a b l e items c h e c k e d on the
s c a l e by the i n d i v i d u a l to s e c u r e o v e r a l l v a l u e of the
inventory.

169

Table 10.— Grade Levels of Interns W h e r e Attit u d e s W e r e
Developed.
Pea r s o n C o r r e l a t i o n Co e f f i c i e n t b e t w e e n the PrePost- T e s t Responses of Interns is 0.5842
(N = 24)
Pre-Test
No. of Interns

Post-Test
No. of Interns

1

1

1

2

0

0

3

2

1

4

0

3

5

0

1

6

2

3

7

5

8

3

0

9

6

5

10

2

2

11

3

1

12

0

1

13

0

0

14

0

3

Grade Level

3

170

Table 11.--Interns 1 Feelings About Arithmetic

in G e n eral.

N = 24
C o r r e l a t i o n B e t w e e n P r e - T e s t F e e l i n g s and P r e - T e s t
A v e r a g e S cores w a s 0.7256
Correlation Between Post-Test Feelings and Post-Test
A v e r a g e Scores w a s 0 . 6 6 8 1
Feeling About Arithmetic
Extreme Dislike

Extreme Like

in G e n e r a l

Pre-Test

Post-Test

1

0

0

2

1

0

3

3

1

4

2

1

5

2

0

6

5

7

7

2

4

8

0

2

9

6

5

10

2

1

11

1

3

171
3.

Aspects of arithmetic liked or d i s l i k e d ;

On the

p arts asking for aspects of arithm a t i c like o r disliked.
T here w e r e various responses.
challenge,
problems,

application,

liked included

sat i s f a c t i o n on corr e c t so l u t i o n of

m a t h e m a t i c a l games.

story problems,

The aspects

The aspects

liked in c l u d e d

teachers, m e m o r i z a t i o n of rules, proofs,

long division and boredom.
4,

A v e r a g e grade in a r i t h m e t i c ;

No useful in f o r m a t i o n

was o b t a i n e d .
The E v a l u a t i o n of I n s t ruction
The e v a l uation of the T e a c h e r Corps Ma t h e m a t i c s
I n s t r uctional p r o g r a m occurs in two stages:
of m e t h o d of i n s t r uction
toward ma t h e m a t i c s

(three levels)

(three levels)

(1) Th e e f f e c t

and entry attitude

upon the mathemat i c s

a c h i e vement at the end of i n s t r u c t i o n was determined.

This

w a s an att e m p t to answer the question:
Is it the case that students w i t h s p e c i f i e d entry
behaviour

(context)

learn m o r e m a t h e m a t i c s

a p a r t i c u l a r m e t h o d of i n s t r u c t i o n

(external)

from

(internal)?

The a n s w e r to this q u e s t i o n provides a co l l e c t i o n of c o n ­
textual i n f o r mation in process

(interim)

formative e v a l u ­

ation and this may help in s p e c i f y i n g the limits of the
pr o ducts in terms of e n t r y b e h a v i o u r and terminal m a t h e ­
matics achievement.
(three levels)
levels)

(2) The effect of m e t h o d of in s t r u c t i o n

and e n t r y m a t h e m a t i c a l apti t u d e

(three

upon the -attitude towards m a t h e m a t i c s at the end

172
of i nstruction was also determined.

This again p r o v i d e s

contextual i n f o r mation in the process

formative e v a l u a t i o n

in terms of entry m a t h e m a t i c a l aptitude and terminal
attitude toward mathematics.
In a d d i t i o n to the T e a c h e r Corps g r o u p two other
groups

(described in chapter III) w e r e involved in this

part of the study.

The three groups ha v i n g three d i f f e r e n t

methods of instruction.
Data collected through the a d m i n i s t r a t i o n of the
two e q u i v a l e n t forms of Dossett's test of m a t h e m a t i c a l
un d e rstandings and the Dutton's A t t i t u d e Inventory w e r e
ut i l i z e d in compa r i n g the groups m a t h e m a t i c a l under s t a n d i n g s
and attitude toward arithmetic.
Hypotheses T e s t e d
Hypothesis Cl
a.

When a linear adjust m e n t is m a d e for the e f f e c t of
varia t i o n due to di f f e r e n c e s in p r i o r m a t h e m a t i c a l
aptitude,

as m e a s u r e d by Dossett's pre-test,

there

w i l l be no s i g n i f i c a n t differ e n c e in m a t h ema t i c s
achievement,

as m e a s u r e d by Dossett's post-test,

bet w e e n the methods of instruction.
T h a t is, there w i l l be no treat m e n t effect.
b.

W h e n a linear adjust m e n t is made for the effect
v a r i a t i o n due to d i f f erences in p r i o r m a t h e m a t i c a l
aptitude,

as m e a s u r e d by D o s s ett's pre-test,

there

w i l l be no s i g n i f i c a n t d i f f e r e n c e in m a t h e m a t i c s

173
a c h i e v e m e n t , as m e a s u r e d by Dossett*s post-test,
b e t w e e n the entry a t t i t u d e s .
That is,
c.

there w i l l be no attitude effect,

W h e n a linear adjustment is made for the eff e c t of
v a r i a t i o n due to d i f f e rences in p r i o r m a t h e m a t i c a l
aptitude,

as m e a s u r e d by Dossett's pre-test,

there

w i l l be a constant difference in m a t h e m a t i c s
achievement,

as m e a s u r e d b y Dossett's post-test,

b e t w e e n the met h o d s of instruction at all levels
entry of attitude.
T h a t is, there w i l l be no treat m e n t by attitude
interaction.
Findings
Hypothesis C l :

The two-way analysis of covariance

technique was u t i l i z e d for the analysis of data.

The

assump tions a n d the limitations of the study w e r e di s c u s s e d
in c h a p t e r III.

The scores of t w e n t y - f o u r interns in G^,

twenty -one students in G ^ f and e i g h t e e n students in

on a

test of b a s i c m a t h e m a t i c a l understandings w e r e used in the
analysis.

Data are p r e s e n t e d in table

13.

The analysis o f covariance is summar i z e d in table 12.
T h e F-t e s t s for t r e a tment effect,

atti t u d e e f f e c t

a n d t r e a t m e n t by attitude i n t e r a c t i o n w e r e 10.4406,
a nd 0 . 6318 respectively.
(p < .0002)

3.3821,

The first was h i g h l y si g n i f i c a n t

and the se c o n d was m a r g i n a l l y sig n i f i c a n t

174
Table 12.--Summary of Anal y s i s of C o v a r i a n c e for the Groups
on the T e s t of Ba s i c Math e m a t i c a l Understandings.
Source of
V a r ia t i o n

D.F.

MS(Adjusted)

Instruction
(Group)

2

230.1307

10.4406*

P < .0002

A t t i tude

2

74.5476

3.3821*

P < ,0415

I n s t ruction x
Attitude

4

13.9257

.6318

P < .6421

Error

53

Total

F

Significance
Proba b i l i t y

22.041970

61
* S i g n i f i c a n t at a =

.05.

(p < .0415), w h i l e the third one was not s i g n ifica n t at
ct =

.05 level.

Thus

H ypoth e s i s Cl(a)

was reje c t e d at

.05 level of s i g n i f i ­

was reje c t e d at

.05 level of s i g n i f i ­

cance .
H ypoth e s i s Cl(b)
cance .
Hypothesis Cl(c)

was not rejected.

The above results s h o w that there are s t a t i s t i c a l l y
s i g n i f i c a n t differences b e t w e e n met h o d s of i n s t r u c t i o n on
one h a n d a n d attitude levels o n the ot h e r hand.

Scheffe's

m e t h o d was used to find the d i r e c t i o n of the sig n i f i c a n c e
of differences.
Pairw i s e c o m p a r i s o n among groups revea l e d that,

Table 13.— Groups Mean Scores on the Test of Basic Mathematical Understandings.

Treatment Group

G1

Attitude
Pre-Test

G2
Post-Test

Pre-Test

G3
Post-Test

Pre-Test

Post-Test

S.D.

Mean

S.D.

Mean

S.D.

Mean

S.D.

Mean

S.D.

Mean

S.D.

High

31.50

9.856

35.875

8.935

32.571

5.623

37.857

5.305

39.333

4.844

32.500

6.411

Medium

27.125 5.817

37.875

3.944

36.857

3.848

40.571

3.780

37.833

6.080

35.500

5.468

Low

23.75

30.750

7.285

28.000

6.952

32.429

4.504

31.500

5.01

30.000

3.794

7.544

175

Mean

176
1.

the adju s t e d p o s t - t e s t mean score o f G^ on basic
m a t h e m a t i c a l u n d e r standings is s i g n i ficantl y h i g h e r
than that of G 3 ;

2.

the adjusted p o s t - t e s t m e a n score of G 2 on b a s i c
m a t h e m a t i c a l u n d e rstandings is s i g n i f i c a n t l y h i g h e r
than that of G^;

3.

the adju s t e d p o s t - t e s t mean scores of G^ and G 2 on
b a s i c m a t h e m a t i c a l u n d e r standings are not s i g n i f i ­
cantly different.
P airwise c o m p a r i s o n among attitude levels r e v e a l e d

that
1,

the adjusted p o s t - t e s t m e a n score of students w i t h
high entry attitude on b a s i c m a t h e m a t i c a l u n d e r ­
s tan d i n g is not s i g n i ficantly hi g h e r than that of
students w i t h m e d i u m entry attitude;

2,

the adju s t e d post- t e s t m e a n score of students w i t h
h i g h entry attitude on b a s i c m a t h e m a t i c a l u n d e r ­
s t a n d i n g is not s i g n i f i c a n t l y hi g h e r than that of
stu d e n t w i t h low entry attitude;

3,

the adju s t e d p o s t - t e s t m e a n score of students w i t h
m e d i u m entry attitude on b a s i c m a t h e m a t i c a l u n d e r ­
standings is n o t sign i f i c a n t l y h i g h e r than that of
students w i t h low entry attitude.

The a p p a r e n t c o n t r a d i c t i o n b e t w e e n the a p o s t e r i o r results
and Hypothesis 01(c)

w i l l be d i s c u s s e d below.

177
Hypotheses C2
a.

W h e n a linear adjust m e n t is made for the ef f e c t of
v a r i a t i o n due to d i f f e rences in pr i o r attitude
toward mathematics,

as m e a s u r e d by Dutton's p r e ­

test, there w i l l be no si g n i f i c a n t differ e n c e in
attitude toward mathematics,
Dutton's post-test,

as m e a s u r e d by

bet w e e n the met h o d s of

instruction.
That is, no t r e a tment m a i n effect.
b.

W h e n a linear a d j u s t m e n t is m a d e for the ef f e c t of
v a r i a t i o n due to d i f f e rences in p r i o r attitude
toward mathematics,
test,

as m e a s u r e d by Dutton's p r e ­

there w i l l be no s i g n i f i c a n t d i f f e r e n c e in

a ttitude toward mathematics,

as m e a s u r e d by

Dutton's post-test, b e t w e e n the entry m a t h e m a t i c a l
aptitude.
T hat is, there w i l l be no aptitude m a i n effect.
c.

W h e n a linear adjust m e n t is m a d e for the ef f e c t of
v a r i a t i o n due to d i f f e rences in p r i o r attitude
t oward mathematics,
test,

as m e a s u r e d by Dutton's p r e ­

there w i l l be a const a n t d i f f e r e n c e in a t t i ­

tude toward mathematics,
post-test,

as m e a s u r e d by Dutton's

b e t w e e n the m e t h o d s of i n s t r u c t i o n at

all levels of entry m a t h e m a t i c a l aptitude.

178
Findings
Hypotheses C 2 ;

T h e scores of 24 interns

students in G 2 , and 18 students in
Inventory

on Dutton's A t t i t u d e

(attitude toward arithmetic)

analysis.

in G.^, 21

w e r e used in the

Data are p r e s e n t e d in table 15.

T he analysis of covariance is summar i z e d in table 14.
The F- t e s t for treatment effect,

aptitude e f f e c t

and treatment by aptitude interaction w e r e 0.5 307,
0.1262 respectively.

The second was s i g n i f i c a n t

3.5024,

(p < 0.0373)

w h i l e the first and the third w e r e n o t s i g n i f i c a n t at
a = 0.05 level.

Thus

Hypothesis C2(a) was not rejected.
H ypothesis 02(b) was rejected at

.05 level of s i g n i f i ­

cance .
H ypothesis 0 2 (c) w a s n o t rejected.
These results show that only entry m a t h e m a t i c a l
a p t itude effect is significant.

Scheffe's m e t h o d w a s used

to d e t e c t the significant differences among pairs of
a p t itude

(group)

m e a n scores.

Pair w i s e c o m p a r i s o n among the m a t h e m a t i c a l aptitude
levels m e a n scores on attitude toward m a t h e m a t i c s r e v ealed
that
1.

the adjus t e d p o s t - t e s t m e a n score of the group w i t h
high e n t r y m a t h e m a t i c a l aptitude on attitude
to w a r d m a t h e matics

is not s i g n i f i c a n t l y d i f f e r e n t

179
T a b l e 14.— Summary of the Analysis of C o v a r i a n c e for the
Scores of Groups on Atti t u d e T o w a r d Arithmetic.
S o urce of
Variation

D.F.

I n s t r u ction
(Group)

2

92.0286

Aptitude

2

607.3783

Instruction
x Aptitude

4

21.8801

Error
Total

53

F

MS(Adjusted)

Signi ficance
Pr o b a b i l i t y

0.5307

p < .5914

3.5024*

p < .0373

0.1262

p < .9724

173.418232

61
*S i g n i f i c a n t at a =

.05

from that of the g r o u p w i t h low entry m a t h e m a t i c a l
aptitude.
2.

the adjusted p o s t - t e s t mean score of the gr o u p w i t h
m e d i u m entry m a t h e m a t i c a l aptitude on attitude
toward m a t h e m a t i c s

is s i g n i f i c a n t l y d i f f e r e n t from

that of the g r o u p w i t h how entry m a t h e m a t i c a l
aptitude.
3.

the a d j u s t e d p o s t - t e s t m e a n score of the group w i t h
h i g h entry m a t h e m a t i c a l aptitude on attitude
toward m a t h e m a t i c s is 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
from that of the g r o u p w i t h m e d i u m en t r y m a t h e ­
m a t i c a l aptitude

(tables

14,

15).

Table 15.— Groups Means Scores on Attitude Toward Arithmetic.

Treatment Group
Gi

Aptituae
Pre-Test
Mean

S.D.

G2
Post-Test

Pre-Test

G3
Post-Test

Pre-Test

Post-Test

Mean

S.D.

Mean

S.D.

Mean

S.D.

Mean

S.D.

Mean

S.D.

High

64.00

13.918

68.875

14.730

55.00

17.981

63.143

10.777

73.667

3.615

67.667

10.652

Medium

53.50

17.246

66.625

16.707

60.714 15.713

69.714

14.997

62.833 16.315

71.167

5.154

Low

48.00

24.413

53.125

19.895

39.286 28.270

47.714

28.511

57.333 11.237

53.833

16.951

181
Q u a n t i t a t i v e E v a l u a t i o n of E n v i r o n m e n t s

ol: Learning
T h e scores of the s u b j e c t s on the a t t i t u d e scales
d e v e l o p e d b y the I n t e r n a t i o n a l P r o j e c t for the E v a l u a t i o n
o f E d u c a t i o n a l A c h i e v e m e n t in M a t h e m a t i c s
w e r e used.

(Husen,

T h e i n s t r u m e n t e v a l u a t e s subjects'

1967)

perceptions

of,
1.

learning mathematics

as a p r o c e s s

2.

the d i f f i c u l t i e s

3.

the p l a c e o f m a t h e m a t i c s

4.

the s c h o o l and s c h o o l

5.

m a n a n d his e n v i r o n m e n t

(B^),

in learning m a t h e m a t i c s
in s o c i e t y

learning

(B^),

(B^) ,

(B^),

(B^).

T h i s p a r t of the st u d y is b a s e d on the a s s u m p t i o n that if
the level o f e n t r y a t t i t u d e t o w a r d m a t h e m a t i c s does n o t
a f f e c t t h e i r scores on the instrument,

then their s cores

is a r e f l e c t i o n of their l e a r n i n g e n v i r o n m e n t s .
Hypotheses Tested
D.

(a)

T h e r e w i l l be no s i g n i f i c a n t d i f f e r e n c e on the
m e a n s c o r e s on H u s e n 's A t t i t u d e Scales due to
methods

of i n s t r u c t i o n .

T h a t is,

t h e r e w i l l be n o t r e a t m e n t m a i n

effect.
(b)

T h e r e w i l l be n o s i g n i f i c a n t d i f f e r e n c e on m e a n
s c o r e s o n H u s e n ' s A t t i t u d e Sc a l e s due to e n t r y
attitude

level.

T h a t is,

there w i l l be n o a t t i t u d e m a i n effect.

182
(c)

T h e r e w i l l be a c o n s t a n t d i f f e r e n c e b e t w e e n the
m e a n scores of G^,

G 2 , G^ on H u s e n ' s A t t i t u d e

Sc ale s at all levels of e n t r y a t t i t u d e t o w a r d
mathemati c s .
T h a t is, t h e r e w i l l b e n o t r e a t m e n t by a t t i t u d e
interaction.
Findings
A

3 x 3 x 5 factorial design wit h repeated measures

on the last fa ct or w a s used.

Multivariate analysis tech­

ni que w a s e m p l o y e d in the analysis.

A s u m m a r y of the m u l t i ­

v a r i a t e a n a l y s i s of v a r i a n c e is g i v e n in t a b l e
T h e m u l t i v a r i a t e F for m e t h o d e f f e c t
2.2210 w h i c h was h i g h l y s i g n i f i c a n t

16.

(row)

(p < .0223).

was
B a s e d on

the r e s u l t of this analysis,
H y p o t h e s e s D(a) w a s

r e j e c t e d at

.05 level of s i g n i f i ­

ca n c e .
We m i g h t t h e r e f o r e c o n c l u d e that there is s t r o n g e v i d e n c e
of n o n - c h a n c e d i f f e r e n c e s d u e to the m e t h o d s o f i nst ruc tio n.
E x a m i n a t i o n of the r o w - t o t a l v e c t o r s
that,

(table 17)

in dic a t e s

a m o n g the three m e t h o d s of i n s t r u c t i o n s , p e r c e p t i o n s

of e n v i r o n m e n t s of l e a r n i n g as m e a s u r e d by five s u b - s c o r e s
of A t t i t u d e s Sca les d e v e l o p e d by the I n t e r n a t i o n a l P r o j e c t
for the E v a l u a t i o n of E d u c a t i o n a l A c h i e v e m e n t in M a t h e m a t i c s
(Husen,

1967),

is h i g h e s t in G^ a p a r t f r o m

w h e r e G 2 's s c o r e exc e e d s G^' s score.
m e a n score for G^ is highest,

How eve r,

sub-score
the t ota l

t h e n f o l l o w e d by t hat o f G 2

183
T a b l e 1 6 .--Summary of M u l t i v a r i a t e Anal y s i s of V a r i a n c e
(Repeated M e a s u r e s Design) on P e r c e p t i o n of
Le ar n i n g Environments.
Mu l t i v a r i a t e
Sources

D.F.
F

Method

(G)

Attitude

(A)

Group x At ti t u d e

(GxA)

Subjects w i t h Groups
(S: GxA)
(Measures)

M

P Less Than

2

2.2210

.0223

2

.8198

.6104

4

1.2519

.2189

54

4

34.06

.0001

M x G

8

2.2637

.0286

M x A

8

.7568

.6414

16

1.1492

. 3154

M x G x A
S

X

M

: G x A

(table 17).

A p p l i c a t i o n of Scheffe's m e t h o d to the least-

square es tim a t e s of effects r e ve a l e d that the m e a n score
for G^ is s tat ist ic all y d i f f e r e n t from each of the m e a n
scores of G 2 and G 3 w h e r e a s

the m e a n scores for G 2 and G^

are not st ati s t i c a l l y d i f f e r e n t

from each oth er

This

i n t e r est ing result w ill be i n t e r p r e t e d later.
The m u l t i v a r i a t e F for entry att i t u d e ef f e c t
(column)

was 0.8198 w h i c h was n o t s i g n i f i c a n t

(p < .6104).

B a s e d on this ana lysis Hyp oth e s i s D(b) w a s not rejected.
The i m p l i c a t i o n is that the p e r c e p t i o n of the groups'
e n v i r o nme nts of lea rning as m e a s u r e d by five sub-scales of

184
Table 17.— Mean Scores of Group on Perception on Learning Environments.
Measures (Dependent Variables)
Total

Groups

B,
1

G1
G2

B_
2
11.95873

0.208

9.7143

10
9.7222

B„
4

3

Q.95833

8.9444

G3

B

13.6667

B

5

8.625

52.41636

8.8095 12.7619

6.4286

47.71429

8.1111 12.0555

8.5

47.33326

Hunsen's in s t r u m e n t is p rob a b l y not aff ected by their entry
at titude to w a r d mathematics.

This result is very useful in

i n t e r p r e t i n g the p r e c e d i n g result w h i c h is the m a i n result
of i n t e r e s t in this part of the study,

it is the assumption

made at the b e g i n n i n g o f this section.
The m u l t i v a r i a t e F for m e t h o d of instru cti on by
at titude

(G x A)

i n t e r a c t i o n was

s i g n i f i c a n t at a = 0.05
H y p o t h e s i s D(c)

1.2 519 w h i c h was not

(p < .2189).

B a s e d on this result,

that there w i l l be a co ns t a n t difference

b e t w e e n the m e a n - s c o r e s of G 1 ,

, G 3 on H u s e n 's Att itu de

Scales at all levels of entry attitude to war d m a t h e m a t i c s
was not rejected.

We m a y c onclude that the p e r c ep tio ns of

the e nv i r o n m e n t s of lea rning by the three groups is ind e­
p e n d e n t of the three entry attitu de

levels t oward m a t h e ­

ma t i c s .
I n t e r p r e t a t i o n and D i s c u s s i o n of Fi nd i n g s
A s s u m i n g that out c o m e of learn ing d epends on the
e n v i ro nme nts of learning it is logical to start the

185
M ea s u r e by T r e a t m e n t Interaction
(Disordinal)

12
in

(U

oil

o

w

10

7-

is attitude toward m a t h e m a t i c s as a pr ocess
B 2 is a ttitude toward the d i f f i c ult ies in learning m a t h e ­
m atics
is a ttitude toward the p l a c e of m a t h e m a t i c s in so cie ty
B^ is a ttitude toward school and school learning
B 5 is attitud e to ward m a n and his e n v i r o n m e n t

186
i n t e r p r e t a t i o n of results

from e v a l u a t i o n o f learning

environments.
En v i r o n m e n t s o f L e a r n i n g
N o n r eje c t i o n of H y p o t hes is D(c)

indica tes that the

re l a t i o n s h i p b e t w e e n the p e r c e p t i o n of le ar n i n g e n v i r o n ­
ments of subjects

in groups G^, G 2 , G 3 is the same across

all levels of e n t r y att itu de tow ard m a t h e m a t i c s .

This

suggests that the r e s p ons es of subjects on H usen's A t t i ­
tude Sca le w i t h i n each g rou p w e r e u n a f f e c t e d by t hei r ent ry
at titude t o w a r d mathematics.

The p rob a b l e i m p l i c a t i o n of

this is that their p e r c e p t i o n s
environ men ts o f learning.

fairly est im a t e s their

This i m p l i c a t i o n is h o w e v e r

co nfi rme d by a c c e p t a n c e of H y p o t h e s e s D(b)
entry a ttitude effect.

that there is no

Results of H y p o t hes es D(b)

and

(c)

imply th at the s u b j e c t s ’ p e r c e p t i o n of their env i r o n m e n t s
of l earning as m e a s u r e d by Hun sen's A t t i t u d e Scales is a
fair e v a l u a t i o n of their le arning environments.
The result of H y p o t h e s i s D(a)
cation of results o f H y p o t h e s e s D(b)

c omb i n e d w i t h i m p l i ­
and

(c) in the p r e c e d i n g

p a r a g r a p h show that the e n v i r o n m e n t of learnin g of G^ is
s i gn i f i c a n t l y s u p e r i o r t han e a c h of those of G 2 and G 3 w h i l e
that of G 2 is n o t statistically,
that of G 3 «
of

In o t h e r w o r d s

ap p r e c i a b l y b e t t e r than

the e n v i r o n m e n t s of le arn ing

w h i c h was ma s t e r y approach,

laborato ry- ori ent ed,

co nte nt and m e t h o d i n t e g r a t e d tog e t h e r w i t h s u b s t an tia l
cl in i c a l experien ce pro v i d e s a lea r n i n g e n v i r o n m e n t b e t t e r

187
than that of laboratory-oriented,

c o n t e n t - m e t h o d integrated,

t o g e t h e r w i t h one h o u r p e r w e e k clinical experience; whereas
the latter does not p r o v i d e a le arning e n v i r o n m e n t b e t t e r
than the r eg u l a r p r o g r a m w h i c h is p art i a l l y laboratoryo r i e n t e d m a t h e m a t i c s co nt e n t s e p a rat ed from m ethods course
w h i c h is l abo ratory-oriented.

T he n o n — apprec iab le d i f f e r ­

ence b e t w e e n the last two o b t a i n e d from this study m i g h t be
due to the fact that the p res s u r e was too m u c h for the
students in G £ .

Th is group,

as P ° i n-ted out in

ch apter III, h ad their m a t h e m a t i c s e d u c a t i o n p r o g r a m in
Fall q u a r t e r in s i x - q u a r t e r h o u r class m e e t i n g
hour cli n i c a l e x p e r i e n c e in school)
program,

(and one

w h a t the reg ula r

G 3 , had in e i g h t - q u a r t e r h o u r class meetings.

This p r e s s u r e o b s c u r e s the di f f e r e n c e b e t w e e n n o n - ma ste ry
l a b o r a t o r y - c o n t e n t m e t h o d i nte gr a t e d p r o g r a m for the regular
co n t e n t - m e t h o d s e p a r a t e d program.
agrees w i t h Hollis

(19 71)

However,

the findings

study that laboratories o r g a n ize d

to p ro v i d e p e r s o n a l and i n d i v i d u a l i z e d as sis t a n c e are
h e l p f u l to learners that are ei t h e r cultural ly or ac a d e m i ­
cally disadvantaged.
Re lat ed Q u e s t i o n ;

A r e l a t e d q u e s t i o n is

Is the d i f f e r e n c e b e t w e e n groups*

pe r c e p t i o n s of

le arning e n v i r o n m e n t the same across all me as u r e s
subsets)?

(five

T he m u l t i v a r i a t e F for this m e a s u r e by m e t h o d

(treatment)
(p < 0.0286)

i n t e r a c t i o n was
in table

16.

2.2637 w h i c h was sig nif ica nt
This indicates a sig nif ica nt

188
interaction,

thus the di ffe ren ce in p e r c e p t i o n of e n v i r o n ­

ments of learning b et w e e n the groups'
each as pect of environments.

is not the same for

Mean score of the groups on

each s ubset of Husen's At titude Scales are in table 17.
F i g u r e shows that the interac tio n is disordinal.

This

disordi nal i nteraction shows that only in some cases is
the e nvi ron men ts of learning as p e r c e i v e d by
to others.

In fact

is supe r i o r in all cases except on

the le arning of m a t h e ma tics as pr ocess
e x c e e d e d by G 2 .

su perior

The scores of

(B ^) w h e r e they w e r e

and Gg on this subset

i n dicate that they vi e w e d m athematics as fixed and given
once and for all time

(a low score) w h i l e group G 2 vi e w e d

ma t h e m a t i c s re lat i v e l y more as s ome thing developing,
growing,

and cha nging

(a high score)

this m i g h t be a

reflect ion of efforts of two faculty members on the group.
Th e figure shows that, G 3 's p e r c e p t i o n of e n v i r o n m e n t of
learning is inferio r in all cases except on at titude toward
man and e nv i r o n m e n t w h e r e they e x c e e d e d G 2 *
It can be c o n c l u d e d that a m a t h e m ati cs edu cat ion
p r o g r a m for e l e m e n t a r y teachers w h i c h uses Bloom's m o d e l of
m a s t e r y learning, w h i c h is l aboratory-oriented,

content-

m e t h o d - i n t e g r a t e d w i t h substa nti al clinical ex per ien ce
provide s an e n v i r o n m e n t o f learning w h e r e trainees * p e r ­
c e ption of m a t h e m a t i c s

learning,

in c o n t e m p o r a r y society,

the role of m a t h e mat ics

the school and school learning,

the r e l a t i o n s h i p of m a n to his e n v i r o n m e n t is m o r e f avour­
able than the p e r c e p t i o n s of trainee in ei t h e r

(1 ) a

189
p r o g r a m w i t h eve ryt h i n g except m a s t e r y approach and sub ­
stantia l clinical experience,

or

(2 ) a p r o g r a m similar to

the p r e s e n t r egular ma the mat ics e d u c ati on co mpo n e n t of
el eme n t a r y edu cat ion at the Mic hi g a n State University.
E v a l u a t i o n of Instruction
The rej ection of Hypothe sis Cl(a)

and subsequ ent a

p o s t er ior i comparisons suggested that there was strong
evidenc e of non chance differ enc es due to me t h o d of i n­
struction.

S u b s e que ntl y a p ost er i o r o comparisons,

via

least square est imates effects revealed a s i g n i fi can t
d i ffe ren ce in m a t h e m ati cs a c h i e vem ent b etw een G^ and
favour of G^,

and b e t w e e n G 2 and G 3 in favour of G 2 b ut not

b e t w e e n G^ and G 2 .
1.

in

Thus,

The interns ba s i c ma th e m a t i c a l understa ndi ngs at
the e n d of i n s t r uct ion is n o t signif ica ntl y d i f f e r ­
ent fro m that of G 2 .

The i m p l i c ati on of this is

that w i t h many of the interns

from low-income b a c k ­

gr oun d and p o o r m a t h e ma tic s ba c k g r o u n d

(test of

fitness of m o d e l - A N C O V A - s u p p o r t s t h i s ) , it is
p o s s i b l e to b r i n g t h e m to the same level of b a s i c
m a t h e m a t i c a l und e r s t a n d i n g via Bloom's m o d e l of
m a s t e r y learning,

as students in the regular

p r o g r a m w h o r ece i v e d some c o n t e n t - m e t h o d integratedl a bor a t o r y - o r i e n t e d instruct ion
learning).

(except for m a s t e r y

This p art agrees w i t h re sea rch findings

on Bloom's m o d e l of ma s t e r y learning

(Block,

1971).

190
The r es ult is also c o n s i s t e n t w i t h findings of
A s t i n et al.
theory

(1964)

(1972)

and Aus ub e l ' s re ver s i b i l i t y

that aca dem ic def ici e n c y a s s o c iat ed

w i t h s o c i o - e c o n o m i c a l l y d i s a d v a n t a g e d ch ild ren
disappears at the co llege if the d i s a d v a n t a g e d is
in a p p r o pri ate p r o g r a m
appropri ate m e t h o d
2.

(Astin,

(Ausubel,

19 72)

or taught by

1964).

The s i g n i f ica nt d i f f e r e n c e b e t w e e n G-^ and G 3 on one
hand and G_ and G
on the o t h e r h a n d e x t e n d e d the
2
3
results of W a s z l y (1970) study to n o n d i s a d v a n t a g e d
and show,
a.

in particular,

that the d i s a d v a n t a g e d group,

u n d e r mastery,

c o n t e n t - m e t h o d i n t e g r a t e d approach,

could

reach a hi g h e r level of m a t h e m a t i c a l u n d e r ­
st andings for elementa ry teachers than a
r e g u l a r g r o u p in the p r e s e n t r eg u l a r p r o g r a m
at the M i c h i g a n State University;
b.

that the c o n t e n t - m e t h o d int egr a t e d course is
ca pable of p r o v i d i n g a be t t e r m a t h e m a t i c a l
u n d e r s t a n d i n g s for re gu l a r e l e m e n t a r y teachers
than the p r e s e n t reg u l a r ma t h e m a t i c s e d u c a t i o n
c o m p o n e n t of their program.

The rej ec t i o n of H y p o t h e s i s 01(b)
(p < .0415).

was m a r g i n a l

T he s u b s e q u e n t a p o s t e r i o r i p r o c e d u r e us ing

S c hef f e ' s m et h o d failed to detect s i g n i f ica nce in pai rwi se
c o m p a r i s o n of the at titude levels.
two reasons:

first Scheffe's

T his m i g h t be due to

tec hnique is known to be we ak

191
for p air w i s e com par iso n be c a u s e of its w i d e confidence
internal and esp eci a l l y w h e n the si gn i f i c a n c e is very
m a rg i n a l as in this case.
out of t w e n t y-f our in

R a n d o m d r o p p i n g of six subjects
and three in G 2 in o r d e r to use a

mo re p o w e r f u l a p o s t e r i o r i T u r k e y - t e c h n i q u e is not desirable
for w e shall only be using 60 p e r c e n t of the interns w h o
or igi n a l l y signed up for the program.

The se con d m i g h t be

due to w e a k r e l a t i o n s h i p b e t w e e n mat hem ati cs ac hie vem ent s
and attitude toward m a t h e m a t i c s that all r ese a r c h efforts
have shown so far co nt r a r y to the feelings of m a t h e m a t i c s
educators.
The accept anc e of Hypoth esi s Cl(c)

o f no t rea tme nt

by entry level i n t e r a c t i o n is a c o n f i r m a t i o n that the
findings u n d e r H ypo the sis Cl(a)

are i n d e p e n d e n t of the ent ry

at titude levels.
Th e acceptan ce of H y p o t h e s i s C2(a),

that there is

no s i g n i fi can t d iff er e n c e am o n g the groups on the attitude
toward m a t h e m a t i c s at the e n d of i n s t r u c t i o n is c o n s i s t e n t
w i t h studies c arr i e d out by V a n c e

(1969)

a n d Jo h n s o n

(1970)

on the e f f e c t of laborat ory ap pr o a c h on attitu de m e a s u r e s
though co ntrary to the claims of a dvo cates of laboratory
approach.
The r e j e c t i o n of Hyp oth e s i s C2(b)

t hat there is no

si gni fic ant d i f f er enc e on att itude of groups w i t h d i f f e r e n t
entry m a t h e m a t i c a l ap titudes is an im por tan t result.

A

po s t e r i o r i c o m p a r i s o n shows there is si g n i f i c a n t d i f f e r ­
ence only in the at titude of s ubjects w i t h m e d i u m and low

192
entry ma t h e m a t i c a l ap titude in favour of the group w i t h
m e d i u m aptitude.

The results in dicate that students w i t h

high entry ma t h e m a t i c a l aptitu de have m e d i u m attitude toward
ma t h e m a t i c s at the end of instruction, w h i l e w i t h initial
m e d i u m m a t h e m a t i c a l a pti t u d e have h i g h at titude toward
mathematics,

and that studen ts w i t h lowest m a t h e m a t i c a l

aptitude at the b e g i n n i n g of i n s t r uc tio n w i l l h ave lowest
attitude toward m a t h e m atic s at the e n d of instruction.
ac cep tan ce of Hypothes is C2(c)

The

of no t r e a t m e n t by aptitude

i n t e r a c t i o n shows that the r e s u l t of Hypoth esi s C2(b)

holds

for any m e t h o d of instruction.
Ev alu a t i o n of Learners
The rej ec t i o n of the m u l t i v a r i a t e hyp oth e s i s and
as soc i a t e d univar iat e H ypo th e s i s Bl, B2 that there w i l l be
no s i g n i fi can t d i f f e r e n c e bet w e e n the p o s t - t e s t means and
p r e - t e s t m e a n s of interns on Dosse tt' s and Dutton's
in str ume nt and the fo reg o i n g di s c u s s i o n show that the
m a t h e m a t i c s e d u c a t i o n c o m p on ent of the e i g h t cycle T e a c h e r
Corps p r o g r a m meets the needs of the interns.
We shall now turn to an i n t e r e s t i n g test p a r t of
the study w h i c h is an a tt e m p t to assess co nt r i b u t i o n of
d i f f e r e n t units and aspects of att i t u d e of ma t h e m a t i c s to
their te rminal bas ic m a t h e m a t i c a l a n d at titude toward
arithmetic.

193
A s s e s s m e n t of the C o n t r i b u t i o n of the Lea r n i n g
Units to the Basic M a t h e m a t i c a l
U n d e r s t a n d i n g s o f Interns
Results of H y p o t h e s i s Cl h ave shown that the b a s i c
m a t h e m a t i c a l u n d e r s t a n d i n g s of the interns
h i g h e r than t hat of students

is si gni f i c a n t l y

in th e r eg u l a r met hod s class

w h i l e it is n o t sig n i f i c a n t l y hi g h e r than that of students
in the o t h e r e x p e r i m e n t a l
class.

(content-method integrated)

A n u m b e r of pos s i b i l i t i e s arise:

case that in fact

It m i g h t be the

and G 2 are s i g n i f ica ntl y b e t t e r than

b e c a u s e G 3 r ece i v e d their m a t h e m a t i c s content,

at least,

a t e r m e ar l i e r than the m e t h o d s class w h e n their m a t h e ­
ma tic al u n d e r sta nd ing s was assessed.

If this a r g u m e n t is

tenable it w o u l d m e a n m a t h e m a t i c s me th o d s co ntains no
m a t h e m a t i c s and it w o u l d b e u nn e c e s s a r y to m a k e F o u n d a t i o n
of A r i t h m e t i c

(Mathematics

201)

a p r e - r e q u i s i t e for the

m e thods c o u r s e — M a t h e m a t i c s Met h o d s
(Education 3 2 5 E ) .

for E l e m e n t a r y Grades

If this is n o t the case,

the result of

Hy pot hes is Bl shows t h a t the p o s t - t e s t m ean score of G^ is
s i g n i f i c a n t l y h i g h e r than the p r e - t e s t me an score on the
b a s i c test of m a t h e m a t i c a l u nde rst and ing , w h i c h indicates
that p r o b a b l y this ap p r e c i a b l e g ain in m a t h e m a t i c a l u n d e r ­
standings is due to the e f f e c t of instruction;

and the

di ffe r e n c e shown in H ypo th e s i s Cl is due to d i f f e r e n c e in
instruction.
As it was p o i n t e d o u t un d e r the m e t h o d o l o g i c a l
limitations and w e a k n e s s of the de sign o f the study that
o n e-g r o u p p r e - t e s t — p o s t - t e s t d e s i g n does not e s t a bl ish

194
cause and e f f e c t relation,

following P unc h

(1971),

corre­

lation analysis and ste pwise r e g r e s s i o n techniques w e r e
em pl o y e d to assess h o w w e l l the scores of interns on the
learning units p r e d i c t their score on the m ath ema tic s
ac h i e v eme nt and to es ti m a t e w h a t p r o p o r t i o n of the p o s t ­
test score on b a s i c mat h e m a t i c a l u n d e r s t a n d i n g can be
a c c o un ted for by the learning u n i t s .
An aly sis
Table 18 shows the simple correl ati ons b e t w e e n
scores on the l e a r n i n g units,

the pre- and p ost -t e s t s on

b a s i c m a t h e m a t i c a l understandings.

The correl ati ons of the

me asu res w i t h p o s t - t e s t on m a t h e m a t i c a l un der s t a n d i n g s in
de c r e a s i n g order of m a g n i t u d e are
.6377,

.5809,

.6695,

.4153 for Numeration,

.6636,

Fractions, M u l t i p l i ­

cation and Di vis i o n of Whole Numbers,

Pre -te st of Bas ic

M a t h e m a t i c a l Understa ndi ngs , Measurement,
S u b t ra cti on of W h o l e Numbers.

.6675,

A d d i t i o n and

It is int ere sti ng to note

that o f all i nstruments a d m i n i s t e r e d on the interns
(including a ttitude scales)

these six me asu r e s hav e the

h i g h e s t six correlat ion s w i t h the p o s t - t e s t score on B a s i c
M a t h e m a t i c a l understandings.

O t h e r correla tio ns w i t h p o s t ­

test scores on b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g s are as
follows:
-0.0717,

.4138,

.3583,

.3288,

.3218,

.2510,

.1162,

.0829,

-0.1224 for E n j o y m e n t of M a t h e m a t i c s Scale, P o s t ­

test of D utton's Att i t u d e

Inventory,

Pre - t e s t of Dutton's

A t t i t u d e Inventory, A t t i t u d e on D i f f i cul tie s of L e a r n i n g

195
T a b l e 1 8 . — C o r r e l a t i o n C o e f f i c i e n t s B e t w e e n P o s t - t e s t , in
Basic Mathematical Understanding, Attitude
Toward Arithmetic and Other Variables (Measures).
V a r No.
POSTACH

1

1.00000

POSTATT

2

.35834

1.00000

PREACH

3

.63771

.32986

1.00000

PREATT

4

.32883

.80444

.41205

Bl

5

.25100

-.03463

.04612

B2

6

.132176

.21419

.08472

B3

7

.11619

-.10894

.07443

B4

8

-.07170

-. 206 32

.03876

B5

9

- .12 239

.42298

.12427

E

10

.41381

.79361

.49106

V

11

.08288

.05811

-.02241

PU1

12

.58094

.36795

.27834

PU2

13

.66948

.26431

.48262

PU3

14

.41532

-. 0 0 4 6 6

.29352

PU4

15

.56357

.25204

.29823

PUS

16

.66752

.28415

.44596

1
POSTACH

2
POSTATT

2
PREACH

196
Mathematics, Attitude Toward Mathematics

as a P r o c e s s ,

A t t i t u d e T o w a r d the P l a c e o f M a t h e m a t i c s in S o c i e t y , V a l u e
of M a t h e m a t i c s S c a l e , A t t i t u d e T o w a r d S c h o o l a n d S c h o o l
Le arn i n g ,

A t t i t u d e T o w a r d M a n a n d His E n v i r o n m e n t .

F r o m t h e s e c o r r e l a t i o n s , it ca n be c o n c l u d e d that
a p a r t f r o m the l e a r n i n g units th e a t t i t u d e sc a l e t h a t has
the g r e a t e s t c o r r e l a t i o n w i t h m a t h e m a t i c s
the

" E n j o yme nt, "

metic.

and n o t D u t t o n ' s

a c h i e v e m e n t is

Attitude Toward Arith­

In f act the E n j o y m e n t o f M a t h e m a t i c s

l a s t of all m e a s u r e s

S c a l e w a s the

a n d the o n l y a t t i t u d e s c a l e w h i c h h as

a s t a t i s t i c a l l y s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n w i t h the
p o s t - t e s t score on b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g s
a =

.05 level

(N=24).

the n e x t chapter,

at

T h o u g h w e shall r e t u r n to this in

it s h o u l d be n o t e d t h a t the a n a l y s i s of

these c o r r e l a t i o n s s u p p o r t s A i k e n

(19 72b)

contention that

w e s h o u l d be t a l k i n g a b o u t d i f f e r e n t a spe c t s o f a t t i t u d e
toward mathematics

th at c o n t r i b u t e s to m a t h e m a t i c s

achieve­

ment and not just general attitude toward mathematics.
The regression analysis

showed that wh e n Numeration

a n d A d d i t i o n a n d S u b t r a c t i o n of W h o l e N u m b e r s w e r e n o t
d e l e t e d f r o m th e r e g r e s s i o n equ at i o n , w i t h F r a c t i o n sc o r e s
a n d a c o n s t a n t l a t e r a d d e d to t h e eq uat i o n ,
variance

t he a n a l y s i s of

for o v e r a l l r e g r e s s i o n g ave an F v a l u e 11,5 996

which was highly significant
i n d i c a t e s sc ore s o n th ese

(p < 0.0005)

(table 19).

Th is

three m e a s u r e s s i g n i f i c a n t l y

p r e d i c t the s c o r e o n p o s t - t e s t o f b a s i c m a t h e m a t i c a l u n d e r ­
st anding.

T h e y h a v e a m u l t i p l e c o r r e l a t i o n of 0.7969 w h i c h

Table 19.— Analysis of Variance for Overall Regression of Post-test in Basic Mathematical Under­
standings and Other Variables.

Sum of Squares

Deg of Freedom

797.17159623

Error

458.16173710

20

1255.33333333

23

Total (About Mean)

Cases
24

265.723865

Sig

11.5996

0.0005

22.90808687
197

Regression (About Mean)

Mean Square

Multiple Corr Coefs
R2
.6350

R
.7969

R Bar 2
.5803

R Bar
.7518

Standard Error of
Estimate

198
was s i g n i f i c a n t l y d i f f e r e n t from zero, and that they jointly
a c c o u n t e d for 63.50 p e r c e n t of the variance in the Dossetts
post-test.

H ow e v e r A d d i t i o n and Sub tra cti on of Wh ole

Nu mbers was f ound n o t to contrib ute si gni fic ant ly to the
p o s t - t e s t score in m a t h e m a t i c a l u nde r s t a n d i n g

(p < .605).

A s s e s s m e n t of the C o n t r i b u t i o n of Di ffe ren t
A s p e c t s of A t t i t u d e to w a r d Mathemat ics
and Sc hool L e a r n i n g to General
^ At ti t u d e t o w a r d Ar ith met ic" "
The results of Hypothes is C2 indicates no s i g n i f i ­
cant d i f f e r e n c e on attit ude t oward ar ith met ic b e t w e e n the
interns and o t h e r groups,

y e t test of the fitness of the

m o d e l i n d i c a t e d that a s i g n i f i c a n t p r e c i s i o n was g a i n e d by
u s i n g a nal y s i s o f c ova ri a n c e m o d e l

(P = 35.2510, p < .0001)

i m pl y i n g that the groups w e r e s i g n if ica ntl y di ffe r e n t in
e n t r y a t t i t u d e tow a r d arithmetic.

Results of Hypothe sis B2

also s h o w e d that the interns at titude toward a rit hme tic
s i g n i f i c a n t l y inc re a s e d at the end of instruction.
r esult o f the fitness of the model.

The

Hypoth ese s C2 and B2

jointly imply t h a t the attitudes of the interns w h i c h w ere
d i f f e r e n t f r o m those of o t h e r groups w e r e b ro u g h t to the
same level as others.

This p a r t o f the study at tempts to

find o u t w h a t factors c o n t r i b u t e d to this gain.
By u s i n g the scores o f the interns on all m e a s u r e s
and att i t u d e scales,

the p r o d u c t - m o m e n t cor relations

be t w e e n the p o s t — test score on attitu de toward ar ith met ic
and each of the o t h e r mea s u r e s and att itude scale we re
found to be

.804,

.794,

.423,

.368,

.358,

.330,

.284,-.264,

199
.252,

.214,

0.058,

-0.005,

-0.035,

-0.109,

-0.206.

The

first three c o r r e lat ion coeffici ent s are found to be s t a ­
tistical ly dif fe r e n t from zero and they are for cor rel ati ons
b e t w e e n the p o s t - t e s t score on attitude toward m a t h ema tic s
and p re- t e s t score o n a tti tud e toward mathematics,

Enjoyment

of Ma the mat ic s Scale, A t t i t u d e To war d Ma n a nd His E n v i r o n ­
m e n t respectively.
and

Though the next two correlat ion s

.36 8

.358 are not s t a t i stic all y s i g n i f ica nt it is i nteresting

to note that they are for n o n - a tti tud e m eas u r e s

(table 18).

Re gre ssi on analysis showed that w h e n attitude
toward m a t h e ma tic s as a process score and the "Enjoyment"
scale score w e r e not to be de le t e d from the regress ion
eq ua t i o n p r e d i cti ng the attitude at the end of instruction,
analysis of varian ce

for ove r a l l reg res s i o n gave an F value

of 18.4379 w h i c h was signif ica ntl y high
a m u l t i p l e c o r r e lat ion of
from zero,

(p < .00005)

wi th

.8570 s i g n i f i c a n t l y d i f f ere nt

and that they jointly acc ou n t e d for 73.44 p e r ­

cent of the v a r i a t i o n

(table 20).

The o ve r a l l regress ion

in clu d e d p r e - t e s t score on a ttitude w h i c h c o n t r i b u t e d
s i gni f i c a n t l y

(p < .011), E n j o y m e n t A t t i t u d e Scale which

co n t r i b u t e d s i g n i f i c a n t l y

(p < .0 2 1 ) and att itude toward

ma t h e m a t i c s as a p rocess and a con sta nt b o t h of w h i c h did
not co ntr i b u t e s i g n i f i c a n t l y
tively) .

(p < .693, p < .203 r e s p e c ­

Table 20.— Analysis of Variance for Overall Regression of Post-test Attitude Toward Arithmetic and
Ohter Variables.

Sum of Squares

Deg of Freedom

Regression (About Mean)

543S.28468099

Error

1966.34031901

20

7404.62500000

23

Cases

24

1812.76156033

Sig

18.4379

0.0005

98.31701595

200

Total (About Mean)

Mean Square

Multiple Corr Coefs
R2
.7344

R
.8570

R Bar 2
.6946

R Bar
.8334

Standard Error of Estimate
9.91549373

201
Summary of Findings
A n alysis of the d a t a co lle c t e d d ur i n g this study
re ve a l e d the follow ing results:
1.

The m e a n p o s t - t e s t score of the interns was s i g n i f i ­
cantly hi g h e r than the mean pr e-t e s t score on the
c r i t e r i o n - r e f e r e n c e d measures in:
(b) Numeration,

2.

(a) Measurement,

(c) A d d i t i o n and S u b t r a c t i o n of

Wh o l e Numbers,

(d) M u l t i p l i c a t i o n and D i v i s i o n of

W h o l e Numbers,

and

Of the 24 interns
(92%)

(e) Fractions.
inc luded in the analysis,

re ached m ast e r y level

23

(96%)

reached mas t e r y

21

(87.5%)

(80%)

level

in Me asurement,

(80%)

rea c h e d m a s t e r y level

and Su b t r a c t i o n of W h o l e Numbers,

22

in Numeration,

(80%)
20

in A d d i t i o n

(83%)

reached

m a s t e r y level on M u l t i p l i c a t i o n and Di vi s i o n of
W h o l e Numbers,

and 16

(67%)

r eached m a s t e r y

level

on Fractions.
3.

T her e wa s s i g n i f i c a n t d i f f e r e n c e on a tes t of m a t h e ­
m a t i c a l u n d e r s t a n d i n g s b e t w e e n the p o s t - t e s t scores
of the interns and t hei r p r e - t e s t scores.

4.

The re w a s a s i g n i f i c a n t i m p r o v e m e n t on an arithmet ic
at titude i nve ntory b e t w e e n the p o s t - t e s t scores of
the in terns and their p r e - t e s t scores.

5.

The a d j u s t e d mean p o s t - t e s t score of the interns,
was s i g n i f i c a n t l y h igh er than the ad ju s t e d me an
p o s t - t e s t score of a group of p r o s p e c t i v e teachers

202
in the r e g u l a r t e a c h e r e d u c a t i o n program,

G^,

on a

t e s t of m a t h e m a t i c a l u n d e r s t a n d i n g s .
6.

T h e a d j u s t e d m e a n p o s t - t e s t score o f the g r o u p of
prospective elementary

t eac her s w h o h a d the e x p e r i ­

mental content-method integrated mathematics
e d u c a t i o n , G 2 , was

signi fi can tly h i g h e r than the

a d j u s t e d m e a n p o s t - t e s t score o f p r o s p e c t i v e
in the r e g u l a r t e a c h e r e d u c a t i o n program,

tea che r

G ^ t on a

test of m a t h e m a t i c a l u n d e r s t a n d i n g s .
7.

T h e r e was

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

the

a d j u s t e d m e a n p o s t - t e s t sco res of the i n t e r n s , G ^ ,
an d the g r o u p of p r o s p e c t i v e e l e m e n t a r y tea ch e r s w h o
h a d the e x p e r i m e n t a l c o n t e n t - m e t h o d i n t e g r a t e d
m a t h e m a t i c s educati on,

G 2 , on a test of m a t h e m a t i c a l

understandings.
8.

T h e r e was a m a r g i n a l 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
th e a d j u s t e d m e a n p o s t - t e s t sc ore s of three e ntr y
attitude

levels.

The d i r e c t i o n of this s i g n i f i c a n t

c o u l d not be d e t e c t e d by S c h e f f e 's
9.

T h e r e was no m e t h o d of i n s t r u c t i o n

method.
by a t t i t u d e

level i n t e r a c t i o n on b a s i c m a t h e m a t i c a l u n d e r ­
standings .
10.

T h e r e wa s no s i g n i f i c a n t d i f f e r e n c e on the adjusted,
m e a n s of p o s t - t e s t sco res of th e three i n s t r u c t i o n a l
gr oup s on a t t i t u d e

11.

toward a r i t h m e t i c .

The a d j u s t e d m e a n p o s t - t e s t s c o r e of the g r o u p w i t h
medium entry mathematical aptitude

on a t t i t u d e

203
toward is signif ica ntl y h igher than the ad jus ted
p o s t - t e s t score of the group w i t h low entry m a t h e ­
ma tical aptitude on attitude toward mathematics.
There was no signific ant differe nce b etw een other
pairs.
12.

There was no s i g n i fic ant interact ion bet wee n methods
of instruct ion and m a t h e mat ica l aptitude on attitude
toward mathematics.

13.

There was signific ant differ enc e b etween the interns'
pe rce pti on of their learning e n v i r o n m e n t and each
of the other group's p e r c ept ion of their learning
environments, mo s t l y in favor of the Interns
learning environm ent s

(see 16 b e l o w ) .

T her e was no

signifi can t difference b etween other two groups.
14.

There was no group

(methods)

by entry attitude

in t e r a cti on on p e r c e p t i o n of learning environment.
15.

Th ere was group
interaction,

(method)

by m e a s u r e of p e r c e p t i o n

w i t h interns c o n s i s ten tly sc ori ng

higher except on view of mathem ati cs

as process

where they were e xceeded by the second ex p e r i m e n t a l
group, G 2 .
16.

There were s i g n i fic ant correlations b e t w e e n

(a) each

of the pos t- t e s t scores on the c r i t e r i o n - r e f e r e n c e d
me asures and the pos t- t e s t score on b a s i c m a t h e m a t i ­
cal understanding,

(b) pre -t e s t and p o s t - t e s t

scores on basic m a t h e m a t i c a l understandings,

and

204
<c ) score of Enjo y m e n t of M a t h e m a t i c s A t t i t u d e
Scale and pos t- t e s t on ba s i c m a t h e m a t i c a l u n d e r ­
standings .
17.

Scores on p o s t - t e s t criterion mea s u r e s on Numeration,
A d d i t i o n and S u b t ra cti on of Whole Nu m b e r s and
F rac tions ac cou n t e d for more than 6 3 per c e n t of
the score on basic m a t h e m a t i c a l understandings.

18.

T h e r e was s i g n i f i c a n t c o r r e l a t i o n b e t w e e n each of
p r e- t e s t scores on at titude toward ma thematics,
scores of E n j o y m e n t of M a t h e m a t i c s Scale and scores
on a ttitude toward m an and his e n v i r o n m e n t and the
p o s t - t e s t scores on a ttitude toward arithmetic.

19.

Scores on p r e - t e s t scores on attit ude toward
ma t h e m ati cs

and E n j o y m e n t of M a t h e mat ics

ac cou nte d

for about 73 p er c e n t of the score on at tit ude
toward mathematics.

CHAPTER V
SUMMARY,

CONCLUSIONS, A N D R E C O M M E N D A T I O N S
Summary

Du rin g the 19 60s in the U n i t e d States of A m e r i c a
m a n y educato rs
teachers

felt the need for ref o r m in p r e p a r a t i o n of

for inner-c ity schools.

In re sponse to this the

Congress e s t a b l i s h e d the Tea c h e r Corps in 1965.
or ig i n a l purpose s of the program,
a u t h o riz ing legislation,
op portunities
low-income

were

The

as p r o v i d e d in the

(1 ) to s t r e ngt hen ed u c a t i o n a l

for chi ldren in areas w i t h c o n c e n t r a t i o n s of

families,

(2 ) to att r a c t and pre p a r e p ers ons

to b ecome teachers in such areas through c o o r d i n a t e d wo rkstudy experiences,
versities,

schools,

and

(3)

to en cou r a g e co lleges and u n i ­

and state depart men ts of e d u c a t i o n to

w o r k together to br o a d e n and improve t e a c h e r - e d u c a t i o n
programs.
Th is study was

the initial e v a l u a t i o n of the m a t h e ­

matics e d u c a t i o n co mpo n e n t the eighth cycle T e a c h e r Corps
p r o g r a m at the Mic h i g a n State U n i v e r s i t y c o n d u c t e d du rin g
the 1973-74 a cademic year.

205

206
P urp ose
The m a j o r purpose of the study was to pro v i d e both
the " i n t r i n s i c 11 and "pay-off" parts of the p rocess and
p r o d u c t of the mathematics c u r r i c u l u m and i n s t r u c t i o n u sin g
internal,
cycle.

external,

and contextual sources of the eighth

Specifically,

g a t i o n so ugh t to
of ma t h e m a t i c s

the intrins ic aspect of the i n v e s t i ­

(1 ) analyze and ev aluate the ade qua cie s

c o n t e n t - m e t h o d int egr a t e d c o m p o n e n t of the

p r o g r a m in m e e t i n g the needs of the p r o s p e c t i v e elementa ry
sc h o o l teachers

( int e r n s ) ; (2 ) pro vid e a c rit ica l ap pra isa l

of i n s t r u c t i o n a l methods and the clinical e x p e r i e n c e s in
the program.

The p ay- off aspect of the study s o u g h t to

(3) e v a l u a t e - l e a r n i n g the ef f e c t of i n s t r u c t i o n as p r e s c r i b e d
by the m a t h e m a t i c s e duc a t i o n component of the T e a c h e r Corps
p r o g r a m on the interns,

in rel ation to s p e c i f i e d c o m p e ­

te ncies and assess of the interns a chieved a degree of
m a s t e r y o v e r these competencies;

(4) e v a l u a t e - l e a r n e r s

the

ef f e c t i n s t r u c t i o n on the b a s i c m at h e m a t i c a l u nde rs t a n d i n g s
of the interns;

(5) e v a l u a t e - l e a r n e r s - e f f e e t of i ns t r u c t i o n

on the interns'

att itudes toward mathematics;

(6 ) e v a l u a t e

i n s t r u c t i o n by co mpa r i n g the m a t h e m a t i c a l u n d e r s t a n d i n g
and att itude to ward m a t h e m a t i c s of the interns w i t h the
m a t h e m a t i c a l und ers t a n d i n g s and attitudes of o t h e r students
enrolled

[a]

in an oth er e x p e rim ent al program,

[b] in the

re g u l a r e l e m e n t a r y teacher m a t h e m a t i c s e d u c a t i o n program;
(7) co mpare interns'

and students'

e n v i r o n m e n t s of learning;

p e r c e p t i o n s of their

(8 ) assess

the c o n t r i b u t i o n of

207
d i f f e r e n t units of the m a t h e m ati cs in s t r u c t i o n to the bas ic
m a t h e m a t i c a l und ers tan din gs of the interns;

and

(9) d e t e r ­

mine the relati ons hip b e t w e e n the attitude toward m a t h e ­
m atics and interns'

p e r c ept ion s of ma the mat ics

learning,

enjoyment, value and environments.
Students

in the Study
T h r e e groups of students w e r e in vol ved in the

study.

T he students in the T e a c h e r Corps pro g r a m s

met for six hours p e r w e e k in Fall term 19 73,

(interns)

four hours

p e r w e e k in the W i n t e r and Spring terms of 19 74.

They

c o v e r e d five learning units wh i c h were m a t h e m a t i c s contentm e t h o d i n t e g r a t e d taught under l a b o r a t o r y - o r i e n t e d w i t h
m a s t e r y lea rning approach.

They spent four hours daily in

e l e m e n t a r y school, w h e r e they w e r e p r o v i d e d w i t h clinical
e x p e r i e n c e under their team leaders and faculty members.
T w e n t y - f o u r ou t of thirty w h o o r i g in all y e n t e r e d the
p r o g r a m w e r e used.

The second group con si s t e d of twenty-

one st udents w h o w e r e random ly se lec ted in Fall,

19 73 and

given c o n t e n t - m e t h o d i n t e gr ate d i n s t r uct ion si m i l a r to that
of i nterns b u t w i t h o u t mas ter y-a ppr oac h.

They m e t for only

si x hours per w e e k in the laboratory and spent one h o u r per
w e e k on cl ini cal e x p e r i e n c e in elemen tar y schools.
third group of students,

The

G^, had the regular co nt e n t and

m e t h o d s epa rated m a t h e m a t i c s e d u c a t i o n program.

They wer e

used for the study du r i n g the Fall t e r m w h e n they were
ha v i n g the me thods course.

The con t e n t course is a

208
pr e-r equ isi te for the me tho ds course.

Only e igh t e e n out of

all the students that c omp let ed all the tests w e r e used in
the analysis.
Li ter atu re Review
Re sea rch efforts on the nature of the disadvantaged,
the d ifficulties e n c u m b e r e d by the i nner-city schools,

the

root of the problems that led to e s t a b l i s h m e n t of T eac her
Corps p r o g r a m w e r e r eviewed in ch a p t e r II.

The w o r k of

A u s u b e l and others led to the premise that there are
critical differen ces bet w e e n the skills req ui r e d to teach
in low-income schools and m i d d l e -cl ass schools b u t this
pr emise does not imply that unique principl es of training
are inv olved in the two d iff ere nt settings.
research also show that the differen ces
experiences,

Results of

in values, p r i o r

and environm ent s among ch ildren from various

income, ethnic,

and racial subgroups are so g r e a t that

teachers need special training in order to apply the
principles and fashion the pr oce d u r e s that w i l l

facilitate

learning process.
Ef forts of p r o f e s s i o n a l groups in e x c h a n g i n g ideas
and d i s s e m i n a t i n g i n f o r m a t i o n on le arning and i nst ruc tio nal
theories finally led to theor eti cal

foundation of the

em erging p rac tices in m a t h e m a t i c s in str uct ion al pr o c e d u r e s
for the disadvantaged.
w o r k of Bruner,

Carroll,

M a j o r p a r t of this rests on the
A u s u b e l and Bloom.

The use of

laboratory a ppr o a c h in the t eaching of m a t h e ma tic s to

209
c h i l d r e n and p r o s p e c t i v e sc hoo l t e a c h e r s in g e n e r a l and
d i s a d v a n t a g e d c h i l d r e n in p a r t i c u l a r is b e g i n n i n g to have
m o r e s u p p o r t i v e r e s e a r c h evidence.
In the last s e v e n years e d u c a t o r s h a v e realized,
m o r e then ever,

the i m p o r t a n c e of u sin g e v a l u a t i o n in

m a k i n g j u d g m e n t and d e c i s i o n s a b o u t a program.

This has

led to the r e c o g n i t i o n of c u r r i c u l u m e v a l u a t i o n as an art,
and a p r o f e s s i o n .

Ef fo r t s are b e i n g m a d e to e s t a b l i s h a

f r a m e w o r k a nd b u i l d a the ory of e v a l u a t i o n that w o u l d be
u n i v e r s a l l y a c c e p t e d to p r a c t i c i n g e v a l u at ors .
p r o b l e m c u r r e n t l y r e c e i v i n g a t t e n t i o n is
instruments
evaluation.

A related

the d e v e l o p m e n t of

tha t s h o u l d be us ed in c o l l e c t i n g dat a for such
P o p h a m a n d oth ers h ave c o l l e c t e d r e a d a b l e

p a p e r s in m e a s u r e m e n t l i t e r a t u r e t hat s u m m a r i z e s e a r l i e r
e f f o r t s o n the subject.
C o n c e r n e d w i t h the p r o b l e m o f t e a c h e r training,
mathematics

e d u c t o r s h a v e n ot a g r e e d o n the m a t h e m a t i c s

c o n t e n t o f b o t h the e l e m e n t a r y an d s e c o n d a r y t e a c h e r s e d u ­
cation programs.

R e l a t e d is sue s s til l to be r e s o l v e d

i n c l u d e e m p h a s i s o n m a s t e r y of m a t h e m a t i c a l c o n t e n t as
o p p o s e d t o c o v e rag e,

the e x t e n t c o m p u t a t i o n a l skill as

o p p o s e d to c o n c e p t u a l u n d e r s t a n d i n g is sought,
c o n c r e t e ver sus
geometry
grades.

a b s t r a c t in t e a c h i n g m a t h e m a t i c s ,

(kind a n d quantity)
Active

the use of
the

in e l e m e n t a r y an d s e c o n d a r y

r e s e a r c h is g o i n g to re s o l v e s c a n t y c o n t r a ­

d i c t o r y r e s e a r c h r e s u l t s ava ila ble .

Research

f ind ing s on

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

210

a n d a t t i t u d e of e l e m e n t a r y
d i s t u r bing.

sch o o l t e a c h e r s

The e f f e c t of e n t r y behavi o u r ,

an d affective,

are still very
both cognitive

a n d m e t h o d of i n s t r u c t i o n on a c h i e v e m e n t is

s t i l l of i n t e r e s t to r e s e a r c h e r s .
Instrumentations
The f o l l o w i n g i n s t r u m e n t s w e r e d e v e l o p e d or s e l e c t e d
f o r the c o l l e c t i o n of data:
achievement measures
Dossett's

(1) Five c r i t e r i o n - r e f e r e n c e d

(two p a r a l l e l f o r m s ) , (2) M.

J.

test of B a s i c M a t h e m a t i c a l U n d e r s t a n d i n g s

p a r a l l e l forms),

(3)

D u t t o n ' s A t t i t u d e Inventory,

(two

(4) A t t i ­

tude S cales T o w a r d D i f f e r e n t A s p e c t s of M a t h e m a t i c s
d e v e l o p e d by the I n t e r n a l S t u d y of A c h i e v e m e n t in M a t h e ­
ma tics,

and

(5)

d e v e l o p e d b y L.
Hypotheses
The

E n j o y m e n t a n d V a l u e of M a t h e m a t i c s S cales
R. Aiken.

and F i n d i n g s
following multivariate hypothesis

associated univariate hypotheses were

and its

t e s t e d to ass e s s

the

e f f e c t o f the i n s t r u c t i o n a l p r o g r a m on the a c h i e v e m e n t of
i n t e r n s on the p r e s c r i b e d m a t h e m a t i c a l c o m p e t e n c i e s :
a.

T h e p o s t - t e s t m e a n s s c o r e w i l l e x c e e d the p r e - t e s t
me a n s s c o r e of the inte r n s

(G^)

on the c r i t e r i o n -

r e f e r e n c e d measures.
H y p o t h e s i s A and its a s s o c i a t e d u n i v a r i a t e h y p o t h e ­
ses w e r e not r e j e c t e d .

211
T h e f o l l o w i n g two h y p o t h e s e s w e r e t e s t e d to a s s e s s
changes

in the i n t e r n s ' b a s i c m a t h e m a t i c a l k n o w l e d g e and

a t t i t u d e t o w a r d arithmetic:
Bl.

T h e p o s t - t e s t m e a n s c o r e of the int e r n s w i l l
e x c e e d th e i r p r e - t e s t m e a n s c o r e on D o s s e t t ' s

test

of m a t h e m a t i c a l u n d e r s t a n d i n g .
B2.

T h e p o s t - t e s t m e a n score of the int e r n s w i l l e x c e e d
t h e i r p r e - t e s t m e a n s c o r e on D u t t o n ' s

arithmetic

a t t i t u d e inventory.
H y p o t h e s e s Bl and B2 w e r e n o t rejected.
The following hypotheses were

t e s t e d to c o m p a r e the

i n t e r n s a n d s t u d e n t s e n r o l l e d in the r e g u l a r t e a c h e r e d u ­
c a t i o n program.

The p u r p o s e is to d e t e r m i n e

of m e t h o d of i n s t r u c t i o n
toward mathematics

{three

(three

(1)

the e f f e c t

l e v e l s ) and e n t r y a t t i t u d e

levels)

upon the m a t h e m a t i c s

a c h i e v e m e n t at the e n d of i n s t r u c t i o n , (2 ) the e f f e c t of
m e t h o d of i n s t r u c t i o n
aptitude

(three l e v e l s ) a n d m a t h e m a t i c a l

(three levels)

on the a t t i t u d e t o w a r d m a t h e m a t i c s

at the e n d of i n s t r u c t i o n .
Cl

(a)

T h e r e w i l l b e n o t r e a t m e n t m a i n e f f e c t on m a t h e ­
matics

(b)

T h e r e w i l l b e no a t t i t u d e m a i n e f f e c t o n m a t h e ­
ma t i c s

(c)

achievement.

achievement.

T h e r e w i l l be n o t r e a t m e n t b y a t t i t u d e i n t e r ­
action on mathematics

H y p o t h e s e s Cl(a)
significance.

and

achievement.

(b) w e r e r e j e c t e d at

.05 level of

H y p o t h e s i s Cl (c) w a s n o t r e j e c t e d .

212
C2

(a)

There will be no treatment m a i n effect on
attitude toward mathematics.

(b)

There w i l l be no aptitude m a i n effect on
attitude toward mathematics.

(c)

There w i l l be no treatment by aptitude int e r ­
action on attitude toward mathematics.

Hypothesis C2(a)
r e j e c t e d at

was not rejected.

Hypothesis C2(b)

.05 level of significance.

was

Hypothesis C2(c)

was not rejected.
The

following hypoth e s i s w i l l be tested to compare

the interns and o t h e r two groups of students in the regular
t e a cher e d u c a t i o n p r o g r a m on five different aspects of
attitudes.
D

(a)

There w i l l be no treat m e n t main effect on p e r ­
c eption of learning environments.

(b)

There w i l l be no entry attitude m a i n effect on
p e r c e p t i o n of learning environments.

(c)

There w i l l be no treat m e n t by entry attitude
i n t e r action on p e r c e p t i o n of learning e n v i r o n ­
ments .

H ypoth e sis D(a) was rejec t e d at

.05 level of significance.

H ypothesis D(b) was not rejected.
r e j e c t e d at

Hypothesis D(c)

was

.05 level of significance.

S t a t i s t ical A n a l y s i s
One w a y m u l t i v a r i a t e analysis and the corre s p o n d i n g
univa r i ate a analysis of variance were used for testing

213
Hypotheses A and B,

A 3 x 3 Analysis of C o v a r i a n c e was

used for each of Hypotheses Cl and C2.

A 3 x 3 mul t i v a r i a t e

A n a lysis of V a r i a n c e was used to test Hypotheses D.

Step­

w i s e regression method was used to estim a t e the contri b u t i o n
of l e a rning units to the terminal m a t h e m a t i c a l u n d e r ­
standings of the interns.

Stepwise r e g r ession m e t h o d was

also used to assess the contributions of the interns p e r ­
ceptions of m a t h e matics
learning,

learning,

school and school

e n j o y m e n t of mathematics,

value of m a t h e m a t i c s to

the interns attitude toward mathematics.
Five p e r c e n t level of sig n i f i c a n c e was used in
a c c e p t i n g or r e j e c t i n g the research hypotheses.
Conclusions
The

following conclusions

are ba s e d on the findings

of this study.
1.

Analysis of the m a t h e m a t i c s educa t i o n c o m p o n e n t of
the interns

ind icated that the m a t h e matics co n t e n t

is not m e a n i n g f u l l y d i f f e r e n t from the con t e n t in
the regular m a t h e m a t i c s e d u c a t i o n for el e m e n t a r y
teachers at the M i c h i g a n State University.
2.

O n the c r i t e r i o n - r e f e r e n c e d meas u r e s the interns
show si g n i f i c a n t gains in a c h i e vement

(p < .0 0 0 1 )

on all measures.
3.

O v e r 90 p e r c e n t of the interns reached m a s t e r y
level on M e a s u r e m e n t and Numeration,

the perce n t a g e

of students w h o reached mas t e r y on the remain i n g
units d e c r e a s e d g r a d ually to 67 percent

(minimum)

on F r a c t i o n s .
The interns sh o w e d s i g n i f i c a n t gains on a test of
b a s i c m a t h e m a t i c a l understandings.
The interns showed a s i g n i f i c a n t gain in attitude
t oward m a t h e m a t i c s .
W ith initial differences allowed for,

the interns

group scored sign i f i c a n t l y hig h e r on a test of
b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g that did a group
of students in the reg u l a r content, m e t h o d se p a r a t e d
m a t h ematics program.
W i t h initial differences

allowed for the group of

students in the c o n t e n t - m e t h o d integr a t e d m a t h e ­
matics p r o g r a m scored s i g n i f i c a n t l y h i g h e r than the
group of students in the regu l a r c o n t e n t - m e t h o d
separ a t e d m a t h e matics program.
W i t h initial d i f f e rences allowed for, there was no
si g n i ficant d i f f e r e n c e in the b a s i c m a t h e m a t i c a l
u n d e r standings b e t w e e n the interns and the groups
of students in the c o n t e n t - m e t h o d i n t e g r a t e d
ma t h e m a t i c s program.
W i t h initial d i f f e rences all o w e d for,

there w a s

statistically s i g n i f i c a n t d i f f e r e n c e on a test of
basis m a t h e m a t i c a l u n d e r standings of groups w i t h
d i f f erent

(three)

entry levels of attitude toward

215
m a t h e m a t i c s b u t the d i f f e r e n c e is n o t m e a n i n g f u l l y
large to d e t e r m i n e the d i r e c t i o n of s i g n i fic a n c e .
10.

With initial differences

a l l o w e d for,

there w a s no

m e t h o d of i n s t r u c t i o n by e n t r y a t t i t u d e

level

i n t e r a c t i o n on t e r m i n a l b a s i c m a t h e m a t i c a l u n d e r ­
standings .
11.

With initial differences

allowed

for,

there w a s 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 the a t t i t u d e of all
the g r o u p s
12.

toward m a t h e m a t i c s .

W i t h ini t i a l d i f f e r e n c e s

allowed

for,

there was

s i g n i f i c a n t d i f f e r e n c e o n l y b e t w e e n the grou p s w i t h
m e d i u m e n t r y m a t h e m a t i c a l a p t i t u d e and low e n t r y
m a t h e m a t i c a l a p t i t u d e in f a v o u r of the former.
13.

With initial differences

a l l o w e d for,

there w a s no

m e t h o d of i n s t r u c t i o n by a p t i t u d e i n t e r a c t i o n o n
attitude toward mathematics.
14.

The interns'

g r o u p p e r c e p t i o n o f th e i r e n v i r o n ­

m e n t s of l e a r n i n g is m o r e

favourable

c e p t i o n of e a c h o f o t h e r g r o u p s .

than the p e r ­

There was 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 the o t h e r two groups'
p e r c e p t i o n s of their e n v i r o n m e n t s of learning.
15.

T h e r e w a s no g r o u p by e n t r y a t t i t u d e i n t e r a c t i o n on
p e r c e p t i o n of l e a r n i n g e n v i r o n m e n t s .

16.

T h e l e a r n i n g units c o n t r i b u t e d s i g n i f i c a n t l y to
the t e r m i n a l b a s i c m a t h e m a t i c a l u n d e r s t a n d i n g s of
the interns.

216
17.

Entry attitude toward m a t h e matics and the enj o y m e n t
of the m a t h e m a t i c s class con t r i b u t e d signific a n t l y
to the terminal attitude of the interns toward
mathematics.
O n the basis of changes in m a t h e matical achievement

on the c r i t e r i o n - r e f e r e n c e d measures and the tests of basic
m a t h e m a tical understandings and attitudes the results
indicate that m o s t of the interns,
cational mathematical)

though w i t h p o o r

(edu­

b a c k g r o u n d can reach a reasonable

level of competency equal and prob a b l y hi g h e r than that of
pr e s e n t regular students in ma t h e m a t i c s

for p r o s p ecti v e

el e m e n t a r y school teachers in laboratory-oriented,
m e t h o d integrated,

content-

p r o g r a m w i t h s u b s t antial clinical

e xperi e nce if given s u f f i c i e n t time and help.
I n t e rpretation and D i s c u s s i o n of Findings
A s s u m i n g that outcome of learning depends on the
e nviro n ments of learning it is reasonable to start the
i n t e r p r etation of results f r o m e v a l u a t i o n of learning
environments.
E n v i r o n ments of Learning
N o n rejection of Hypothesis D(c)

indicates that the

r e l a t i o nship betw e e n the p e r c e p t i o n of learning e n v i r o n ­
m ents of subjects in groups G^, G 2 r G^ m i g h t w e l l be the
same across all levels of e n t r y attitude toward mathematics.
This suggests that the responses of subjects on Husen's
A t t i t u d e Scale w i t h i n each group w e r e u n a f f e c t e d by their

217
entry attitude toward mathematics.
of this

The p r o b a b l e i m p l i c a t i o n

is that t h e i r p e r c e p t i o n s f a i r l y e s t i m a t e s t h e i r

e n v i r o n m e n t s of l e a r n i n g .

This

i m p l i c a t i o n is h o w e v e r

c o n f i r m e d by a c c e p t a n c e of H y p o t h e s e s D(b)
e n t r y a t t i t u d e effect.

Res u l t s

t h a t there is no

of H y p o t h e s e s D(b)

and

(c)

i m p l y t h a t the s u b j e c t s ' p e r c e p t i o n o f t h e i r e n v i r o n m e n t s
of l e a r n i n g as m e a s u r e d by H u s e n ' s A t t i t u d e S c a l e s
fair e v a l u a t i o n o f th e i r

is a

learning environments.

T h e r e s u l t of H y p o t h e s i s
c a t i o n o f resu l t s of H y p o t h e s e s

D(a)
D(b)

combined with impli­
and

(c)

in the p r e c e d i n g

p a r a g r a p h show t h a t the e n v i r o n m e n t o f l e a r n i n g of G^ is
s i g n i f i c a n t l y s u p e r i o r than e a c h of those of G 2 and G^
w h i l e t h a t of G 2 is n o t s t a t i s t i c a l l y ,
G3.

In o t h e r w o r d s

the e v i d e n c e

b e t t e r than that o f

indicates

t h a t the

e n v i r o n m e n t s of l e a r n i n g of G-^ w h i c h w a s m a s t e r y app r o a c h ,
laboratory-oriented,

c o n t e n t and m e t h o d i n t e g r a t e d t o g e t h e r

with substantial clinical experience provides

a learning

e n v i r o n m e n t b e t t e r than that of l a b o r a t o r y - o r i e n t e d ,
content-method integrated,

together with one hour per week

c l i n i c a l e x p e r ience;

whereas

the

latter does not p r o v i d e a

significantly better

l e a r n i n g e n v i r o n m e n t t h a n the r e g u l a r

p r o g r a m w h i c h is p a r t i a l l y l a b o r a t o r y - o r i e n t e d m a t h e m a t i c s
c o n t e n t s e p a r a t e d f r o m m e t h o d s c o u r s e w h i c h is l a b o r a t o r y oriented .

The non-appreciable difference between

two o b t a i n e d

f r o m this s t u d y m i g h t be due to the

the p r e s s u r e w a s
group,

too m u c h

for the s t u d e n t s in G 2 .

G 2 , p o i n t e d o u t in c h a p t e r

the

last

fact that
This

III, h a d t h e i r m a t h e m a t i c s

218
e d u c a t i o n p r o g r a m in Fall qua r t e r in six-quarter h o u r class
meeting

(and one h o u r clinical experi e n c e in school)

the r e g u l a r program,
meetings.

what

, h a d in e l e v e n - q u a r t e r h o u r class

This p r e s s u r e obscures the d i f f e r e n c e betw e e n

n o n - m a s t e r y laborator y - c o n t e n t m e t h o d integr a t e d p r o g r a m
from the regular c o n t e n t - m e t h o d separated program.
the findings agrees w i t h Hollis

(1971)

However,

study that la b o ­

ratories o r g a n i z e d to pro v i d e pers o n a l and indivi d u a l i z e d
a s s i s t a n c e are helpful to learners that are eit h e r culturally
or a c a d e m i c a l l y disadvantaged.
Related Q u e s t i o n :

A rel a t e d q u e s t i o n is:

Is the differ e n c e b e t w e e n groups'

p e r c e ptions of

l earning e n v i r o n m e n t the same across all m e a s u r e s
subsets)?

(five

The m u l t i v a r i a t e F for this mea s u r e by met h o d

(treatment)

i n t e r a c t i o n was
11.

2.2637 w h i c h was sig n i f i c a n t

(p < 0.0286)

in table

This indicates a si g n i f i c a n t

interaction,

thus the differ e n c e in p e r c e p t i o n of e n v i r o n ­

ments of learning b e t w e e n the groups'
each a s pect of e n v i r o n m e n t s .

is not the same for

M e a n score of the groups on

each s ubset of Husen's A t t i t u d e Scales are in table 17,
F igure shows that the i n t e r a c t i o n is disordinal.

This

d i s o r d i n a l i n t e r action shows that only in some cases
the e n v i ronments of learning as p e r c e i v e d by
others.

is

supe r i o r to

In fact G-^ is supe r i o r in all cases ex c e p t on the

learning of ma t h e m a t i c s as process
e x c e e d e d by G 2>

(B^

w h e r e they w e r e

The scores of G^ and G^ on this subset

219
indicate that they vi e w e d m a t h e matics as fixed and g i v e n
once and for all time

(a low score)

while group G 2 viewed

m a t h e m a t i c s relatively more as somet h i n g developing,
growing,

and changing

(a high score)

this m i g h t be a

r e f l e c t i o n o f efforts of two faculty members on the group.
The figure shows that, G 3 's p e r c e p t i o n of e n v i r o n m e n t of
learning is infe r i o r in all cases ex c e p t on attitude t o w a r d
man and e n v i r o n m e n t w h e r e they e x c e e d e d G 2 .
It can be concluded that a m a t h e m a t i c s e d u c a t i o n
p r o g r a m for e l e m entary teachers w h i c h uses Bloom's m o d e l of
m a s t e r y learning, w h i c h is laboratory-oriented,

content-

me t h o d - integr a t e d w i t h substantial clinical e x p e r i e n c e
p r o vides an e n v i r o n m e n t of learning w h e r e t r a i n e e s ’ p e r ­
ception of m a t h ematics
c o n t e m p orary society,

learning,

the role of m a t h e m a t i c s in

the school and school learning,

r e l a t i o nship of m a n to his e n v i r o n m e n t is more
than the p e r c e p t i o n s of trainee in ei t h e r

the

favourable

(1) a p r o g r a m

w i t h e v e r y t h i n g exc e p t mas t e r y approach and sub s t a n t i a l
clinical experience,

or

(2 ) a p r o g r a m sim i l a r to the p r e s e n t

r e gular ma t h e m a t i c s e d u c a t i o n c o m p onent of e l e m e n t a r y e d u ­
cation at the M i c h i g a n State University.
E v a l u a t i o n of Instruction
The rejec t i o n of Hypothesis Cl(a)

and s u b s e q u e n t

a p o s t e r i o r i comparisons sugge s t e d that there w a s strong
e v i dence of nonchance d i f f e rences due to m e t h o d of
instruction.

S u b s e q u e n t l y a p o s t e r i o r comparisons,

via

220
l east square estimates effects reve a l e d a significant
d i f f e r e n c e in m a t h e m a t i c s
favour of G^,
between
1.

achievement b e t w e e n

and

in

and between G 2 and G 3 in favour of G 2 b u t n o t

and G 2 »

T-us,

The interns b a s i c m a t h e m a t i c a l understandings at
the end o f i n s t r uction is not sign i f i c a n t l y d i f f e r ­
ent from that of G 2 •

The i m p l i cation of this is

that w i t h m a n y of the interns from low-income b a c k ­
g r o u n d and p o o r m a t h e m a t i c s b a c k g r o u n d
fitness of m o d e l —A N C O V A - s u p p o r t this)

(test of
it is p o s s i b l e

to bring them to the same level of ba s i c m a t h e ­
m a t i c a l u n d e r s t a n d i n g via B l o o m ’s m o d e l of ma stery
learning,

as students in the regular p r o g r a m w h o

recei v e d some c o n t e n t - m e t h o d i n t e g r a t e d - l a b o r a t o r y o r i e n t e d i n s t r uction

(except for m a s t e r y l e a r n i n g ) .

This p a r t agrees w i t h research findings on Bloom's
m o d e l of mas t e r y learning

(Block,

1971).

The result

is also c o n s i s t e n t w i t h findings of A s t i n et al.
(1972)

and Ausub e l ' s reve r s i b i l i t y theory

(1964)

that academic defici e n c y a s s o c i a t e d w i t h s o c i o ­
e c o n o m i c a l l y d i s a d v a n t a g e d chil d r e n disapp e a r s at
the col l e g e if the d i s a d v a n t a g e d is in approp r i a t e
program
method

(AStin,
(Ausubel,

1972)

or taught by appropriat e

1964).

The open, e f f e c t i v e and

ave r a g e c o m m u n i c a t i o n s a m o n g students

(G^)

un d e r

co o p e r a t i v e goal s t r u c t u r e instruction m i g h t have
c o n t r i b u t e d to the p e r f o r m a n c e of G 1 in the study.

221
2.

The s i g n i f i c a n t differ e n c e b e t w e e n

and G^ on one

hand the G£ and G^ on the o t h e r hand e x t e n d e d the
results of W a s z l y
and show,
a.

(1970)

study to n o n d i s a d v a n t a g e d

in particular,

that the d i s a d v a n t a g e d group,

under mastery,

c o n t e n t - m e t h o d integr a t e d approach,

could

reach a hi g h e r level of m a t h e m a t i c a l u n d e r ­
stand i n g s

for e l e m e n t a r y teachers than a

r eg u l a r g r o u p in the p r e s e n t regu l a r p r o g r a m
at the M i c h i g a n State University;
b.

that the c o n t e n t - m e t h o d i n t e g r a t e d course is
c ap a b l e of p r o v i d i n g a b e t t e r m a t h e m a t i c a l
u n d e r s t a n d i n g for r e g u l a r e l e m e n t a r y teachers
than the pres e n t regular m a t h e m a t i c s e d u c a t i o n
c o m p o n e n t of their program.

The r e j e c t i o n of Hypoth e s i s Cl(b)
(p < .0415).

was m a r g i n a l

The s u b s e q u e n t a p o s t e r i o r i p r o c e d u r e using

Scheffe's m e t h o d failed to de t e c t sign i f i c a n c e in p a i r w i s e
c o m p a r i s on of the a t t i t u d e levels.
two reasons:

This m i g h t be due to

first S c h e f f e ' s t e c h n i q u e is k n o w n to be w e a k

for p a i r w i s e c o m p a r i s o n bec a u s e of its w i d e co n f i d e n c e
i nterval and e s p e c i a l l y w h e n the s i g n i f i c a n t is very
m a r g i n a l as in this case.

R a n d o m d r o p p i n g of six subjects

out of t w e n t y - f o u r in G^ a n d three in G 2 in order to use a
more p o w e r f u l a p o s t e r i o r i T u r k e y - t e c h n i q u e is not de s i r a b l e
for we shall only be using 60 p e r c e n t of the interns w h o
o r i g i n a l l y sig n e d up for the program.

T h e se c o n d m i g h t be

222
due to w e a k r e l a t i o n s h i p b e t w e e n m a t h e m a t i c s
and attitude toward mathematics

achievements

that all r e s e a r c h e f f o r t s

h a v e s h o w n so far c o n t r a r y to the feelings o f m a t h e m a t i c s
educators.
T h e a c c e p t a n c e o f H y p o t h e s i s Cl(c)

of no t r e a t m e n t

by e n t r y level i n t e r a c t i o n is a c o n f i r m a t i o n that the
findings u n d e r H y p o t h e s i s Cl(a)

are i n d e p e n d e n t of the

e n t r y a t t i t u d e levels.
T h e a c c e p t a n c e o f H y p o t h e s i s C2(a),
no s i g n i f i c a n t d i f f e r e n c e
toward mathematics
with studies

that there is

a m o n g the g r o u p s on the a t t i t u d e

at the e n d of i n s t r u c t i o n is c o n s i s t e n t

c a r r i e d out b y V a n c e

(1969)

and J o h n s o n

(1970)

on t h e e f f e c t of l a b o r a t o r y a p p r o a c h on a t t i t u d e m e a s u r e s
t h o u g h c o n t r a r y to the cl a i m s of a d v o c a t e s of l a b o r a t o r y
approach.
out,

It m i g h t b e t h e case as P h i l l i p s

t h at the p e r i o d s

(1973)

pointed

of i n s t r u c t i o n w e r e not s u f f i c i e n t l y

long e n o u g h to a f f e c t a t t i t u d e of the student,

T h e r e j e c t i o n of H y p o t h e s i s C2(b)

especially

that there is 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 of groups w i t h d i f f e r e n t
e n t r y m a t h e m a t i c a l a p t i t u d e s is an i m p o r t a n t result.
posteriori

c o m p a r i s o n shows

A

there is s i g n i f i c a n t d i f f e r ­

ence o n l y in the a t t i t u d e of s u b j e c t s w i t h m e d i u m and low
e n t r y m a t h e m a t i c a l a p t i t u d e in f a v o u r of the g r o u p w i t h
m e d i u m aptitude.

The results indicate

that s t u d e n t s w i t h

h i g h e n t r y m a t h e m a t i c a l a p t i t u d e have m e d i u m a t t i t u d e
toward mathematics

at the e n d o f i n s t r u c t i o n ,

while with

223
i n i t i a l m e d i u m m a t h e m a t i c a l aptit u d e h a v e high a t t i t u d e
toward mathematics,

and that s t u d e n t s w i t h lowest m a t h e ­

m a t i c a l a p t i t u d e at the b e g i n n i n g of i n s t r u c t i o n w i l l h a v e
lowest a t t i t u d e

t o w a r d m a t h e m a t i c s at the end of i n s t r u c t i o n .

T he a c c e p t a n c e o f H y p o t h e s i s C2(c)

of no t r e a t m e n t by

a p t i t u d e i n t e r a c t i o n shows that the r e s u l t of H y p o t h e s i s C2
(b) h o l d s

for any m e t h o d of instruction.

Evaluation of Learners
T h e r e j e c t i o n of the m u l t i v a r i a t e h y p o t h e s i s

and

a s s o c i a t e d u n i v a r i a t e H y p o t h e s i s Bl, B2 that t h e r e w i l l be
n o 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 the p o s t - t e s t m e a n s and
p r e - t e s t m e a n s of int e r n s

on D o s s e t t ' s and D u t t o n ' s

instru­

m e n t a n d the f o r e g o i n g d i s c u s s i o n s h o w that the m a t h e m a t i c s
e d u c a t i o n c o m p o n e n t of the ei g h t c y c l e T e a c h e r C o r p s p r o g r a m
meets

the needs o f the interns.
While

ment only

92 p e r c e n t r e a c h e d m a s t e r y

83 and 67 p e r c e n t s

last t w o topics.

level on M e a s u r e ­

reached mastery

level on the

T h i s m i g h t be e x p l a i n e d in two ways.

the H a w t h o r n e ' s e f f e c t o f

First

la b o r a t o r y a p p r o a c h t u r n e d t h e m

o n at the b e g i n n i n g o f the program.

Secondly,

the e n t i r e

T e a c h e r C o r p s p r o g r a m s e e m e d to b e w e l l o r g a n i z e d at t h a t
time.

H o w e v e r , in S p r i n g t e r m there w e r e a d m i n i s t r a t i v e

difficulties

in s c h e d u l i n g

the m a t h e m a t i c s

class.

not c e r t a i n , until very l a t e , that mathematics
b e held,

t h o u g h this w a s

time this w a s

settled,

It w a s

class w o u l d

in the o r i g i n a l proposal.

By the

a n o t h e r class has b e e n s c h e d u l e d

224
to use the m a t h e matics
for the interns.

laboratory at the hours avail a b l e

W h i l e ev e r y effort was made to b r i n g the

n e c e s s ary mater i a l s to the c l a s s r o o m used this term, m o s t
of the interns
setting.

felt that this was not truly a laboratory

This feeling m i g h t have some sort of n e g a t i o n of

H a w t h o rne's e f f e c t — that they w o u l d not do w e l l in n o n ­
laboratory s e t t i n g — on the interns.
the interns

(not the instructor)

This e x p e c t a t i o n of

m a n i f e s t e d into a m o d i f i e d

form of J a c o b s o n - R o s e n t h a l effect on the interns'

achieve­

ments this term.
A p a r t from the p e r f o r m a n c e of interns on Fractions
w h i c h m i g h t also be affe c t e d by the end of term's pr e s s u r e
on the interns

from other courses

in their schedule,

the

o v e r a l l p e r f o r m a n c e seems to follow findings on pe r c e n t a g e
of students that attain mas t e r y under Bloom's m o d e l of
m a s t e r y learning

(Peterson,

1972).

Recomme n d a t i o n s
The f o l l owing r e c o m mendations are ba s e d on the
i n v e s t igator's i n t e r p r e t a t i o n of the results of this study
and p e r s o n a l o b s e r v a t i o n s in c a r r y i n g out the p r e s e n t
evaluation.
R e c o m m endations for Ac tions
a.

Based o n the Study
It is re c o m m e n d e d that p r o v i s i o n of e n r i c h m e n t

activities be co ntinue d w i t h the interns w h o d e m o n st r a t e
the d e s ire a n d ability.

225
It is r e c o m m e n d e d that m o r e Geom e t r y be inc l u d e d
in the T e a c h e r Corps M a t h e m a t i c s E d u c a t i o n Program.
It is r e c o m m e n d e d that the p r e s e n t m e t h o d of
i n s t r uctio n - m a s t e r y appr o a c h l a b o r a t o r y - o r i e n t e d - c o n t e n t me t h o d i n t e g r a t e d w i t h s u b s t a n t i a l time in clinical
e x p e r i e n c e — be continued w i t h i n c r easing i n t e r jecti n g
lectures.
It is r e c o m mended that e l e m entary p r o b a b i l i t y and
statistics be incl u d e d in the Tea c h e r Corps program,
b.

B a s e d on P e r s o n a l O b s e r v a t i o n s
It is r e c o m m e n d e d that the faculty members

involved

with T e a c h e r Corps spend more time in schools w i t h the
interns.
In order to a c h i e v e this type of i n v o l v e m e n t from
co llege faculty p e o p l e cer t a i n sacrifices s e e m eminent.
Perhaps the m o s t i m p o r t a n t of these is money.

The cost of

train i ng teachers via field e x p e r i e n c e d b a s e d programs

in

w h i c h college faaulty s p e n d time in p u b l i c schools w o u l d
m o s t likely b e g r e a t e r than the p r e s e n t n o n f i e l d ba s e d
programs.

It is also qu i t e likely that this field-

e x p e r i e n c e - b a s e d p r o g r a m w o u l d require additional p e r s o n n e l
to be avail a b l e for supervision.
It is r e c o m m e n d e d that the t e a m leaders be r e q uired
or e n c o u r a g e d to take some g r a d u a t e courses
m a t h e m a t i c s educa t i o n

in el e m e n t a r y

to i m p r o v e their compet e n c y in the

s u p e r v i s i o n of the i n t e r n s .

226
It is r e c o m mended that interns b e tra i n e d in
q u e s t i o n i n g technique w h i c h w i l l enable them to succeed in
d i r e c t i n g c h i l d r e n ’s learning.
It is r e c o m mended that interns b e en c o u r a g e d to
develop interest in reading the A r i t h m e t i c T e a c h e r .

It is

r e c o m m ended that this reading be required not just optional.
It is r e c o m m e n d e d that w h e r e v e r the Indiana
m a t e rials are to be used efforts should b e m a d e to secure
the a c companying t a p e s .
It is r e c o m me nded that class attendance be made
compulsory for the i n t e r n s .
R e c o m m endations for
F uture R e s e a r c h
It is r e c o mmended that the rel a t i o n s h i p b e t w e e n the
p r e s e n t regular mat h e m a t i c s educat i o n s p r o g r a m and a w e l l s t r u c t ured l a b o r a t o r y - o r i e n t e d m a t h e m a t i c s c o n t e n t - m e t h o d
integrated

(with i n t e r j e c t i n g lectures)

p r o g r a m for

p r o s p e c t i v e elemen t a r y teachers b e e x a m i n e d in depth to
d e t e rmine d i f f e rences in m a t h e matics

a c h i e vement and

k n o w l e dge of teaching e l e m e n t a r y school m a t h e m a t i c t o p i c s .
It is rec o m m e n d e d that the e f f e c t of the m a t h e ­
m a tical i n s t r u c t i o n on the cognitive and affective
b e h a v i o u r s of elemen t a r y school c h i l d r e n taught b y the
p r e s e n t interns be studied.

227
It is r e c o m m e n d e d that more research should be
c o n d u c t e d on treatment b y aptitude i n t e r action on m a t h e ­
m a t i c s achievement,

t reat m e n t by attitude interactio n an

achievement.
It is r e c o m m e n d e d that m o r e studies sh o u l d be
c o n d u c ted on treatment by a t t i t u d e interaction on attitude,
t r e a tment by aptitude i n t e r a c t i o n on attitude.

BIBLIOGRAPHY

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Astin,

H, S., Astin, A. W . , B i s c o n t i , A. S., and
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Hi g h e r E d u c a t i o n and the D i s ­
a d v a n t a g e d S t u d e n t . Washington, D.C.:
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Astin, A l e x a n d e r W . , and P a n o s , Robert J.
"The E v a l u ­
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Washington, D.C.:
American Council
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D a v i d P.
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C u l t u r a l l Depr i v e d Child,"
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D e v e l opm ent. "
Re v i e w of E d u c a t i o n a l Research,
43, 2 (1973), 2X V — 36.

167.

Scriven, M.
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ation."
E v a l u a t i o n C o m m e n t , 3 (1972), 1-4.

168.

Shah,

169.

Simmon,

170.

Skinner, B. F.
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171.

Slauchter, C. H.
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A m o n g Ne g r o Ch ild r e n . "
Jo u r n a l of N e g r o E d u ­
c a t i o n (Spring 1969) , 10*>-11.
1

172.

Sowell,

173.

Stake,

174.

_________ .
"Objectives, Priorities, and O t h e r J u d g m e n t
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R e v i e w of E d u c a t i o n a l Research, 40
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175.

Staropoli, C har l e s J., and Heitzmann, W i l l i a m R.
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The T e a c h e r E d u c a t o r , 8 , 3 (1973).

176.

Stiles,

S. A.
" Sel ected G eom e t r i c C o n c e p t T a u g h t to
Children, Ages Seven to Eleven."
The A r i t h ­
m e t i c T e a c h e r , 16, 2 (February 1969), 119-28.
G. B.
"Comments on Implicat ion s of C r i t e r i o n Referenced Measurement."
Jo ur n a l of E d u ­
c a ti o n a l M e a s u r e m e n t , 6 , 4 ri'^9) , X ^ 9 - ^ 0 .

Katye O., and Hodgin, K a t h a r i n e W.
"A 'Head
S t a r t 1 P r o g r a m in S tu d e n t T e a c h i n g in M a t h e ­
ma t i c s ."
T h e M a t h e m a t i c s T e a c h e r , 65
(November 1972), 617-20.
" ™
R. E.
"The C o u n t e n a n c e of E d u c a t i o n a l E v a l u ­
ation."
Tea c h e r s C oll e g e Record, 68 (1967),
523-40.

L i n d l e y J.
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cation."
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_

245
177.

Stodolsky, S. S., and Lesser, G.
" Lea rning Patterns
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Ha r v a r d Ed u c a t i o n a l
Review, XXXVII (1967), 5T(T-93.

178.

Suppes,

P.
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S c i e n t i f i c A m e r i c a n , 215 (1966), 206-21,

179.

Todd, R. M.
"A M a t h e m a t i c s Course for E l e m e n t a r y
Teachers:
Does It Improve U n d e r s t a n d i n g and
At tit u d e ? "
The A r i t h m e t i c Teacher, 13
(1966),
198-366.

180.

Unkel,

181.

Vane, J u l i a R.
"Relation of E a r l y School A c h i e v e m e n t
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182.

Vo ntress, C. E.
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Phi Delta K a p p a , X L V (1963), 77-81.

183.

Walbesser, H. H., Jr., and Carter, H. L.
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18 4.

Walberg,

185.

W a s h b u r n e , C. W.
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to I n d i v i d u a l i z i n g I n s t r u c t i o n and P rom oti ons ."
J o u r n al of E d u c a t i o n a l R e s e a r c h , 5 (19 22),
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186.

Webster,

187.

Wes tbury, I.
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E s t h e r R.
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Washi ngt on, D . C . : EDRS MF. H C .
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A b s t r a c t e d and a n a l y z e d by A r m s t r o n g
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S. W . , and Lund, S. E. T.
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246
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189.

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190.

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191.

Wolf,

E. P., and Wolf, L.
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192.

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193.

Airasian, Peter S.
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194.

Block, James H.
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and T i m e V a r i a b l e s . "
Ph.D. d iss ertation,
U n i v e r s i t y of Chicago, 1970.

19 5.

Dave, R. A.
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E n v i r o n m e n t a l Pro c e s s V a r i a b l e s that are
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Ph.D.
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196.

Dossett, M. J.
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School Te ach e r s . "
Do ct o r a l dis ser tat ion ,
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197.

Howard, V i v a n Gordon.
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D i s s e r t a t i o n Abstrac ts, 31A
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247
198.

Johnson,

R. E.
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Ph.D. dissertation. Uni ver sit y of Minnesota,
1970.

19 9.

Peskin, A. S.
"Teacher U n d e r st and ing and A tti tud e
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Doc toral d i s s e r ­
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200.

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Dis ser tat ion
A b s t r a c t s , 25 (March 1965):3162-3.

201.

Shakrani, M. S.
"A Formative E v a l u a t i o n of the
M a t h e m a t i c s C omp o n e n t of an E x p e r i men tal
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202.

Trevino,

Bertha.
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Ma t h e m a t i c s in the Pri mary Grades."
Ph.D.
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1968, D A 2 9 — 521—A.

203.

Vance,

204.

Ward,

205.

Williams, M a c k e Albert.
"Concept D e v e l o p m e n t in
M e a s u r e m e n t at the Nu rs e r y School Level, w i t h
a N e w Man i p u l a t i v e L e a r n i n g Aid."
Ed.D.
dissertation, C o l o r a d o State College, 1969,
D A 3 0 :4166—A.

J. A.
"The Effects of a M a t h e m a t i c s Lab ora t o r y
P r o g r a m in Grades 7 and 8 : A n E x p e r ime nta l
Study."
Do cto ral dissertation. Uni ver sit y of
Alberta, 1969.
Ro b e r t Leland.
"The E f f e c t of Ins tru cti on and
Age on the C l a s s i f i c a t o r y Be ha v i o r of
C u l t u r a l l y D e p r i v e d Pr e-s c h o o l Children."
Ed.D. dissertation, Indiana University, 1969,
D A 30:4294—A .

248
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206.

Abedor, A.
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P ape r
p r e s e n t e d at the C o n f e r e n c e on Four Case
Studies in Fo rma t i v e Evaluation, Bloomington,
Indiana, July 16-17, 1971.

207.

Anderson, R. C.
"The C o m p a r a t i v e Fi e l d Experience:
A n I l l u s t r a t i o n High School Biology."
In
The P r o c e e din gs of 1969 Invita tio nal C o n f e r ­
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Edu­
ca tio nal T e s t i n g Service, 1969.

20 8 .

Atkinson, R. C.
"Com put er- bas ed In str uct ion in
Initial Reading."
In P r o c e edi ngs of the
1967 Invita tio nal C o n f e r e n c e on Te s t i n g
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Princeton, N.J.:
E d u c a t ion al
T e s t i n g Service, 196 8 .

209.

Beilin,

210.

Bloom,

211.

Bourmuth, J o h n R.
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Fi n a l Report/USOE, C o n t r a c t No. O E C — 3-7_
0700052-0326, 1969.

212.

Eash, M. J.
"Develop ing an I nst ru m e n t for the
A s s e s s m e n t of I n s t r u c t i o n a l M a t e r i a l s
(Form I V ) . 1 P a p e r p r e s e n t e d at the m e e t i n g
of the A m e r i c a n E d u c a t i o n a l Re seach A s s o c i ­
ation, Minneapolis, M a r c h 19 70.

213.

Edwards,

Harry, and Gotkin, Lassar, G.
"Psychological
Issues in the D e v e l o p m e n t of Ma t h e m a t i c s
C u r r i c u l a for So cially D i s a d v a n t a g e d Children."
In SMSG Report of a C o n f e r e n c e on M a t h e mat ics
E d u c a t i o n for B e l o w A v e r a g e A c h i e v e r s ,
A p r i l 19 64.
It has also b e e n p u b l i s h e d in
P a s s o w et al,, e d s ., E d u c a t i o n of the D i s ­
advantaged:
A Book o'£ R e a d i n g s . N e w Y o r k :
Holt, Ri neh art and Winston, I n c .
B. S.
"An I n t r o d u c t i o n to Mas t e r y Le arn ing
Theory."
Pap er read to A E R A S y m p o s i u m
"Schools, Society, and M a s t e r y Learning,"
h e l d on 28 F ebr u a r y 19 73.
New Orleans, as
p a r t of the P r o c e edi ngs of the A E R A A nnu al
Meeting.

R. R.
"The P r e d i c t i o n of Success in Remedial
M a t h e m a t i c s C ou r s e s in the P u b l i c C o m m u n i t y
J u n i o r Col lege."
Mathematics Education
Reports at the N a t i o n a l C ou n c i l of T eachers of
M a t h e m a t i c s 50th A n n u a l Meeting, Chicago,
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249
214.

Fifer,

C.
"Social Class and C u l t u r a l Group D i f f e r ­
ences in D iverse M e n t a l Abiliti es. "
Paper
pr e s e n t e d at the Invitational C o n f e r e n c e on
T e s t i n g Problems, N e w York, 1964.

215.

Forbes,

J. E.
"Reaction to Survey of Projects."
In
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March
1970, pp. 29-32.

216.

Glasser,

R.
"Adapting the Ele men t a r y School C u r r i c u ­
lum to Individual Performance."
P roc eed ing s
of the 196 7 Invitational Con fer e n c e on
Te st i n g P r o b l e m s . Princeton, N . J . : E d u ­
cationa l T est i n g Service, 1968.

217.

Goodwin,

W. L., and Sanders, J. R.
"The Effects on
Pupil P e r f o rm anc e of P r e s e nta tio ns Made on
an Audio- bns ."
Paper p r e s e n t e d at the me eti ng
of the Na ti o n a l Co uncil on M e a s u r e m e n t in
Education, New York, Feb ruary 19 71.

218.

Hatfield, L arry L.
For N C T M C o m m i s s i o n on T eac her
Education.
"Some Issues and Problem s in
Ma t h e m ati cs Te ach er E ducation."

219.

Henderson, Judith E.
"An Introd uct ion to Teaching."
U n i t I, Educ a t i o n 200, Co llege of Education,
M i c h i g a n State University, 19 73,

220.

Kersh,

221.

Lan sin g S chool D i s t r i c t and M i c h i g a n State University.
L a n s i n g School D i s t r i c t 8th Cycle Tea che r
Corps P r o p o s a l . N o v e m b e r 1972,

222.

Leiderman, G l o r i a F.
"Mental D e v e l o p m e n t and
L e a r n i n g Ma t h e m a t i c s in S l o w - L e a r n i n g
Children."
In SMSG Report of a C o n f e r e n c e on
M a t h e m a t i c s E d u c a t i o n for B e l o w A v e r a g e
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223.

Moore,

E. Mildred.
"A Study of Mas t e r y L e a r n i n g in
El eme n t a r y M athematics."
M a t h e mat ics E d u ­
ca t i o n Reports at the Na ti o n a l C o u n c i l o£
Teachers of Ma t h e m a t i c s 50th A n n u a l Meeting,
Chicago, 111., Ap r i l 16-19, 1972, Erie, Ohio.

J. W . , Mahan, J. M . , and Ritts, C. A.
"An
E v a l u a t i o n of the C o n t i n u o u s Progre ss C o n ­
cept of Instruc tio n w i t h U n i v e r s i t y S t u d e n t s . ”
Paper read at the annual m e e t i n g of AERA,
Chicago, Feb rua ry 196 8 .

250
224.

Na ti o n a l A s s e s s m e n t of Edu cat ion al Progress,
Mathe­
ma
t
i
c
s
Obiectives.
A
n
n
Arbor,
Mich.:
NAEP,
rTru^ --------

225.

Peterson, Penelope.
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M a s t e r y - L e a r n i n g Strategies."
Un p u b l i s h e d
manuscr ipt, Intern ati ona l A s s o c i a t i o n for
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226.

Reynolds, L., and Light, *J.
"Procedures for the
Fo rma t i v e Eva lua t i o n of an In div i d u a l i z e d
M a t h e m a t i c s Curric ulu m."
P a p e r p r e s e n t e d at
the C o n f e r e n c e on F o u r Case Studies in
Fo rma t i v e Evaluation, Bloomington, Indiana,
July 16-17, 1971.

227.

Riesin, A u s t i n H.
"Excessive Ar ous al Effects of
S t i m u l a t i o n after Early Deprivation,"
Se nsory Deprivation:
A S y m p o s i u m Held at
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Ha r v a r d tTniversity P r e s s , 1961, pp. 34-40.

228.

Sherman, J. C.
"Appli cat ion of R e i n f o r c e m e n t
P r inc i p l e s to a C ol l e g e Course."
Pa per
p r e s e n t e d at the an n u a l m e e t i n g of the AERA,
N e w York, Fe bru a r y 19 67.

229.

Solonon,

230.

Tate,

231.

Wasylyk, E.
"A L a b o r a t o r y A p p r o a c h to M at h e m a t i c s
for L o w Achievers:
A n E x p e r i m e n t a l Study,"
A w o r k i n g paper, U n i v e r s i t y of A lberta, 1970.

232.

Weiss, J.
"Formative C u r r i c u l u m Evaluation:
In N e e d
o f M e t h o d o l o g y ."
P a p e r p r e s e n t e d at the
m e e t i n g o f the A m e r i c a n E d u c a t i o n a l Res ear ch
A s soc iat ion , N e w York, Fe bru ary 1971.

P., e t al.
Se nsory Deprivation:
A Symposium
H e l d at H a r v a r d Un ive r s i t y M e d i c a l Schoo.
Cambridge, Mass.:
Harvard U n i v e r s i t y P r e s s ,
1961.

S. A.
"Evaluation in the Social Studies:
A
C r i t i c a l R e v i e w of the Li terature."
Pitts­
burgh, Penn.:
U niv er s i t y of P i t t s b u r g h
S c h o o l of Education, In ter nat ion al and D e v e l o p ­
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251
233.

Wilkinson, Jack.
"The La b o r a t o r y M e t h o d of T e a c h i n g
St rategy in Grades 5-7."
Pa p e r p r e s e n t e d at
the N C T M m e e t i n g in Winnipeg, Manitoba,
Oc to b e r 17, 19 70.

234.

Woodby,

L. G.
"Survey of Projects."
In SMSG Report
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T5^25.--- ----------

AP P E N D I C E S

APPENDIX A
SET O F F IVE PRE- A N D P O S T - T E S T FORMS OF THE
CRITERION-REFERENCED MEASURES

APPENDIX A
SET OF FIVE PRE- AND POST-TEST FORMS OF THE
CRITERION-REFERENCED MEASURES
MEAS UREMENT— P RE-TEST
N a m e ________
I need 45 square feet of a carpet to cover a floor. If it is
sold in strips 24 inches wide, how long a strip do I need?
Select a grade level {K-2, 3-4, or 5-6). Name four manipulative
aids that would be useful in teaching a measurement unit.
Which of the following is the shortest?
a.
d.

20 centimeters
one yard

b. 30 inches
e. one decimeter

c.

one meter

A wire is 20 centimeters long. it is bent to form a rectangle.
What is the maximum area that can be completely enclosed by the
wire?
Suppose the area of a triangle is fixed at 30 square units. Let
its base be 'b* and its height be 'h'. Pick four different
volumes of 'b' and calculate the corresponding volumes of 'h*.
Graph the points. Use this to find the base of a triangle whose
area is 30 square units and height is two— five units.
What is the slope of your graph?

253
6.

Find the area of the following figures:

(a)

(b)

(c)

(d)

Two trees 50 meters apart are represented by two points which are
four centimeters apart on a map. How far apart are two trees
which are represented by two points six centimeters apart on the
map?

a.
d.

25 meters
75 meters

b.
e.

500 meters
80 meters

c.

60 meters

8.

A book is found to be as long as twenty-one paper clips. The
same book is as long as nine equal pencils. If a paper clip is
found to be two centimeters long, how long is the pencil?

9.

If a car travelled at an average speed of twenty-five m.p.h . , how
long would a journey of 175 miles take?
a.
d.

10.

17 hours
4 hours

b.
e.

7 hours
5 hours

6 hours

It is now a belief that practical work should preceed computational
practice at early grades. With this in mind, outline a lesson
plan for an introductory lesson on weight that will show the need
for a standard unit.

254
MEASUREMENT— POST-TEST
Name _______
Student: No.
1.

Suppose the perimeter of a rectangle
Let x centimeters be its width and y
Pick five different volumes of x and
volumes of y. Tabulate your results

is fixed at 24 centimeters.
centimeters be its length.
calculate the corresponding
and graph the points.

What is the shape of the graph?
What are the dimensions of the rectangle with the fixed perimeter
of 24 that has the largest area?
2.

A book is found to be twenty-one paper clips long. The same book
is seven (equal) pens in length. If a pen is two centimeters long,
how long is a clip?

3.

Design a lesson plan which will teach the formula L x W = A to a
group of three to five learners. Identify the pre-requisite
mathematical concepts and/or skills for this lesson.

4.

Which of the following is the longest?

5.

a.

one meter

b.

one yard

c.

d.

30 inches

e.

one decameter

Find the area of thefollowing figures:

(a)

(b)

one
millimeter

(c)

(d)

6.

A triangle, on a geoboard, of area 7 units had 4 boundary nails.
How many interior nails must it have? Represent such a triangle
on these dots,

7.

You are assigned the task of supplying the math lab with materials
that would be useful in teaching a measurement unit. List all of
the materials you would select, quantities of each, and how you
would organise them in the lab.
(Assume you want enough materials
for at least three classrooms to be able to use at the same time).

8.

Forty unequal pieces of stones are to be ordered by a scientist
according to their weights. If the only thing available to him is
just a balance (without any known weight), what is the minimum
number of weighings he has to do to achieve his objective?

9.

In a weighing exercise where a given block of bronze is the unit,
it is found that two such blocks balanced three aluminum discs,
two paper-clips balanced one aluminum disc. What weight should be
assigned to a paper clip?

10.

(a)
Describe an activity that when graphed produces points which do
not lie on a straight line. What is the slope of such graph.
(b)
Give an example of a measurement activity in which the transitivity
property must be used to achieve the desired results. Explain
how it must be used.

256
Name ____________
NUMERATION— PRE-TEST

Student No.______

1.

If you were judging a culture's numeration system (the way in
which numbers were recorded), list two properties that would be
considered important for the numeration system to have.

2.

In a culture's numeration system the following symbols:
<f>, a, 8,
r. A,
it,A# and x
represent respectively, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 so that
symbol
represents 30.
a.

b.

Which of the following is equal to 7/23?

a.

r/e>

b

D.

TT/Br

E . “f/aB

.

Ti/re

c.

What does A<}>A represent?
A.
D.

12
408

B. 48
E. 21

C. 804

3.

Give an example of a grouping activity designed to develop a
child's understanding of numeral 27.

4.

The decimal expansion of the numeral 35.72 is:
A.

3 (10)

+ 5 (1) - 7 (10) - 2

(1)

B.

3 (10)

+ 5 (1) - 7 (10) - 2

(100)

C.

3 (10) + 5 (10°) - 7 (101) - 2 (10°)

D.

3 (101) + 5 (10°) + 7 (10_1_ + 2 (10-2)

E.

3 (101) + 5 (10°) + 7 (10-1) + 2 (10"°)

In base ten, 421_.
is:
f ive
A.

551

B.

111

D.

100101

E.

35

C.

821

Which of the following is correct?
A.

23.
= 32
ten
seven

B.

23

C.

23.
= 32 .
ten
six

ten

= 32 .
five

= 32 . ,
eight

D.

23

E.

None of the above is correct

ten

257
In which base is 36 + 54 = 1127
A.

five

B.

E.

none of the above

C.

four

eight

D.

seven

8.

Name four manipulative aids you would use to teach a unit on
numeration to children and arrange the materials in the order
you would use them with your pupils.

9.

The number represented on this instrument is
A.

132.1 sxx
.

B.

1.231 .
sxx

C.

1321 sxx
Ones

10 ,

D*

1321thirty six

E.

132*^thirty six

Thirty-

Using the labelling shown, represent 427ten on this instrument

Hundreds

Tens

Ones

Tenths

Hundredths

258
Name ______
Student No.
NUMERATION— POST-TEST
PLEASE SHOW AIL WORK
1.

state three important characteristics of a good numeration system.

2.

Jim stated that there are 44 hours in a day.
in base
(a)

four

£b)

five

(c)

six

(d)

three

He must be working
(e)

none of these

3.

What would you do with younsters to enablethem to understand
the significance of the zero in 2035?

4.

Describe an activity that would introduce the concept of decimals
to elementary school youngsters.

5.

(i)

3 (52) + 1 (5^) + 2 (58) + 4

(a)

610.4f_^ve

(b)

(e)

7 5 + 5 + 2 - 2 0

3 1 2 . 4 ^ ve

(5
(c)

isthe expanded notation
50fcen (d)

3124 times five

(ii) In base ten, 124 . . is:
eight
(b)

(1 + 2 +

<c)

1 x 8 + 2 x 82 + 4 x 83

<d)

1 x 1016 + 2 x 108 +

(e)

1 x 82 + 2 X: 81 + 4 x 8°

U)

The numeral 32.
is different from
ten
''

(a)

52

(e)

40 eight
■ L.*.

(ii)
(iii)
7.

(b)

.

six

3211

=

200,
four

(c)

44

seven

<d >

CD

8 X 1 + 8 x 2 + 8 x 4

X

(a)

.0

62five

i r\
1 1 U

60°i0 =

A number is represented using base 6 wood in the following way:
3 cubes

4 flats

5 longs

3 units

Using the smallest number of pieces of base 4 woods, represent
the same number.

of

List four concrete materials you would suggest for use in
numeration. List them in the order in which you would use them
with elementary school children and defend your ordering.
Given that

j— i

+

10 □

3

is an addition problem in which each addend has three digits and
the numerals are base six representation. Find the missing
numerals.
Summarize effective experiences for helping primary pupils learn
the concept of place value.

Name
Student No.
ADDITION AND SUBTRACTION— PRE-TEST
What property or properties of addition are you using when you
check addition by adding from bottom to top after you have added
from top to bottom?
Give one real world situation in the life of a child that would
require subtraction for its solution.
How would you correctly regroup 432 in the following subtraction?
432
-179
List three objectives that are important to attain in the
formation of the concept of number which serve as a prelude to
addition of whole numbers.
Give three models for addition with one example of each model.
List three manipulative aids that are helpful in developing the
concepts of addition and subtraction.
Games can be used to help children master basic combinations of
numbers. Describe one activity that can be used for such a
purpose.
Give an example of a problem which employs the missing addend
approach to subtraction.
List and give examples of 2 properties that hold true for the
addition of whole numbers.
(a)

The inverse of subtraction is division (T or F)

(b)

The set of whole numbers is closed with respect
to subtraction (T or F)

(c)

The property that is illustrated in (17 + 24) + 13 =
17 + (24 + 13) is shown as the parentheses shift
property (T or F)

(d)

Regrouping is needed in every addition problem
(T or F)

(e)

The operations of addition and subtraction ,
on the set of whole numbers, are commutative
(T or F)

261
Name
Student No.
ADDITION AND SUBTRACTION— POST-TEST
Name one visual aid and two manipulative-aids that can be used to
develop the concepts of addition and subtraction. Choose one of
the three and describe briefly how you would use it to help
develop the concept of subtraction.
2 . Show pictorially all steps involved in working the following

problem on the abacus. Assume each place has 5 and only 5 beads
and that each bead must be pictured somewhere. The problem should
be worked as you would have the elementary school child work it.
435 - 14s

Start

3.

Give reasons in terms of the properties of whole numbers that
justifies each step of the following:
(7+12)

+ 23 = 7 + (12 + 23) _________________________
= 7 + (23 + 12) _________________________
= (7 + 23) + 12 ___ _____________________
= (7 + 23) + 12 + 0
= [(7 x 1) + 23] + 12 + 0

262
Outline a lesson designed to introduce to a class of second or
third graders the idea of regrouping or "carrying" in addition,
using the problem
26
+ 17
Describe briefly how you would use the appropriate concrete
material and how you would make the transition from concrete
to abstract (symbolic).
(a)
Complete the table for the addition facts in base five

(b)
Solve the following

235
+ 345
6.

35 + □

- 12,

Identify, with examples, four properties that hold for the
following system. The given set is (0,4 , o) and the operation
is @ .

©
□
A

□
Q
A

A

o

•Ci

o

o

O

a

O

□
A

1.
2.
3.
4.

263
7.

Three approaches to subtraction were presented: comparison,
take-away and missing addend. Give a real world example, from
an elementary school child's world, and the symbolic representation
of each.

8.

Briefly outline the experiences involving addition activities you
would give a child prior to the time when he would work the
problem 348 + 875 in the following way:
348
+875
1223

9.

10.

The text lists 10 objectives for re-addition and subtraction
skills (skills that you would want a child to have pior to
beginning addition or subtraction). Give four of these skills.
You may get credit for your own if they are valid pre-addition
subtraction skills.
Show pictorially all steps involved in working the following base
ten addition problem. Assume each place has 10 and only 10 beads
and that each bead must be pictured somewhere. The problem should
be worked as you would have the child work it.
385 + 468

264
Name _ _ _ _ _
Student No.
MULTIPLICATION AND DIVISION— PRE-TEST
In relating the childs real world and mathematics, the teacher often
starts with a problem which can initially be represented by concrete
models (using real objects, counters, etc.) or pictorial models (using
pictures, diagrams, etc.). Finally the model can be used to translate
a real problem into mathematical symbols. In the first two problems
one of the three parts of the process outlined above will be given.
Your job will be to fill in the two missing parts.

Real World
Problem
(For Child)

Model
(Concrete or
Pictorial)

Symbols

24

t

Q

=8

2.
Real World
Problem
(For Child)

Model
(Concrete or
Pictorial)

There were 9 people in Jeryx's
club; if each person paid 3 cents
for dues, how much would be paid
all together?

265
3.

Express the numbers 26 and 32 in expanded notation and then show
how you apply the distributive law to solve the multiplication
problem 26 x 32.

4.

Explain how addition is related to multiplication and how you
might use this relationship to introduce the idea of multiplication
to kids.

5.

Explain how subtraction is related to division.

6.

What is the least common multiple of 24 and 36? What is the
greatest common divisor of 24 and 36? (Show work)

7.

Construct the multiplication table for single digit numbers in
base five. Use this table to solver
a.

102

b.

22

t

3 =O

c
8.

Pick some manipulative aid and explain how you would use it to
teach any concept you choose related to multiplication or division
of whole numbers.

9.

Define the following:

10.

2.

a.

The division algorithm

2.

b.

Factor

1.

c.

Multiple

After each property stated below write if the property holds for
the set of whole numbers for the operation given. Support your
answers with examples.
a.

Multiplication— Commutative Property

b.

Division— Commutative Property

c.

Division— Identity Property

d.

Multiplication— Inverse Property

_______________

266
Name ______
Student No.
MULTIPLICATION AND DIVISION POST-TEST
1.

Complete the following with TRUE or FALSE.
FALSE, explain why.

If your answer is

a.

|0, 1, 2^ is not closed under multiplication

_____

b.

The least common multiple of 84 and 126 is 42

_____

c.

21 is a multiple of 3 and a factor of 105

d.

The rectangular array, subtraction, the set,
and the numer line all provide models for
multiplication of whole numbers

e.

(3 x a) ^ 3 = a

2.

Describe three plausible "thinking strategies" elementary school
children might use in finding the answer to 9 x 7 = ?

3.

Describe how you would develop the concept of multiplication using
a Real World Problem— Model-Symbol strategy. Be sure to include
an example as you describe your strategy.

4.

Pick two properties of whole numbers for the operation of
multiplication and describe how they can be used to help children
in early multiplication learning.

5.

Multiply 123 by 38 usine one of the following non-standard
algorithms:
Lattice Method
Russian Method
Doubling

6.

The following demonstrates one person's calculations using a
"transitional" algorithm for division of whole numbers:
78

| 2267
J -780
1482
-780
702
-390
312
-156
156
-156
0

10
10
therefore, 2262
5
2
2
+___
29

t

78 = 29

a.

Discuss the advantages of having elementary school children
use this method.

b.

Why does this method work?

a.

The set of whole numbers is not closed under division.

b.

If n and d are whole numbers with a greater than or equal to
d and d not equal to zero then, there exists unique whole
numbers q and r such that n = (q x d) + r. What is the
restriction on r for

c.

Define or describe the following:
1.

Algorithm

2.

Least Common Multiple

3.

Greatest Common Factor

(give mathematical evidence)

Table 1

NOTE:

Table 2

is the inverse of CE>

a.

From TABLE

1, solve completely 3(2)__ = 1

b.

From TABLE

1* solve completely 3 © 2 = _

c.

From TABLE 1, solve completely ___ (5)2 = 3

d.

From TABLE 2, solve completely

e.

From TABLE 2, solve completely 2©

3 © ___ = 4
_

= 1

Why?

268
9.

10.

Discuss the Weaver article "Big Dividends From Little Interviews"
and include the following:
a.

Brief summary of article

b.

One fact that Miss Watkins (who is actually a red-hot Mama)
found out about any of the six kids that helped her assess
their needs.

c.

One advantage and one disadvantage of using her technique.

Show, by example, 3 plausible errors that elementary school
children might make in attempting to solve multiplivation (or
division) problems using the STANDARD ALGORITHM.

269
Name ______
Student No.

FRACTION PRE-TEST
1 . Briefly describe a strategy used to introduce fractions to

elementary school children.
2.

Explain how to develop the equivalence class ^2/3, 4/6, 6/9, . . .J
using a rectangle as an aid.

3.

Explain how you could use the idea of equivalence classes of
fractions to find the solution to 1/3 + 3/4.

4.

List five different types of manipulates that would be helpful in
developing understandings in a fraction unit. State briefly
how each might help.

5.

Which is larger 21/55 or 34/89?

6.

When dividing fractions, say, 2/5 ^ 3/7 the algorithm states:
invert the divisor and multiply; that is, 2/5 -f 3/7 = 2/5 x 7/3.
Explain why that produces the correct answer.

7.

Using a rectangular model show how to represent the problem
2/3 x 3/4. Show also how the rectangular model shows the solution.

8

.

9.

Why?

Given the set of fractions greater than zero (numbers of the form
a/b where a and b are natural numbers) state all the properties,
with examples, that are true for multiplication.
a.

State a number greater than 2/3 but less than 3/5.

b.

Exactly how many 3/4's are there in 1?

c.

Express .412 as a fraction.

d.

Express .121212 . . .

e.

(2/3 v 3/ 4 ) t 7/8 *

as a fraction.

1 0 . Show 1/5 + 2/3 using the number line as a model.

270
Name _______ _____________
Student No. _ _ _ _ _ _ _ _ _ _ _

FRACTIONS POST-TEST
1. In beginning a unit on fractions one would usually start with
examining the meaning of 1/2 where a is a non-zero whole number.
Briefly describe the concepts that would follow and indicate the
ORDER in which they should be taught. Be sure the list is a
complete overview of those concepts (and their order) that would
be included in a unit on fractions.
2.

Explain how to develop the equivalence class ^3/5,
12/20, . .
using the rectangle as a model.

3.

Explain how you would use the idea of equivalence classes to find
the solution to 1/3 + 2/5.

4.

Solve the following ("reduce" answers to lowest terms)
a.

13/756 + 17/504 =

b.

14 2/3 * 9 7/8 =

6/10, 9/15,

(show work)

5.

In adding 2/7 and 1/3 the phrase "selecting a proper form of one"
can be used. Explain exactly what this means.

6.

A child asks: "When you reduced 6/8 to 3/4, you said that you
divided both 6 and 8 by two to get the answer. What gives you
the right (mathematically) to do this?" Answer his question.

7.

Give an explanation why inverting the divisor and then multiplying
produces the correct answer when dividing fractions.

8.

Using a rectangular model show how to represent 1 2/3 x 2 3/4.
Use the model to solve the problem.

9.

List three different types of manipulatives that would be helpful
in developing a unit on fractions. State briefly how each might
help.

10.

Explain how to solve 2/3 v 3/4 using Cuisenaire Rods.

APPENDIX B
T E S T OF BASIC M A T H E M A T I C A L UND ERS T A N D I N G S
FORM A

(PRE-TEST)

A N D F ORM B

(POST-TEST)

APPENDIX B
A TEST OF BASIC MATHEMATICAL UNDERSTANDINGS
FORM A (PRE-TEST) AND FORM B (POST-TEST)

A Test of Basic Mathematical Understandings
Prepared B y :
Dr. Mildred Jerline Dossett
Michigan State University
East Lansing, Michigan
1964
Directions;
This test is designed to measure your understanding of mathe­
matics. Many of the items relate to the new content in present
programs of mathematics for elementary pupils.
Each of the fifty-five questions is of multiple-choice type and
includes four possible answers. Read each question carefully and
decide which answer fulfills the requirements of the statement. Then
circle the response on the answer sheet to indicate your choice.
Circle only one answer for each question. If you change your
choice, erase your original mark and circle the correct one.
Sample Question;
1.

Which of the following shows the decimal form of the fraction 5/4?
a.

125

b.

12.5

c.

1.25

d.

.125

271

272
Answer Sheet;
1.

a

b

d

Since 1.25 is the correct answer# the letter (c) is circled.
FORM A (PRE-TEST)

1.

When you write the numeral "5" you are writing
a.

the number 5

b.

a pictorial expression

c.

a symbol that stands for an idea

d.

a Hindu-BabyIonian symbol

2. Bill discovered that > means "is greater than" and < means "is
less than." In which of the following are these symbols not
used correctly?

3.

a.

The number of states in the United States < the number of
United States Senators.

b.

The number of states in the United States < the number of
stripes in the flag.

c.

2

d.

3 + a < 5 + a

3

2
> 3

Whentwo Roman numerals stand side by side
values are added.
a.

always

b.

sometimes

c.

never

d.

if the base is X

in a symbol, their

273
4.

Which of the following describe/describes our own system of
numeration?
a.

additive

b.

positional

c.

subtractive

d.

introduces new digits for numbers larger than 10
1.

a and b are correct

2.

a and c are correct

3.

a and d are correct

4.
5.

6.

7.

b ( and d are correct

Zero may be used
a.

as a place holder

b.

as a point of origin

c.

to represent the absence

d.

in all of the above different

ofquantity
ways

2,200.02 is shown by
a.

2000

+ 200 + 20

b.

2000

+ 20 + 2/10

c.

2000

+ 200 + 2/100

d.

2000

+ 200 + 200

5840 rearranged so that the 8 is 200 times the size of 4 would be
a.

5840

b.

8540

c.

5048

d.

5408

274
8.

9.

Which of the following does not show the meaning of 423
a.

(4 x 100) + {2 x 10) +3(1} = 423

b.

42 tens + 3 ones = 423

c.

423 ones = 423

d.

4 hundreds + 42 tens + 23 ones = 423

b.

11.

?

A numeral for the X's in this example can be written in many
different bases. Which numerals are correct?
a.

10.

ten

100t
four

XX

X

XX

XX

X

X

X

X

XX

X

^twelve

c.

16

d.

31

X

ten

X

five

1.

a and c are

correct

2.

b and c are

correct

3.

a, b, and c

are correct

4.

all four are correct

A "2" in the third place of a base twelve number would represent
a.

2 x 123

b.

12 x 2 3

c.

12 x 2 12

d.

2 x 122

In this addition example, in what base are the numerals written?
a.

base two

120„

b.

base three

+

c.

base four

d.

none of the above

10 ,

200,

275
12.

13.

14.

15.

16.

About how many tens are there in 6542?
a.

6540

b.

654

c.

65 1/2

d.

6.5

Place or order in a series is shown by
a.

book no. 7

b.

three boxes of matches

c.

a dozen cupcakes

d.

two months

Which of the following indicates a group?
a.

45 tickets

b.

track 45

c.

page 54

d.

apartment No. 7

The sum of any two natural numbers
a.

is not a natural number

b.

is sometimes a natural number

c.

is always a natural number

d.

is a natural number equal to one of the numbers being added

The counting numbers are closed under the operations of
a.

addition and subtraction

b.

addition and multiplication

c.

addition, subtraction, multiplication, and division

d.

addition, subtraction, and multiplication

276
17.

18.

19.

20.

If a and b are natural numbers, then a + b = b +
of
a.

commutative property

b.

associative property

c.

distributive property

d.

closure

if a

x

b =

0

aisan

example

then

a.

must be zero

b.

b must be zero

c.

either a or b must be zero

d.

neither a nor b must be zero

When a natural number is multiplied by a natural number other
than 1 , how does the answer compare with the natural number
multiplied?
a.

larger

b.

smaller

c.

the same

d.

can't tell from information given

Which of the following is the quickest way to find the sum of
several numbers of the same size?
a.

counting

b.

adding

c.

subtracting

d.

multiplication

277
21.

22 .

How would the product in this example be affected if you put the
29 above the 4306 and multiplied the two numbers?
a.

The answer would be larger

b.

The answer would be smaller

c.

You cannot tell until you multiply both ways

d.

The answer would be the same

4306
x26

An important mathematical principle can be helpful in solving the
following example.
28 + 659 + 72 =

□

What principle will be of most help?

23.

24.

a.

the associative principle

b.

the commutative principle

c.

the distributive principle

d.

both the associative and distributive principles

The product of 356 x 7 is equal to
a.

(300 x 50) x (6 + 7)

b.

(3 x 7) + (5 x 7) + (6 x 7)

c.

300 x 50 x 6 x 7

d.

(300 x 7) + (50 x 7) + (6 x 7)

Which of the following is not a prime
a.

271

b.

277

c.

281

d.

282

278
25.

26.

27.

28.

Which of the following numbers is odd?
a.

18 x 11

b.

11 X 20

c.

99 x 77

d.

none of the above

The inverse operation generally used to check multiplication is
a.

addition

b.

subtraction

c.

multiplication

d.

division

The greatest common factor of 48 and 60 is
a.

2 x 3

b.

2 x 2 x 3

c.

2 X 2 x 2 x 2 x 3 x 5

d.

none of the above

Look at the example at the right. Why is the "4"
in the third partial product moved over to places
and written under the 2 of the multiplier?

157
x 246
942
628
314

a.

If you put it directly under the other partial
products, the answer would be wrong.

b.

You must move the third partial product two places to the
left because there are three numbers in the multiplier.

c.

The number 2 is the hundreds column, so the third partial
product must come under the hundreds column.

d.

You are really multiplying by 200.

279
29.

30.

31.

32.

Which of the fundamental properties of arithmetic would you employ
in proving that (a + b) + (a + c) = 2a + b + c?
a.

associative property

b.

commutative property

c.

associative and distributive properties

d.

associative and commutative properties

If N represents an even number, the next larger even number can
be represented by
a.

N +

1

b.

N+

2

c.

N

+

N

d.

2 x

N

+

L

Every natural number has at least the following factors;
a.

zero and one

b.

zero and itself

c.

one and itself

d.

itself and two

It is said that the set of whole numbers has a natural order.
To find the successor of a natural number, one must
a.

add 1

b.

find a number that is greater

c.

square the natural number

d.

subtract 1 from the natural number

280
33.

The paper below has been divided into 6 pieces.

it shows

n r v i t ~i

34.

35.

36.

a.

sixths

b.

thirds

c.

halves

d.

parts

A fraction may be interpreted as:
a.

a quotient of two natural numbers

b.

equal part/parts of a whole

c.

a comparison between two numbers

d.

all of the above

When a common (proper] fraction is divided by a common fraction,
how does the answer compare with the fraction divided?
a.

it will be larger

b.

it will be smaller

c.

it will be twice as

d.

there will be no difference

large

Which algorithm is illustrated by the following sketch?
a.

1 / 2

x

3/4

=

b.

1 / 2

+

3/3

= ?

?

1/4

1/4

1/4

1/4

281
37.

38.

Another name for the inverse for multiplication of a rational
number is the
a.

reciprocal

b.

opposite

c.

reverse

d.

zero

Examine the division example on the right.
Which
sentence best tells why the answer is larger than
the 5?
a.

Inverting the divisor turned the 3/4 upside down.

b.

Multiplying always makes the answer larger.

c.
d.

39.

The divisor 3/4 isless than 1.
Dividing by proper and improper fractions makes the answer
larger than the number divided.

The value of a common fraction will not be changed if
a.

40.

5 v 3/4 = 6 2/3

We add the same number to both terms.

b.

We multiply one term and divide
same number.

c.

We subtract the same amount from both terms.

d.

We multiply both terms by the same number.

The nearest to 45% is
a.

44 out of 100

b.

.435

c.

4.5

d.

.405

the other term by that

282
41.

The principal of a school said that 27 percent of the pupils went
to the museum.
Which statement best describes the expression
"27 percent of the pupils went to the museum"?
a.

It means that 27 children out of every 100 children went
to the museum.

b.

It means that you must multiply the number of children in
the school by 27/100 to find the number who went to the
museum.

c.

If the children were divided into groups of 100 and those
who went to the museum were distributed evenly among them,
there would be in each group 27 who
went to themuseum.

d.

42.

43.

27 percent is the
in percent form.

same as .27— a decimal

Sally completed 2/3 of the story in 12 minutes.
long will it take her to read the entire story?
a.

18 minutes

b.

12 minutes

c.

6 minutes

d.

24 minutes

fraction written

At that rate how

There were 400 students in the school.
One hundred percent of the
children had lunch in the cafeteria on the first day of school.
On the second day 2 boys were absent and 88 children went home for
lunch.
Which of the following equations can be used to find the
percent of the school enrollment who went home for lunch?
a . 400 - 88 = X
b.

x/100 = 88/400

c.

x/88 == 400

d.

400 - 90 = X

283
44.

What can be said about y in the following open sentence if x is
a natural number?
x + x + 1 = y

45,

46.

47,

a.

x < y

b.

x > y

c.

x = y

d.

x / y

Which one of the following fractions will give a repeating decimal?
a.

1/2

b.

3/4

c.

5/8

d.

6/11

Which of the following is not an open sentence?
a.

7

b.

h - 5 = 9

c.

c/1 - 30 = 6

d.

n - 3

+ 2

=

1~1

For a mathematical system consisting of the set of odd numbers and
the operation of multiplication.
a.

the system is closed

b.

the system is commutative

c.

the system has an identity element

d.

all of the above are correct

284
48.

Measurement is a process which
a.
b.

compares an object with some known standard or accepted
unit
tries to find the

exact amount

c. is never an exact measure
d.

49.

chooses a unit and then gives a number which tells how
many of that unit it would take
1.

a and b are correct

2.

a and c are correct

3.

a, b, and d are correct

4.

a, c, and d are correct

The set of points sketched below represents a
<r~---------------

a.

line

b.

ray

c.

line segment

d.

none of the above
contain?

51.

a.

four

b.

six

c.

eight

d.

ten

The set of points on two rays with a common end-point is called
a.

a triangle

b.

an angle

c.

a vertex

d.

a side of a triangle

285
52.

If a circle is drawn with the points of a compass 3 inches apart,
what would be 3 inches in length?
a.

circumference

b.

diameter

c.

area

d.

radius

53. The solution set of an open sentence may consist of

54.

55.

a.

two or more numbers

b.

no numbers

c.

only one number

d.

any or all of these

Consider a set of three objects.
can be arranged?
a.

nine

b.

eight

c.

seven

d.

six

How many sub-sets orgroups

If two sets are said to be equivalent, then
a.

every element in the first set can be paired with one
and only one element in the second set

b.

every element in one set must also be an element in
the second set

c.

they are intersecting sets

d.

one must be the null set

286
FORM B (POST—TEST)
1.

Which
refer
a.

4^ can be written on the blackboard

b.

Regardless of what symbol we use, we are thinking about
the number 2

c.

A pencil is used for writing.

d.
2.

3.

of the underlined words or signs in the followingsentences
to symbols rather than the things they represent?

The number 16 is the same as the number 7 + 9 ,

When we use the = symbol between two terms (as 2 + 2 = 4 ) , we
mean that both terms represent the same concept or idea. Which of
the following is not correctly stated?
a.

3 + 4 = 5 + 2

b.

5 + 2 = 7 and 7 = 5 + 2

c.

(5 + 2) x 3 = 7 x 3

d.

7 = 7

1.

a and b are correct

2.

a and c are correct

3.

a, b. and c are correct

4.

a, b, c, and d are correct

If the Roman system of numertion were a "place value system" with
the same value for the base as the Hindu-Arabic system, the first
four base symbols would be
a.

I, X, C,

and M

b.

I, V, X,

and L

c.

X, L, C, and M

d.

X, C, L, and D

287
4.

5.

6.

Which of the following does not describe a characteristic of our
decimal system of numeration?
a.

It uses zero to keep position when there is an absence
of value.

b.

It makes a ten a standard group for the organization of
all numbers larger than nine.

c.

It makes 12 the basis for organizing numbers larger than
eleven.

d.

It uses the additive concept in representing a number of
several digits.

In the numeral 7,843, how does the value of the 4 compare with
the value of the 8?
a.

2 times as great

b.

1/2 as great

c.

1/10 as great

d.

1/20 as great

In the numeral 6,666 the value of the 6 on the extreme left as
compared with the 6 on the extreme right is
a. 6,000 times as great

7.

b.

1,000 times as great

c.

the same since both are sixes

d.

six times as much

Which of the following statements best tells why we write a zero
in the numeral 4,039 when we went it to represent "four thousand
thirty-nine"?
a.

Writing the zero helps to fill a place which would other­
wise be emptyand lead to misunderstanding.

b.

The numeral would represent "four hundred thirty-nine" if
we did not write the zero.

c.

Writing the zero tells us not to read the hundreds' figure.

288
d.

8.

9.

1.

a and b are correct

2.

a and c

are correct

3.

a and d

are correct

4.

a, b, and d are correct

Below are four numerals written in expanded notation.
is not written correctly?
a.

4(ten)2 + 9(ten)1 + 3(ones) = 493^
ten

b.

3 (seven)3 + 6 (seven)1 + l(one) = 363

c.

4(twelve)2 4- S(twelve)1 + e(one) = 45e

d.

2 (five) 2 + 2 (five)1 + 4 (one) = 224

Which one

seven
twelve

five

If you are permitted to use any or all of thesymbols
0, 1, 2, 3/
4, and 5 for developing a system of numeration with aplace value
system of numeration similar to ours, a list of all possible bases
would include t
a.

base one, two, three, four, five, and six.

b.

base two, three, four, five, and six.

c.

base two, three, four, and five.

d.
10.

Zero is used as a place-holder to show that there is no
number to record in that place.

base one, two, three, four, and five.

About how many hundreds are there in 34,870?
a.

3 1/2

b.

35

c.

350

d.

3,500

289
11.

In what base are the numerals in this multiplication example
written?
a.

base five

34^
23
■■
i24?
70^

b.

base eight

c.

base eleven

d.

you can’t tell

;

1024 ■?

1 2 . Which of the following are correct?

13.

14.

a.

In the symbol

5 3, 5 is the base and 3 is the exponent.

b.

In the symbol

5 3, 3 is the base and 5 is the exponent.

c.

53 = 5 x 5 x 5

d.

5a = 3 x 3 x 3 x 3 x 3
1.

aand d are correct

2.

b and c are correct

3.

a and c are correct

4.

b and d are correct

In the series of numerals 1....17, 18, 19, 20, 21,..., what term
best applies to 19?
a.

nominal

b.

ordinal

c.

composite

d.

cardinal

Examine the following illustration:

<1

4

Which of the following does the above best illustrate?
a.

The idea of a cardinal number.

b.

The use of an ordinal number.

c.

A means for determining the cardinal number of the set by
counting with ordinal numbers.

d.

None of the above.

290
15.

16.

17.

The quotient of any two whole numbers
a.

is not a natural number

b.

is sometimes a natural number

c.

is always a natural number

d.

is a natural number less than one of the numbers being
divided

The integers are closed under the operations of
a.

addition

b.

subtraction

c.

multiplication

d.

division
1.

a and b are correct

2.

a and c are correct

3.

a, b, and c are correct

4.

a, b, c, and d are correct

A student solved this example by adding downj then he checked his
work by adding up.
Add 34
+ 52
86

+
Check

34
52
86

It could be classified as an example of
a.

the distributive principle

b.

the associative principle

c.

the commutative principle

d.

the law of compensation

291
18.

19.

20.

21.

The statement "the quotient obtained when zero is divided by a
number is zero" is expressed as
a.

a/0 = 0

b.

0/a = 0

c.

0/0 = a

d.

a/a = 0

When a natural number is divided by a natural number other than
1, how does the answer compare with the natural number divided?
a.

larger

b.

smaller

c.

one-half as large

d.

can't tell from information given

If you had a bag of 350 marbles to be shared equally by 5 boys,
which would be the quickest way to determine each boy's share?
a.

counting

b.

adding

c.

subtracting

d.

dividing

If the multiplier is x, the largest possible number to carry is
a.

x

b.

x + 1

c.

0

d.

x - 1

292
22.

23.

Which one of the following methods could be used to find the
answer to this example?
17*^612
a.

Multiply 17 by the quotient.

b.

Add 17 six hundred times.

c.

The answer

d.

Subtract 17 from 612 as many times as possible. The
answer would be the number of times you were able to
subtract.

would be the sum.

Which one of the following would give the correct answer to this
example?
2.1

x21

24.

25.

a.

The sum of

1 x 2.1 and21 x 2.1.

b.

The sum of

10 x 2.1 and 2 x 2.1.

c.

The sum of 1 x 2.1 and 20 x 2.1.

d.

The sum of 1 x 2.1 and 2 x 2.1.

Which would give the correct answer to 435 x 563?
a.

Multiply 439 x 3, 439 x 60, 439 x 5 and then add the answer.

b.

Multiply 563 x 9, 563 x 3, 563 x 4 and then

c.

Multiply 563 x 9, 563 x 39, 563 x 439 and thenadd the
answer.

d.

Multiply 439 x 3, 439 x 60, 439 x 500 and then add the
answer.

addthe answer.

Which of these numerals are names for prime numbers?
a.

3

b.

4/2

293
d.

26.

27.

28.

29.

9-2
1.

a is correct

2.

a and c are correct

3.

a, b, and d arecorrect

4.

a, b, c, and dare

correct

Let x represent an odd number; let y represent an even number.
Then x + y must represent.
a.

an even number

b.

a prime number

c.

an odd number

d.

a composite number

The inverse operation for addition is
a.

addition

b.

subtraction

c.

multiplication

d.

division

The least common multiple of 8, 12, and 20 is
a.

2x2

b.

2 x 3 x 5

c.

2 x 2 x 2 x 3 x 5

d.

2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5

Which statement best tells why we carry 2 from the second column?
a.

If we do not carry the 2, the answer would be 20
less than the correct answer.

b.

Since the sum of the second column is more than
20, we put the 2 in the next column.

251
161
252
271
935

294

30.

c.

Since the sum of the second column is 23 (which has two
figures in it), we have room for the 3 only, so we put 2
in the next column.

d.

Since the value represented by the figures in the second
column is more than 9 tens, we must put the hundreds in the
next column.

The operations which are associative are
a.

addition

b.

subtraction

c.

multiplication

d. division

31.

1.

a and b are correct

2.

a and c are correct

3.

a, b, and c are correct

4.

a and d are correct

Which of the following is an even number?
a.

(100}..
three

b.

(100)c .
five

c.

(100)

seven

d ' (200)five
32.

The fact that a + b (b + c) is exactly equal to (c + b) + a is an
example of
a.

distributivity

b.

commutativity

c.

closure

d.

associativity

295
33.

34.

35.

36.

Observe the drawing on the right. When the circle is cut into
equal pieces, the size of each piece
a.

decreases

as the number

of pieces increases

b.

increases

as the number

of pieces decreases

c.

increases

as the number

of pieces increases

d.

decreases

as the number

of pieces decreases

1.

a

and b are correct

2.

a and c are correct

3.

b and c are correct

4.

b and d are correct

The symbol 3/4 may be used to represent the idea that
a.

3 is to be divided by 4

b.

3 of the 4 equal parts are being considered

c.

3 objects are to be compared with 4

d.

all of the above

objects

When a whole number is multiplied by a common (proper) fraction
other than one, how does the answer compare with the whole
number?
a.

it will be larger

b.

it will be smaller

c.

there will be no difference

d.

you are not able to tell

Which of the addition examples is best represented by the shaded
parts of the diagram below?
a.

1/2 + 1/3

b.

2/3 + 3/4

c.

2/3 + 1/4

d.

1/3 + 1/3

+

296
37.

_2
We can change the denominator of the fraction 3
without changing the values of the
4
fraction by
5
a.

39.

adding 5/4 to the numerator and denominator

b.

subtracting 5/4 from the numerator and the denominator

c.

multiplying both the numerator and the denominator by 5/4

d.
38.

dividing the numerator and the denominator by 5/4

What statement best tells why we "invert the divisor and multiply
when dividing a fraction by a fraction?
a.

It is an easy method of finding a common denominator and
arranging the numerators in multiplication form.

b.

It is an easy method for dividing the denominators and
multiplying the numerators of the two fractions.

c.

It is a quick, easy, and accurate method of arranging two
fractions in multiplication form.

d.

Dividing by a fraction is the same as multiplying by the
reciprocal of the fraction.

If the denominator of the fraction 2/3 is multiplied by 2, the
value of the resulting fraction will be
a . half as large

40.

to the number "1"

b.

double in value

c.

unchanged in value

d.

a new symbol for the same number

i may also be written as
a.

.45

b.

45/100

c.

45 x 100%

d.

.450
1.

a

and b are correct

2.

a

and c are correct

3.

a

and d are correct

4.

a,

b, and d are correct

297
41.

.5 and .27 are illustrations of "decimal fractions." They could
be written as "common fractions" in the form of 1/2 and 27/100#
respectively. What is a decimal fraction?
a.

42.

It isanother way of writing percentage.

b.

It is an extension of the decimal number system to the
right of one's place.

c.

A number like .37 1/2 which has both a decimaland
fraction as parts of it.

d.

A number like .2/.56 which is a fraction and has
as either the numerator or denominator or both.

a
a decimal

The ratio of x's in Circle A to x's in Circle B can be shown by
a.

16/4

b.

1/4

c. 1/2
d. 4/16
A
43.

44.

B

Sue paid 20C for 4 apples. Which of the equations below could be
used to find the cost of 1 apple?
a.

4/20= 1/x

b.

x + 4 = 20

c.

x/4 = 20

d.

x - 4 = 20

The decimal for the numeral 6/17 will
a.

be a repeating decimal

b.

not repeat or end since 17 is prime

c.

repeat in cycles of less than 23 digits
1.

a is correct

2.

a and b are correct

3.

a and c are correct

4.

a, b, and c are correct

298
45.

46.

47.

48.

49.

Which of the following statements is not correct?
a.

(-9)

+ 6 = -3

b.

(-5)

+ (-5) =

c.

-8 + 0 = -8

d.

(-8)

-10

+ (9) = “1

Which of the following is a list of all of the factors of 12?
a. 1^

2* 3, 4 r

8 & 12

b.

1,

2 r 3, 4,

6 & 12

c.

1,

2, 3, 4,

&6

d.

2,

3, 4, 6 & 12

Modular arithmetic is
a.

cummutative

b.

associative

c.

distributive with respect to multiplication over addition

d.

all of the above

which of the following is an approximate measure?
a.

35 farms

b.

12 buttons

c.

7 1/2 inches

d.

15 beads

Which of the following does the sketch below represent?
■

—

i.

■

—

a.

line

b.

ray

c.

line segment

-i

— ■ .

299
d.

50.

51.

52.

set of points
1.

a is correct

2.

a, b, and d are correct

3.

a, c, and d are correct

4.

b and d are correct

Which of these triangles are right triangles according to the
length of the sides given?

6"

II

6"

(I

A distinct point is
a.

a point

you can see

b.

a sharp

object

c.

the intersection of two lines

d.

a dot

A clerk sold a lady a roundtablecloth that had aradius of
inches. Which of theformulas can she use todetermine the
around the cloth?
a.

A = TTr

b.

C = TTd

c.

C - 2TTr

d.

A = C/d

14
length

300
53 .

54.

Which of the following best defines a solution set?
a.

A solution set is a set which includes each and every
member that gives a true statement.

b.

A solution set is a single sentence which identifies a
variable that will give a true statement.
,

c.

A solution set is a set containing all the positive
integers, zero, and the negative integers.

d.

A solution set is

a set containing rational numbers.

Examine the following illustration.
S = {0, 1 (-1), 2, (-2), 3,...10}
Which one of the following is not a subset of S?

55.

a.

{+9, + 10}

b.

{0, (-2) , 5}

c.

{3, (-3), 10}

d.

(1, (-1), 6,

10}

If we use the set concept to define the operations for the
counting numbers, addition would be defined in terms of
a.

the intersection of disjoint sets

b.

the union of intersecting sets

c.

the intersection of sets with common elements

d.

the union of disjoint sets

AP P E N D I X C
DU TTO N A R I T H M E T I C A T T I T U D E INVENTORY

APPENDIX C
DUTTON ARITHMETIC ATTITUDE INVENTORY

Name ________

Student Number

Place a check (/) before those statements which tell how you
feel about arithmetic. Select only the items which express your true
feelings— probably not more than five items.
____
____
____
___
___
_____

_ _
___

1. I avoid arithmetic because I am not very goodwith
figures.
2. Arithmetic is very interesting.
3. I am afraid of doing word problems.
4. I have always been afraid of arithmetic.
5. Working with numbers is fun.
6. I would rather do anything else than do arithmetic.
7. I like arithmetic because it is practical.
8. I have never liked arithmetic.
9. I don*t feel sure of myself in arithmetic.
10. Sometimes I enjoy the challenge presented by anarithmetic
problem.
11. I am completely indifferent to arithmetic.
12. I think about arithmetic problems outside of school and like
to work them out.
13. Arithmetic thrills me and I like it better than any other
subject.
14. I like arithmetic, but I like other subjects just as well.
15. I never get tired of working with numbers.
16.

Place a circle around one number to show how you feel about
arithmetic in general.
1
2
3
Dislike

4

5

6

7

8

9

10

11
Like

17.

My feelings toward arithmetic were developed in grades:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, other ___ (circle one).

18.

My average grades made in arithmetic were:

301

A B C D (circle one)

302
19.

List two things you like about arithmetic.
A.
B.

20.

List two things you dislike about arithmetic.
A.

B.

APPENDIX D

AT TI T U D E SCALES TOWARD DIFFERENT
AS PECTS OF MATHEMATICS

APPENDIX D
ATTITUDE SCALES TOWARD DIFFERENT
ASPECTS OF MATHEMATICS
A Study of Attitude of Prospective
Elementary School Teachers
Dear Student:
We are attempting to evaluate the attitudes of prospective
elementary school teachers, such as yourself, toward some aspects of
mathematics, school, and life in general.
Will you please read each statement and circle the response
that reflects your feeling toward that statement?
D, if you disagree, or U, if you are undecided.

A, if you agree,
Please be sure to

circle only one letter for each statement.
The information obtained through this questionnaire will be
kept strictly confidential.

It will be used for research purposes

only.
student's Name
Student's Number

303

304

W
Ui
(K

W
W
S

K
H
H
<

U> Pi
< Ul

U
c
A

U) U

A

D U

2. In our school we got a great deal of practice and drill
until we were almost perfect in our learning.

A

D U

3. The students spent most of their class time listening
to the teachers and taking notes.

M2
q 5
D U

A D D

1. Most school work is memorizing of information.

4.

My mathematics teacher showed us different ways of
solving the same problem.

A

D U

5. Our teachers wanted us to do most of our learning from
the textbook which is used in the course.

A

D U

6. My mathematics teacher did not like students to ask
questions after he had given the explanation.

A

D U

7. My mathematics teacher wanted students
only by the procedures he taught.

A

D U

8. We were expected to learn and discover many ideas for
ourselves.

A

D U

9. We were expected to develop a thorough understanding of
ideas and not just to memorize information.

A

D U

to solve problems

10. Our teachers believed in strict discipline and each
student did exactly what he was told to do.

A

D U

11. Students were encouraged to devise their own projects or
experiments in order to learn on their own.

A

D U

12. My mathematics teacher expected us to learn how to solve
problems by ourselves but helped when we had difficulties.

A

D U

13. In my mathematics classes# students who had original ideas
got better grades than did students who were most careful
and neat in their work.

A

D U

14. Most of our classroom work was listening to the teacher.

A

D U

15. My mathematics teacher required the students not only to
master the steps in solving problems, but also to under­
stand the reasoning involved.

A

D U

16. My mathematics teacher encouraged us to try to find
several different methods of solving particular problems.

305
W

a
M

P.

Eh

w o o ;
S
O g
M 8
S

*c a 5

A

D

U

17.

My mathematics course required more thinking about methods
of solving problems than memorization of rules and
formulas.

D

U

18.

My mathematics teacher wanted us to discover mathematical
principles and ideas for ourselves.

D

U

19.
were expected to develop the methods of solutions for
ourselves.

A

D

U

20.
Various sources and books from which we can learn were
suggested to us.

A

D

U

21.
to give us practice in using a particular rule or formula.

A

D

U

22.

Much of our classroom work was discussing ideas and
problems with the teacher and other pupils.

A

D

V

23.

In mathematics there is always a rule to follow in
solving problems.

A

D

U

24.

A

D

U

25.

A

D

U

26.

A

D

U

27.
become mathematicians and mathematics teachers.

A

D

U

28.
revealed by science.

A

D

U

29.

A

D

U

30.

A

D

u

31.
poverty can be eliminated in the world.

A

D

u
U

32.

I dislike school and will leave just as soon as possible.

A

D

u

33 .

With increased medical knowledge, it should be possible
to lengthen the average life span to 100 years or more.

306
U
w

&
CD

CD
<
C/1
M

a

55

n

g

01

H
U
B

Outside of science and engineering, there is little need
for mathematics (algebra, geometry, etc.) in most jobs.

A

D

u

A

D

U

35.

Mathematics is of great importance to a country*s
development.

A

D

U

36.

The most important reason for studying arithmetic and
secondary school mathematics is that they help people to
take care of their own financial affairs.

A

D

U

37.

Very few people can learn mathematics.

A

D

U

38.

Mathematics help one to think according to strict rules.

A

D

U

39.

Mathematics (algebra, geometry, etc.) is not useful for
the problems of everyday life.

A

D

U

40.

Someday the deserts will be converted into good farming
land by the application of engineering and science.

A

D

U

41.

I am bored most of the time in school.

A

D

U

42.

Almost all of the present-day mathematics was known at
least a century ago.

A

D

U

43.

Education can only help people develop their natural
abilities; it cannot change people in a fundamental way.

A

D

u

44.

I enjoy everything about school.

A

D

u

45.

A thorough knowledge of advanced mathematics is the key
to an understanding of our world in the twentieth century,

A

D

u

46.

School is not very enjoyable, but I can see value in
getting a good education.

A

D

u

47.

It is important to know mathematics (algebra, geometry,
etc.) in order to get a good job.

A

D

u

48.

Almost anyone can learn mathematics if he is willing to
s tudy.

A

D

u

49.

Mathematics is a very good field for creative people to
enter.

A

D

u

50,

Unless one is planning to become a mathematician or a
scientist the study of advanced mathematics is not very
important.

307

w
B
o
<
A

W
8
u
fC
C/1
M
a
D

S3
M
e5
P3
8
s
u 51.
stand a good deal of mathematics.

A

D

u

52.

The most enjoyable part of my life is the time I spen
in school.

A

D

u

53.

Even complex mathematics can be made understandable a
useful to every high school student.

A

D

u

54.

In the near future most jobs will require a knowledge
advanced mathematics.

A

D

u
U

55.

With hard work anyone can succeed.

A

D

u

56.

Almost every present human problem will be solved in
the future.

A

D

u

57.

Almost all pupils can learn complex mathematics if it
is properly taught.

A

D

U
u

58.
58. I like all school subjects.

A

D

U
u

59. There is little place for originality in mathematics.
59.

A

D

u

60.

I enjoy most o f my school work and want to get as mud
additional education as possible.

A

D

u

61.

Only people with a very special talent can learn
mathematics.

A

D

u
U

62. Mathematics will change rapidly in the near future.
62.

A

D

u

63.

Although school is difficult, I want as much educatioi
as I can g et.

A

D

u

64.

In the study of mathematics, if the student misses a J
lessons it is difficult to catch up.

A

D

U
u

65. I find school interesting and challenging.
65.

APPENDIX E
ENJOYMENT AND VALUE OF M A T H EMATICS SCALES

APPENDIX E

ENJOYMENT AND VALUE OF MATHEMATICS SCALES
Two Scales of Attitude Toward Mathematics
by Lewis R. Aiken
Directions:

Draw a circle around the letter(s) that show(s) how closely
you agree or disagree with each statement: SD {Strongly
Disagree), D (Disagree), U (Undecided), A (Agree), SA
(Strongly Agree).
E SCALE: ENJOYMENT OF MATHEMATICS

1. I enjoy going beyond the assigned work and trying
to solve new problems in mathematics.

SD

D

U

A

SA

2. Mathematics is enjoyable and stimulating to me.

SD

D

u

A

SA

3 . Mathematics makes me feel uneasy and confused.

SD

D

u

A

SA

4. I am interested and willing to use mathematics
outside school and on the job.

SD

D

u

A

SA

5. I have never liked mathematics, and it is my most
dreaded subject.

SD

D

u

A

SA

6 . I have always enjoyed studying mathematics
in school.

SD

D

u

A

SA

7. I would like to develop my mathematical skills
and study the subject more.

SD

D

u

A

SA

8. Mathematics makes me feel uncomfortable and
nervous.

SD

D

u

A

SA

9. I am interested and willing to acquire further
knowledge of mathematics.

SD

D

u

A

SA

308

309

E SCALE:

ENJOYMENT OF MATHEMATICS

10. Mathematics is dull and boring because it leaves
no room for personal opinion.

SD

D

U

A

SA

11. Mathematis is very interesting and I have usually
enjoyed courses in this subject.

SD

D

u

A

SA

1. Mathematics has contributed greatly to science
and other fields of knowledge.

SD

D

u

A

SA

2. Mathematics is less important to people than
art or literature.

SD

D

u

A

SA

3. Mathematics is not important for the advance
of civilization and society.

SD

D

u

A

SA

4. Mathematics is a very worthwhile and necessary
subject.

SD

D

u

A

SA

5. An understanding of mathematics is needed by
artists and writers as well as scientists.

SD

D

u

A

SA

6. Mathematics helps develop a person’s mind and
teaches him to think.

SD

D

u

A

SA

7. Mathematics is not important in everyday life.

SD

D

u

A

SA

8. Mathematics is needed in designing practically
eveything.

SD

D

u

A

SA

9. Mathematics is needed inoorder to keep the
world running.

SD

D

u

A

SA

10. There is nothing creative about mathematics;
it's just memorizing formulas and things.

SD

D

u

A

SA

V SCALE:

VALUE OF MATHEMATICS

APPENDIX F
RAW SCORES OF T E A C H E R CORPS INTERNS
ON ALL MEASURES

APPENDIX F
RAW SCORES OF TEACHER CORPS INTERNS ON ALL MEASURES

u
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a
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1
2
3
4
5
6
7
e
9
10
11
12
13
14
15
16
17
IB
19
20
21
22
23
24

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tn
0
£U

ro

»

jj

JJ
<w
to
01

a

40
52
41
36
21
41
38
29
36
23
41
34
32
34
35
28
27
39
36
39
41
29
42
22
B. ”
B ■
B* =

u
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1
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0

&

tn
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6.8
7.3
6.3
4.7
6.2
8.3
5.5
7.1
7.9
7.4
7.1
8.6
7.8
4.9
3.8
7.4
2.6
7.1
7.6
7.1
6.0
3,6
4.5
2.8

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20
44
42
22
13
36
21
28
37
22
30
23
32
33
21
17
16
23
36
32
26
30
33
22

5.3
6.9
7.4
3.3
2.3
7.1
6.8
5.3
6.8
7.5
5.8
8.6
7.6
4.5
35.
8.0
2.0
5.5
7.0
6.3
4.8
3.2
3.9
2.9

u

a.

31
44
33
23
23
31
23
27
38
31
38
38
44
21
15
33
13
29
15
28
30
27
23
IB

30
36
29
39
33
31
31
26
35
27
29
38
40
35
29
31
29
30
21
25
35
30
31
33

40
46
30
25
20
35
35
30
30
20
40
25
35
45
25
15
15
20
50
20
35
5
30
30

d

Jj

M
CQ

10
8
13
12
6
11
10
6
10
8
4
11
10
7
8
6
12
10
5
11
12
7
9
9

0

c*

12
11
13
14
12
12
14
12
14
7
14
10
14
10
9
14
13
12
12
12
14
11
12
9

rt
CO

8
11
8
15
11
8
10
10
6
11
10
6
12
12
10
6
6
8
9
8
9
7
12
8

^
Q

15
14
16
14
14
12
16
15
IB
15
13
7
13
16
12
12
17
15
11
13
13
15
10
12

a

m

B
10
10
8
11
7
5
10
10
11
8
7
16
7
6
10
10
8
B
11
8
6
7
5

4J
to
V

H0
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u
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1
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©

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Attitude Toward Mathematics as Process
AttitudeToward Difficulties of Learning Mathematics
AttitudeToward Place of Mathematics in Society

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36
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31
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44
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39
33
27
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36
26
43
47
11
50
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10
50
43
10
16
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10
23
3
18
30
15
5
9
26
0
30
23
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13
31
23
6

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46
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39
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49
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29
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34
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43
47
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39
40
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16
42
35
34
19
39
25
13
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15
35
19
7
16
33
IB
25
17
23
19
23
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32
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47
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38
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43
36
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28
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48
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26
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31
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19
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24
50
36

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35
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12
10
18
21
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20
7
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42
47
28
35
28
45
36
29
42
18
42
22
17
29
18
28
16
31
35
26
33
25
45
28

* Attitude Toward school and School Learning
= Attitude Toward Man and His Environment
E-Scale = Enjoyment of Mathematics
V-Scale ■» Value of Mathematics

APPENDIX G
R A W S C O R E S O F THE

" C O M P A R I S O N GROUPS"

ON ALL MEASURES

APPENDIX G

RAH SCOPES OF THE "COMPARISON GROUPS" ON ALE MEASURES

Raw Scores of Students In the Content-Method Integrated Program,
*)
N
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M
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5

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0

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Dr

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40
45
31
34
39
39
31
26
33
33
33
28
41
36
33
43
37
45
36
36
47

4.0
7.1
6.9
2.5
6.0
B.3
2.0
2.4
8.6
7.8
7.4
3.2
6.3
7.0
5.0
7.0
5.4
7.8
8.0
5.7
8.0

3
4
5

6

1

e

9

10
11
12
13
14
15
16
17

IB
19

20
21

*>
a
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30
40
35
25
36
23
30
14
31
32
36
30
13
30
38
37
36
35
30
42
41

5.3
6.9
7.5
2.9
2.4
8.3
2.9
2.4
3.2
7.1
7.1
2.4
7.1
1.0
4.8
5.2
4.8
5.2
7.6
6.5
7.9

4
12
8
4
11
11
9
11
10
12
9
8
11
11
9
9
14
7
10
12
12

10
10
8
14
9
14
14
11
7
6
14
7
14
12
9
8
10
6
10
8
9

8
8
10
8
10
10
10
9
11
9
10
6
9
6
11
8
8
6
8
10
10

10
14
12
18
17
17
9
11
9
11
IS
11
13
7
14
13
12
14
12
11
13

3
4
9
B
4
4
6
5
5
4
10
4
11
4
10
3
6
12
8
8
7

» Attitude Toward Mathematics as Process
B~ * Attitude Toward Difficulties Learning Mathematics
B, * Attitude Toward Place of Mathematics in Society
B, “ Attitude Toward School and School Learning
4

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

o
n
o

1
a
0
B.
a
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o
a

w
4J

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a
a

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G
S
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a3

2

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eh

Raw Scores of Students In the Regular Methods Class, G,

a

23
43
35
35
36
40
29
33
27
29
32
30
38
29
30
33
36
24

0
B.

*1
a

a
n

i
•
w
B.
a

a

—

a

u

o

e.i
6.3
7.1
7.1
7.4
7,9
6.7
6.8
7.4
3.3
4.9
3.3
7.3
6.7
6.3
5.9
7.0
6.1

V

a
a
a

36
46
36
37
42
44
38
44
32
27
38
31
44
34
30
30
37
26

&2 • Attitude Toward Kan and His

V
a

e
1
a
it
a.

H

a

"c

a9

m
o

9.0
7.0
6.3
5.8
7,8
7.1
3.9
7.1
7,4
4.6
7.4
4.4
7.8
6.4
6.3
5.6
6.3
6.1

4
6
10
8
11
11
6
14
10
6
11
6
10
B
11
10
13
&

10
11
9
10
7
7
10
8
10
10
13
6
10
14
11
14
12
3

8
9
7
9
6
10
10
6
4
5
10
8
9
12
9
6
11
7

Environment
E-Scale « Enjoyment of Mathematics
V-Scale « Value of Mathematics

15
8
14
16
10
10
13
12
11
7
15
2
12
20
U
18
14
9

8
9
5
9
4
5
10
4
6
10
14
10
10
16
10
8
8
7