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I

73-29,780
SHAKRANI, Mosen Sharif, 1942A FOfMATIVE EVALUATION OF THE MATHEMATICS
COMPONENT OF AN EXPERIMENTAL E L M N E A R Y
TEACHER EDUCATION PROGRAM AT MICHIGAN STATE
UNIVERSITY.
Michigan State University, Ph.D., 1973
Education, teacher training

U niversity M icrofilm s, A X£ROXCompany , A n n A rbor, M ich ig an

A F O R M A T I V E E V A L U A T I O N OF THE M A T H E MA TICS
C O M P O N E N T OF A N E X P E R I M E N T A L E L E MENTARY
TEA CHER E D U C A T I O N P R O G R A M AT
MICHI GAN

STATE U N I V E R S I T Y

By
Mo sen Sharif Shakrani

A THESIS
S ubm itted to
M i c higan State U n i v e r s i t y
in p artial fulf ill men t of the re quirement s
for the deg r e e of

D O C T O R OF P H I L O S O P H Y

C o l l e g e of E d u c a t i o n

1973

ABSTRACT
A F ORMATIVE EVALUA TION OF THE MAT H E M A T I C S
C O M P O N E N T O F AN E X P E R I M E N T A L E L E M E N T A R Y
T E A C H E R ED UCA TI ON P R O G R A M AT
MICHIGAN STATE U N I V E R S I T Y
By
Mosen Sharif Shakrani

This study was a formative eval u a t i o n of the
mathematics

c o m p o n e n t of an e x p e r i m e n t a l elem e n t a r y

educa t i o n p r o g r a m at Michiga n S t ate Uni versity.
cally,

this

ade qu acy of

investi gat ion s ou g h t to:

the future e le m e n t a r y school teachers,
e ffe c t of the
students'
tencies,

Specifi­

(1) evaluate

the mat hematical c o n t e n t in m e e t i n g

te acher

the

the need of

(2) evalu a t e

in structional t reatme nt on the p a r t i c i p a t i n g

p e r f o r m a n c e on the p r e s c r i b e d m a t h e m a t i c a l
(3)

(80 percent)

the

compe­

a s c e r t a i n w hether a s p e c i f i e d de gre e of m a s t e r y
had been attai ned on the p r e s c r i b e d c o m p e t e n ­

cies w i t h i n each m a t hem atical topic,
of i n s t r u c t i o n on the students'
and at titudes,

(4) e v a l u a t e the ef f ect

mathematical understandings

(5) compare stude nts in the e x p e r i m e n t a l

p r o g r a m w i t h studen ts

in the r e g u l a r e l e m e n t a r y

tea che r

e duc a t i o n p r o g r a m in relation to their m a t h e m a t i c a l u n d e r ­
st andings

an d attitudes,

and

b e t w e e n s e l e c t e d variables

(6) d e t e r m i n e the r e l a t i o n s h i p

and a c h i e v e m e n t in mathema tics .

Mosen Sharif Shakrani

The e x p e r i m e n t a l p r o g r a m is fu nde d and st affed by
the "Tr ain ers of T e a c h e r Tra iners"
base d on aspects of
Program

(BSTEP)

(TTT)

P rojec t and is

the B e h a v i o r a l S cience t eacher E d u c a t i o n

d e v e l o p e d at M i c h i g a n State Univ e r s i t y

in

1968.
The m a t h e m a t i c s c o m p o n e n t of the experime nta l
p r o g r a m is c o m p o s e d of two c ourses
of m a t h e m a t i c s w i t h

that integrate

the m e t h o d o l o g y of

in a labora tor y setting.

The

the study

teaching m a t h e m a t i c s

first course, o f f e r e d during

the fresh man year,

e m p h a s i z e s ari thm eti c.

o ffe red during the

junior year,

etry.

includes c l i n i c a l e x p erience w h e r e p r o ­

Each co urse

sp ect ive teachers
mathematical

tr an slat e

concepts,

The se con d course,

emph a s i z e s al g e b r a and g e o m ­

theory

into pract ice by

taught at the University,

teaching

to groups

of ele m e n t a r y s c h o o l children.
Durin g the academic y e a r
integrated c o n t e n t - m e t h o d s

1971-1972,

the

first

c o u r s e was deve lop ed by a te am

of m a t h e m a t i c s e d u c a t o r s an d e l e m e n t a r y school teachers.
The co urse c o m p r i s e d
(1) Mea sur em ent,
Relations,

the f ollowing m a t h e m a t i c a l

(2) N u m e r a t i o n Systems,

(4) W h o l e Numbers,

(7) Re lat io ns and Functi ons,
and

topics:

(3) Sets and Set

(5) Fractions,

(6)

Decimals,

(8) Pr o b a b i l i t y and Statistics,

(9) M a t h e m a t i c a l Systems.
For each topic,

specified;

m a t h e m a t i c a l c o m p e t e n c i e s we re

and to ac h i e v e th ese competencies,

experiences

Mosen Sharif Shakrani
utili z ing m a n i p u l a t i v e and other instructio nal m aterials
w e r e prescribed.
class hours).

Each to pic was c overed in one we ek

At the end of each w e e k

in small groups,

the students,

p l a n n e d instructional designs

aspects of that topic.

These d esigns wer e

(eight
working

to teach

im ple mented

in an el ementa ry sc hool in the s u c ceeding week.
F o r each
re fer enced
assess

topic,

tw o p ara ll el

forms of cr it erion-

tests w e r e d e v e l o p e d by this i n v e s t i g a t o r to

the students'

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

m ati cal compe ten cies .

Th e method by which

c o n s t r u c t e d insu red their content validity.
estimat es of the tests ranged
Th e

tests w e r e

Reliability

from 0.77 to 0.93.

fol lowing instruments we re als o s e l e c t e d

the col l e c t i o n of data:
Un ders tandings,
and

these

for

(1) Test of Basic M a themati cal

(2) Du tto n A r i t h m e t i c A t t i t u d e

Inventory,

(3) A t t i t u d e s Towa rd D ifferent Aspects of Ma t h e m a t i c s

d e v e l o p e d by the Intern ati onal Study of A c h i e v e m e n t in
Mathematics.
T h e e x p e r i m e n t a l gr oup in this study w e r e
e i g h t freshm an e l e m e n t a r y
and wer e

selec ted

E v i dence i n d i c a t e d

thirty-

edu cation majors w h o v o l u n t e e r e d

to p a r t i ci pate

in the e x p e r i m e n t a l program.

that these volun teer s d i d not dif f e r

o the r fr eshman e l e m e n t a r y education majors
tive and a f f e c t i v e be hav iors

from

in their c o g n i ­

toward mathematics.

Mosen Sharif Shakrani
M u l t i v a r i a t e and u n i v a r i a t e analys is of vari anc e
w e r e used in asses s i n g the e f f e c t of the i n t e g r a t e d contentm eth ods co urse upon students*
refer enc ed tests.

The

t-test for corr e l a t e d m e a n s was

use d in tes ting changes
and attitudes.
ficient wa s

p e r f o r m a n c e on the criter ion -

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

understa n d i n g s

P e a r s o n Pr o d u c t Mom e n t C o r r e l a t i o n Coef ­

utilize d

in the r e l a t i o n s h i p analysi s repo rte d

in this study.
Th e

level of s i g n i f i c a n c e was se t at 0.05

testing all hyp o t h e s e s

for

in this study.

Findings of the Study
1.
(p < .005)

The e x p e r i m e n t a l g r o u p m a d e

in mean sc ores from pre- and p o s t - t e s t s during

the int e g r a t e d c o n t e n t - m e t h o d s
r e f erenced
2.
level

si g n i f i c a n t gains

tests

course o n the crit erion-

for all topics exc ept Meas ure men t.

The e x p e r i m e n t a l

g roup a t t a i n e d the maste ry

(achieving at least 80 p ercent of

on the post-test)

the items correct

on the c r i t e r i o n - r e f e r e n c e d

tests

for all

topics e x c e p t M e a s u r e m e n t and M a t h e m a t i c a l Systems.
3.
improvement

The e x p e r i m e n t a l g r o u p s h o w e d s i g n i f i c a n t
( p < .001)

b e t w e e n pre- and p o s t - t e s t means 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 s and on arith ­
m e t i c a t t i t u d e scale,

whil e e n r olled in the i n t egrated

c o n t e n t - m e t h o d s course.

Mosen Sharif Shakrani
4.

With

ex perimental

initia l d i f f e r e n c e s a llowed for,

group,

af ter c o m p l e t i n g o n l y

of t h e i r ma t h e m a t i c s edu cation,

attitudes

in the regular
all

(a)

toward m a t h e m a t i c s

conc ept s and mo re p o s i ­

than a group of students

teac her e d u c a t i o n p r o g r a m who ha d co mp leted

the re quir ed m a t h e m a t i c s
5.

the first p a r t

showed s i g n i f i c a n t l y b e t t e r

u n d e r s t a n d i n g of bas ic m a t h e m a t i c a l
tive

the

education.

T h e r e w e r e s i g n i f i c a n t c o r r e l a t i o n s between:

pre- and p o s 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

measures,

(b) p o s t - t e s t sc ores on the test of m a t h e m a t i c a l

understandings

and the a r i t h m e t i c a t t i t u d e scale,

of h i g h school c ourses

in m a t h e m a t i c s

sc ores on the test of m a t h e m a t i c a l
and p o s t - t e s t scores

and pre-

(c) numb er

and p o s t - t e s t

understandings,

(d) pre-

on 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

test

and high school g r a d e - p o i n t average.
6.

Th e e x p e r i m e n t a l

participation

in cli nic al

laboratory o r i e n t e d

gro up e x p r e s s e d

des ire

for more

expe r i e n c e c o n c u r r e n t with

the

int e g r a t e d c o n t e n t - m e t h o d s courses.

Conclusions
The a c t i v i t y - o r i e n t e d i n t e g r a t e d c o n t e n t - m e t h o d s
c o u r s e c on c u r r e n t w i t h c l i n i c a l e x p e r i e n c e h a d a s i g n i f i c a n t
p o s i t i v e e f f e c t on p r o s p e c t i v e e l e m e n t a r y
cogn i t i v e and affective b e h a v i o r s

teachers'

tow ard math ema tic s.

ACKNOWLEDGMENTS

I wish
Lanier,

to e x press my a p p r e c i a t i o n

chairm an of my Doctoral Committee,

a g e m e n t and understanding.
most

to Dr.

in flu ential
I wish

Fitzgerald,

Dr.

His

for his e n c o u r ­

couns el and

in gu i d i n g this st udy

to exp res s my g r a t i t u d e
Laur e n G. Woodby,

Perry E.

judgmen t w e r e

to completion.
to Dr, W i l l i a m M.

and Dr.

M aryellen T.

M c S w e e n e y , wh o w e r e me m b e r s of my

D o c t o r a l C o m m i t t e e and

gave g e n er ously of th eir time and

talents

t h r ough out this

study.
I wish
Scrivens,

to thank Dr. D a vid O'Neal,

and Mr.

their students

Dani el W i l l i a m s

Robert

for a l l o w i n g me to use

in this study.

Spe cial a p p r e c i a t i o n
Dufour

Dr.

is e x p r e s s e d to Dr.

for v a l u a b l e s uggestions

France

in p r e p a r i n g this m a n u ­

script .
My s i n c e r e

thanks

to Mr. W i l l i a m C.

his p r o o f r e a d i n g and e d i t o r i a l comments.

Har twig

for

TABLE OF CONTENTS
Chapter
I.

II.

Page
INTRODUCTION

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

1

B a c k g r o u n d ...................
The T T T P r o g r a m at M i c h i g a n S t a t e
U n i v e r s i t y ..............
The B e h a v i o r a l S c i e n c e T e a c h e r E d u c a t i o n
P r o g r a m (BS T E P )
.
N e e d for the S t u d y .............................
P u r p o s e of the S t u d y ...........................
F o r m a t i v e E v a l u a t i o n and C u r r i c u l u m
I m p r o v e m e n t ..................................
H y p o t h e s e s .......................................
A s s u m p t i o n s .....................................
L i m i t a t i o n s of the S t u d y ......................
D e f i n i t i o n of T e r m s ...........................

2

R E V I E W O F L I T E R A T U R E .............................
Curriculum Evaluation
........................
Defining Curriculum Evaluation . . .
M e a s u r e m e n t of A c h i e v e m e n t ..........
R a t i o n a l e for the I m p r o v e m e n t of T e a c h e r
E d u c a t i o n P r o g r a m s at the E l e m e n t a r y
L e v e l .........................................
D e s i g n s for P r o g r a m s of T e a c h e r E d u c a t i o n
.
P r e s a g e F a ctors
......................
P rocess F a c t o r s
......................
Product Factors
......................
Myths About Teacher Effectiveness
..........
A p p r o a c h e s to T e a c h e r T r a i n i n g ...............
M a t h e m a t i c a l C o m p e t e n c i e s of E l e m e n t a r y
School Teachers
.............................
M a t h e m a t i c a l C o m p e t e n c i e s of E l e m e n t a r y
School Children
.............................
R e s e a r c h on A t t i t u d e s T o w a r d M a t h e m a t i c s . .
T e c h n i q u e s Used to M e a s u r e
A t t i t u d e s ...........................
A t t i t u d e s and P e r s o n a l i t y
Characteristics
...................

iii

4
5
8
9
10
11
13
14
16
18
18
19
23

29
34
35
37
39
40
41
50
54
57
57
59

Page

Chapter
Teachers* Atti tudes and
E f f e ctiveness To ward Ma t h e m a t i c s . .
............
A t t i t u d e s and A c h i e v e m e n t
A t t i t u d e s and the N e w M a t h e m a t i c s
C u r r i c u l a .............................
S u m m a r y ...........................................
III.

D E S C R I P T I V E F E A T U R E S OF THE S T U D Y ..............

60
63
65
68
71

71
S t u dents in the S t u d y ..........................
E v a l u a t i o n of the E x p erimental P r o g r a m . . . .
73
D e s c r i p t i o n of the M a t h e m a t i c s
C o m p o n e n t of the E x p e r i m e n t a l
T e a c h e r Educati on P r o g r a m
..........
73
A s s e s s m e n t of the M a t h e m a t i c s
C o m p o n e n t of the E x p e r i m e n t a l
T e a c h e r Edu cation Program:
D e v e l o p m e n t of a C r i t e r i o n Ref e r e n c e d L i s t ......................
75
The Int e g r a t e d C o n t e n t - M e t h o d s C o u r s e in
M a t h e m a t i c s ....................................
80
D e s i g n i n g the C u r r i c u l u m of the
Int e g r a t e d Conten(.-Methods
C o u r s e in M a t h e m a t i c s
..............
80
E x p e r i m e n t i n g with the I n t e g r a t e d
C o n t e n t - M e t h o d s Course:
Procedu re
F o l l o w e d ............................
82
C l i n i c a l E x p erience
...................
84
A s s e s s i n g the C o n t e n t of the
Int e g r a t e d C o n t e n t - M e t h o d s Course
.
87
D e v e l o p m e n t of 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 Measures ...................
87
D e v e l o p m e n t of the Te st Instruments
.
89
E v a l u a t i o n of the C r i t e r i o n Ref 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
. .
95
S e l e c t i o n of a Test of M a t h e m a t i c a l
Understanding
.........................
100
S e l e c t i o n of an A t t i t u d e Inventory . . 101
C o n s t r u c t i o n of S t u d e n t Q u e s t i o n ­
nair e
.................................... 102
S t a t i s t i c a l Pro ce dures for the
A n a l y s i s of D a t a ....................102
S i g n i f i c a n c e Level C h o s e n
.............
104
S u m m a r y ......................................... 105

iv

Chapter
IV.

Page
A N A L Y S I S OF DATA A N D R E S U L T S ........................ 107
A n a l y s i s of the M a t h e m a t i c a l Conten t
of the Le arning U n i t s ........................... 108
F i n d i n g s .................................... 113
C o m p a r i s o n of the E x p e r i m e n t a l G roup and
O t h e r Fr eshman Gr oup s on C o g n i t i v e and
A f f e c t i v e B e h a v i o r s To war d M a t h e ma tics . . . 115
I n s t r umentation
........................
116
Data A n a l y s i s ............................. 117
F i n d i n g s .................................... 121
C o n c l u s i o n s ................................125
E v a l u a t i o n of the E x p e r i m e n t a l Group
Per f o r m a n c e 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 e s ............................................126
..........................
126
Hypot hes es Tes t e d
Data A n a l y s i s ...................
128
F i n d i n g s .................................... 129
A n a l y s i s of Tes t R e s u l t s ................. 136
E ff e c t of the E x p e r i m e n t a l P r o g r a m on the
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 s and
Atti t u d e s T o w a r d M a t h e m a t i c s .................
138
Growt h in B a s i c M a t h e m a t i c a l
139
Un d e r s t a n d i n g s .............................
H y p othesis B 1 ............................. 139
Data A n a l y s i s ............................. 139
F i n d i n g s .................................... 141
A n a lysis of Pre- and Post- T e s t
R e s u l t s ............................... 142
C han ges in A t t i t u d e Tow a r d M a t h e m a t i c s . .
143
Hypo the sis B 2 ............................. 143
T r e a t m e n t of D a t a ..................... 144
F i n d i n g s ................................. 144
Relate d Q u e s t i o n s
.....................
14 5
F i n d i n g s ................................. 146
C o m p a r i s o n of E x p e r i m e n t a l G r o u p w i t h a
R egu lar E l e m e n t a r y Te a c h e r E d u c a t i o n Group
on Mathem a t i c a l U n d e r s t a n d i n g s and Atti t u d e s
Towa r d A r i t h m e t i c
. . . .
150
Hyp o t h e s i s C l .......................... 152
F i n d i n g s .................................... 152
H y p othesis C 2 .......................... 154
F i n d i n g s ................................. 154
C o r r e l a t i o n A n a l y s i s .............................
156
E v a l u a t i o n of S t u d e n t R e a c t i o n to the
M a t h e m a t i c s C o m p o n e n t of the E x p e r i m e n t a l
P r o g r a m .........................................160
S u m m a r y ........................................... 166

v

Chapter
V.

Page
SUMMARY, CONCLUSIONS,

A N D RE C O M M E N D A T I O N S

. . .

168

. ......... ............................... 16 8
The M a t h e m a t i c s C o m p o n e n t of the
Experi m e n t a l P r o g r a m .................
169
P u r p o s e .................................... 169
Re view of L i t erature ...................
170
H y p o t h e s e s .................................. 173
The I n t egrated C o n t e n t - M e t h o d s
C o u r s e .................................... 175
Student s in the S t u d y ....................177
I nst rum e n t a t i o n
........................
178
S t a t i s t i c a l An alys is ...................
178
L i m i t ations of the S t u d y ................. 179
Findings of the S t u d y ........................ 180
Analysis of the M a t h e m a t i c a l C o n t e n t
of the E x p e r i m e n t a l P r o g r a m
. . . .
180
C o m p a risons of the E x p e r i m e n t a l G r oup
w i t h O t h e r F r e shman Gr oups on E n t e r ­
ing C o g n i t i v e and A f f e c t i v e
B eha viors To ward M a t h e m a t i c s . . . .
181
Eval u a t i o n of the E x p e r i m e n t a l G r oup
P e r f o r m a n c e on the C r i terionRefe r e n c e d M e a s u r e s ....................182
The Eff e c t of the E x p e r i m e n t a l
P r o g r a 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 s t a n d i n g s and Attit u d e s
T owa r d M a t h e matic s ...................
184
C o m p a r i s o n of the E x p e r i m e n t a l G ro up
wi th a Regul ar E l m e n t a r y Ed uca tion
G r o u p on M a t h e m a t i c a l Und e r s t a n d i n g s
and A t t i t u d e s Tow a r d M a t h e matics .
. 186
C o r r e l a t i o n A n a l y s i s ...................
187
Stude nt Re actions to the M a t h e matics
C o m p o n e n t of the E x p e r i m e n t a l
P r o g r a m ............................. 188
C o n c l u s i o n s .................................... 189
D i s c u s s i o n ............................. 192
Observ a t i o n s .............................
194
R e c o m mendations
.................................
19 8
Recomme n d a t i o n s for Fu ture A c t i o n
.
. 198
Recomme n d a t i o n s for Fu ture Research
. 201

S umm ary

BIBLIOGRAPHY

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

vi

206

Appendix
A.

B.

Page

S E T OF N I N E PRE- AN D P O S T - T E S T F O R M S O F
TH E 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 ............

218

A 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 S
F O R M A (PRE-TEST) A N D FORM B (POST-TEST)

158

..........

2 04

C.

DUTTON A R I T H M E T I C A T T I T U D E

D.

A T T I T U D E SCA L E S T O W A R D D I F F E R E N T A S P E C T S
O F M A T H E M A T I C S .................................... 286

E.

TH E E X P E R I M E N T A L G R O U P E V A L U A T I O N OF
D I F F E R E N T A S P E C T S OF TH E P R O G R A M ............... 292

F.

A S P E C I M E N OF STU D E N T FILE FO R TH E L E A R N I N G
UNIT ON N U M E R A T I O N S Y S T E M S ...................... 295

G.

SC ORES OF S T U DENTS IN THE E X P E R I M E N T A L GROUP
ON THE PRE- A N D P O S T - C R I T E R I O N REFE R E N C E D
T E S T S .................................................306

H.

SC OR ES OF S T U DENTS IN THE E X P E R I M E N T A L GROUP
ON THE 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
A N D DU TTON A R I T H M E T I C A T T I T U D E INVEN T O R Y
. .

30 8

THE E X P E R I M E N T A L G R O U P HIGH S C H O O L B A C K G R O U N D
F ACT ORS A N D FIN AL G R A D E ON T H E C O M B I N E D
C O N T E N T - M E T H O D S COU R S E
........................

310

J.

CORRELATION MATRIX

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

312

K.

N U M B E R OF S T U D E N T S IN THE E X P E R I M E N T A L GROUP
WHO A N S W E R E D THE T E S T ITEMS C O R R E C T L Y ON
THE N I N E PRE- AN D P O S T - 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 .............................

321

I.

I NVENTORY

. .

L.

RAW SCOR E S OF S T U D E N T S IN TH E E X P E R I M E N T A L
GROUP O F E N T R Y C H A R A C T E R I S T I C S .............. 32 4

M.

T E S T SC O RES OF S T U DENTS IN 325E ON 50 PE R C E N T
ITEM SAMPLE O F C R I T E R I O N - R E F E R E N C E D TEST S . .

326

N.

SCO RES O F T HE "C O M P A R I S O N GROUPS " ON E NTRY
D A T A .................................................328

O.

PRE- A N D P O S T - T E S T S C O R E S OF S T U D E N T S IN
R E G U L A R M E T H O D S C O U R S E (32 5E) ON DU TTON
A T T I T U D E S C ALE A N D T E S T OF M A T H E M A T I C A L
UNDERSTANDINGS
..................................

vii

332

Appendix

Page

P.

ONE- WA Y A N A L Y S I S O F V A R I A N C E RELATIVE TO
TESTI NG D I F F E R E N C E S B E T W E E N THE
E X P E R I M E N T A L GROU P A N D T H E "COMP ARI SON
G R O U P S " ........................................... 334

Q.

HO YT REL I A B I L I T Y C O E F F I C I E N T FOR C R I TERIONREF 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 ..............

viii

3 39

LIST OF TABLES
Table
1.

2.

3.

4.

Page
S u g g e s t e d Topics for the M a t hematical
Pr e p a ration of E l e m e n t a r y School Te achers

. . .

78

Reli abil ity C o e f f i c i e n t s 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 Tests .........

98

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 tween Pre- and
P o s t - T e s t Scores of the Students in Regul ar
M eth ods Course (Education 325E) on ItemSample d 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
. . . .

99

Means and Standard D e v iations on Entry
Data for the E x p e r i m e n t a l G r o u p and
Three Fresh man C o m p a r i s o n Groups .................

118

5.

t-Values for M e a n C o m p a r i s o n of Exp eri men tal
G rou p and the Thre e "C omparison Groups" on
Entry C h a r a c t e r i s t i c s ........................ 120

6.

Means and Standard Deviations of Pre- and
P o s t - T e s t Scores on the Nine Cri terion
Measures for the E x p e r i m e n t a l G r o u p
............

130

M u l t i v a r i a t e A n a l y s i s of Varia n c e for the
E x p erimenta l G r o u p on Diff erence s Betwee n
Pre- and Post - T e s t Scores on the Nine
C r i t e r i o n Measures .................................

132

Multiv a r i a t e A n a l y s i s of V a r i a n c e for the
Ex per imental G r o u p on Diff erence s B etween
P o s t - T e s t Scores and Mast ery Level (80
percent) on the Ni ne C r i t e r i o n Measures
. . . .

134

7.

8.

9.

10.

Per c e n t a g e of S t u d e n t s in the E x p e r i m e n t a l
Group (N = 38) A t t a i n i n g the P r e - E s t a b l i s h e d
Ma ste ry Level
..........................
. . . . .

137

Pre- and P o s t - T e s t Results of the
E x perimenta l G r o u p on the T e s t of
M a t h e m a t i c a l Und e r s t a n d i n g s
.....................

141

ix

Table
11.

Page
Pre- and P o s t - T e s t Resu lts of the
E x p e r i m e n t a l G r o u p on the Dutton
A t t i t u d e S c a l e ..............

145

12.

Stud ent s (N = 38) Fee lings A b o u t A r i t h m e t i c
in G e n e r a l .............................................. 146

13.

G r a d e Level W h e r e Students' (N = 38)
Attitudes W e r e D e v e l o p e d ..........................

14 8

14.

A spects of A r i t h m e t i c Students (N - 38) Liked
........................... 149
an d Dislike d M o s t

15.

Pre- and P o s t - T e s t Re s u l t s of the
Experi m e n t a l G r o u p and the R eg ular Me t h o d s
C o u r s e (Education 325E) S t u dents on the Te st
of Mathem a t i c a l U n d e r s t a n d i n g s (MU) and
Dut t o n A r i t h m e t i c A t t i t u d e Scale (AA)
.........

151

Summary 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 the E x p e r i m e n t a l G r o u p and
the Regular (Education 325E) S t u d e n t s on
the TeBt 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
. . . .

153

S ummary of A n a l y s i s of C o v a r i a n c e for the
Sco res of the E x p e r i m e n t a l G r oup and the
Students in the Regul ar M ethods Course
(Education 325E) on D u t t o n A r i t h m e t i c
A t t itude Inventory ..................................

155

Frequen cy D i s t r i b u t i o n of E x p e r i m e n t a l Group
Response to St u d e n t E v a l u a t i o n Q u e s t i o n n a i r e

161

16.

17.

18.

. .

19.

Summa ry of A n a l y s i s of V a r i a n c e for MSU
Ar i t h m e t i c T e s t ....................................... 335

20.

Summ ary of A n a l y s i s of V a r i a n c e for MSU
M a t h e matics T e s t ....................................... 335

21.

Summ ary of A n a l ysis of V a r i a n c e
of Attit u d e s To ward M a t h e m a t i c s

22.

on the Scale
as a P rocess

. .

336

Summ ary of A n a l y s i s of V a r i a n c e
on the Scale
of Attit u d e s Tow a r d D i f f i c u l t i e s of Le a rning
M a t h e m a t i c s ............................................336

x

age
S ummary of A n a l y s i s of V a r i a n c e on the
Sc ale of A t t i t u d e s To war d School and
School Lea rn i n g
...............................

337

Su m m a r y of A n a l y s i s of V a r i a n c e on the
Sc ale of A t t i t u d e s Tow a r d Plac e of
M a t h e m a t i c s in So c i e t y ........................

337

S umm ary of A n a l y s i s of V a r i a n c e on the
Dut t o n A r i t h m e t i c A t t i t u d e Sc ale ............

338

Su mma ry of A n a l y s i s of V a r i a n c e for P r e - T e s t
in M e a s u r e m e n t . . . .

340

S umm ary of A n a l y s i s of V a r i a n c e
in M e a s u r e m e n t . . . .

for P o s t - T e s t

S umm ary of A n a l y s i s of V a r i a n c e
in N u m e r a t i o n
. . . .

for P r e - T e s t

Summa ry of A n a l y s i s of V a r i a n c e
in N u m e r a t i o n
. . . .

for P o s t - T e s t

340
341
341

S ummary of A n a l y s i s of V a r i a n c e for P r e - T e s t
of Sets and Set R e l a t i o n . . . .

342

S ummary of A n a l y s i s of V a r i a n c e for P o s t - T e s t
of Sets and Set R e l a t i o n . . . .

342

Su m m a r y of A n a l y s i s of V a r i a n c e
in W h o l e N umbe rs . . .

for P r e - T e s t

Summary of A n a l y s i s of V a r i a n c e
in W h o l e Number s . . .

for P o s t - T e s t

S ummary of A n a l y s i s of Va r i a n c e
in F r a c t i o n s .........

for Pr e - T e s t

Summary of A n a l y s i s of V a r i a n c e
in F ractions .........

for P o s t - T e s t

Su m m a r y of A n a l y s i s of V a r i a n c e
in D e c i m a l s
.........

for P r e - T e s t

S u m m a r y of A n a l y s i s of V a r i a n c e
in D e c i m a l s
.........

for P o s t - T e s t

343
343
344
344
345

345

Table

Page

38.

Sum ma ry of An alys is of V a r i a n c e for P r e - T e s t
in Re la tion s and F u n c t i o n s ........................... 346

39.

S ummar y of Analysi s of V a r i a n c e for P o s t - T e s t
in Rel atio ns and F u n c t i o n s ........................... 346

40.

Summary of Ana lys is of V a r i a n c e for P r e - T e s t
in P r o b a bility and St atist ics
...................

347

41.

Su m m a r y of A n a l y s i s of V a r iance for P o s t - T e s t
in P r o b a b i l i t y and S t atis tics
....................347

42.

S ummary of Analysi s of V a r i a n c e for P r e -Test
in M a t h e m a t i c a l System s
............................

348

S ummary of An alysis of Va r iance for Post- T e s t
............................
in M a t h e m a t i c a l Syste ms

348

43.

xii

CHAPTER I
INTROD U C T I O N

Our c i v i l i z a t i o n is fast b e c oming mor e and m o r e
techno log ica l

in nature.

We on ly need look at the kinds

of jobs that e xisted fifty,

twenty,

and even

five yearB

ago to see the extreme ly rapid change tow ard a t e c h n o l o g ­
ically e x t r e m e l y c o m p l e x en v i r o n m e n t w h i c h man is c r e ating
on this planet.
(1)

C o n s e q u e n c e s of such a cha nge are twofold:

less and less "unskilled" work w i l l be re quired in the

future as such jobs wi ll be do ne i n c reas ingly by machi nes ,
and

(2) a h i g h l y - t r a i n e d m a n p o w e r is nee d e d to h a n d l e the

new si tu ati on

(95).

Since chang es are b e c o m i n g so rapid,
not p racti cal to train p e r s o n n e l
tions and sp ecific problems,

it is c l e a r l y

to hand l e c l e a r - c u t s i t u a ­

b eca use the chanc es

are that by

the time such pe op le have c o m p l e t e d their education,
t raini ng w i l l be out of date.
trained

their

Wha t the m a n p o w e r nee ds

to be

for is how to learn to adapt to new situations.

This means a flexibl e a p p r o a c h
w hich is rela t i v e l y new,
"education"

in the training,

an a p p r o a c h

sinc e mos t of w h a t goes by the w o r d

up until quite re cently has m e a n t f a c t - l e a r n i n g

and not learn ing to think.

1

2

Th is
training.

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

L e a r n i n g h o w to t hin k mathem a t i c a l l y ,

reason a b o u t
other w i l l

f lexibility espe c i a l l y a pplies

how to

abstract str u c t u r e s built into and arou nd each

soon be

an i m p erative r e q u i rement of ev ery

citizen.
Backgr oun d

In the last fifteen years, m a t h e m a t i c s edu cat ors
from all o v e r the world have b e e n increa s i n g l y c oncerned
with the n e c e s s i t y of impr ovi ng m a t h e m a t i c s e d u c a t i o n
line w i t h

t e c h n o l o g i c a l and s o c i o l o g i c a l chan ges and in

accord w i t h via b l e res ear ch
sciences.

in

findings

in the b e h a v i o r a l

This c o n c e r n has b e e n e vident e s p e c i a l l y at

the e l e m e n t a r y level,

since lea rni ng and m a n i p u l a t i o n of

abstra ct m a t h e m a t i c a l

str u c t u r e s c lea rly b e g i n at the

el eme ntary

level.

In the U n i t e d States,

no less than 32 m a t h e m a t i c s

c u r r i c u l u m p r o j e c t s have been d e v e l o p e d by m a t h e m a t i c s
educators

and n a t i o n a l l y - r e c o g n i z e d a d v i s o r y grou ps or

organizations

(20).

M o s t of these p r o jects

st r essed the

need to c h a n g e n o t onl y the c ontent an d a p p r o a c h to "modern
m ath em a t i c s "

but also meth ods of te aching and teach er

p r e p a r a t i o n in ma the mat ics.
A s a chi ld' s m a t h e m a t i c s
elementary

school,

e d u c a t i o n begi ns at the

it follows that the teacher,

in daily

3

contact w i t h children,

forms an e xtremely imp ort an t link

in the chain of p r o blems p r e s e n t e d by the new n e cessity to
spread m a t h e m a t i c a l e d u c a t i o n
case until now.

However,

far w i d e r than has been the

avail a b l e data sugges t that the

greate st b o t t l e n e c k in o b t a i n i n g sound m a t h e m a t i c a l e d u c a ­
tion for c h i ldren

(and c o n s e q u e n t l y for the future citizen)

is the p r o b l e m of e d u c a t i n g the teach ers w h o are to impart
the rev is ed m a t h e m a t i c a l c u r r i c u l u m to these children.
reason for the d i f f i c u l t y is the almo st

One

"total ignorance"

on the part of the v a s t m a j o r i t y of ele m e n t a r y school t e a c h ­
ers of w h a t m a t h e m a t i c s rea lly

is;

ano ther is that it seems

almost i m p o s s i b l e to introduce a new m a t h e m a t i c a l c u r r i c u l u m
w i t h o u t c o n s i d e r a b l y c h a n g i n g the con d i t i o n s under w hich
child ren

learn.

Hav i n g

l earned their m a t h e m a t i c s in a

mech a n i c a l w a y and ofte n as a skill subject,
e lem e n t a r y

today's

school teac her s may have serious d i f ficu lties

teaching m a t h e m a t i c s m e a n i n g f u l l y to children.
mea n i n g f u l
associated.

tea chi ng and m e a n i n g f u l
T oday's e l e m e n t a r y

Thus,

learning are closely

school teachers mus t have

a clear u n d e r s t a n d i n g of ea ch new m a t h e m a t i c a l concept
presented to c h i l d r e n if they are to succe ed in their
teachi ng

(46).
C o n s c i o u s of these problems,

U n d e r g r a d u a t e P r o g r a m in M a t h e m a t i c s
in 1960,

a list of r e c o m m e n d a t i o n s

the C o m m i t t e e on the
(CUPM)

published,

for the m a t h e m a t i c a l

4

preparation of e l e m e n t a r y school te achers

(8) .

nationwide c o n f e r e n c e s for mat h e m a t i c s e d u c a t o r s
various e d u c a tional ins titu tio ns

Since then*
from

and b a c k g r o u n d were

co nducted to d i s c u s s the p r o b l e m s

inherent to teacher

educ ation and to exp lor e p o s s i b l e avenues to improved
mathemat ica l p r e p a r a t o r y programs.
At M i c h i g a n State Uni versity,

an exper i m e n t a l

progr am for the p r e p a r a t i o n of e l e m e n t a r y school
was rec e ntly designed.

teachers

T h e pr o g r a m is funded and staffed

by the Mi chigan Stat e T r a iners of T eacher T r a ine rs
Project,

and is b a s e d on several

Science T eacher E d u c a t i o n P r o g r a m

(TTT)

aspect s of the Behavioral
(B STEP) mode l developed

at M i c h i g a n State U n i v e r s i t y in 1968.

The TTT P r o g r a m at M i c h i g a n
State Uni ver si ty

In the w o r d s of the

speci ali sts wh o c o n c e i v e d and

de veloped the program:
The basic p u r p o s e of th e M i c higan State
Univer­
sity TTT (Trainers of T e a c h e r Trainers)
proje ct
is to bring about that type of i n s t i t u t i o n a l
change at the U n i v e r s i t y that has the g r e a t e s t
p romise of r e - d e s i g n e d tea cher e d u c a t i o n p r o ­
grams that are far m o r e relevant to the real
w o r l d of local school and c o m m u n i t y than now
is the case. . . (92:1).
The specific need to w h i c h the TT T p roject is
ad dressed is the p r o d u c t i o n of t e a c h e r s w h o
and

are competent

w h o s e d i s c i p l i n e knowl e d g e a nd teaching b e h a v i o r is m o r e

5

relevant to the real w orld of the school,
populate
it

it,

the stu den ts who

and the co mm u n i t y w h i c h surr o u n d s and suppo rts

(92).
A m o n g the m a j o r o b j e c t i v e s of the TT T P roject at

Michi gan State U n i v e r s i t y are the
1.

Invol ve the community,
un iv e r s i t y

following:

the school,

and the

in the t r a i n i n g of teacher s and

teach er trainers.
2.

Include the best e d u c a t i o n a l d e v e l o p m e n t s
on the m o s t critic al e d u c a t i o n a l

3.

Instit u t i o n a l i z e the i m p r o v e m e n t s

and focus

issueB.
from TT T into the

u n i v e r s i t y and into the school.
4.

Develop a competency-based teacher education program
that i n c orporates s e v e r a l aspect s of the M o del
Program B S T E P .

The B e h av ioral S ci ence Tea c h e r
E d u c a t i o n P r o g r a m (BSTEP)

At Michigan

State U n i v e rsity,

a comprehensive pr o ­

gram for 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 was

designed in 1968.

the p r o f e s s i o n a l

The p r o g r a m ce n t e r s

foundations of the teacher e d u c a t i o n p r o g r a m model upon the
beh avio ral sciences,
methods and their

that is "tho se inqu i r i e s

findings

. . . thei r

. . . w h i c h c o n s t i t u t e reliable

and valid sources of e n l i g h t e n m e n t a b o u t the human,
nature a n d his con ditions"

(22:A-3).

his

6

The p r o g r a m m o d e l emp h a s i z e s d e v e l o p m e n t a l clin ica l
experiences w h i c h b e gin

in a p r o s p e c t i v e te acher's

freshman

year and contin ue th roughout his pr eserv ice education.
The p r o g r a m is o r g a n i z e d into five m a j o r c u r ricular
areas and each area is d i v i d e d into various

components.

Mathe matics is one component of the five c u r r i c u l a r areas.
The five cur ricular areas as they relate to the m a t h e m a t i c s
component are:
1.

General L iberal E d u c a t i o n , w h o s e o b j e c t i v e

is to

relate to the gener al e d u c a t i o n ac quired by the
t e a c h e r -trainees a k n o w l e d g e of the h i s torical
d e v e l opment of m a t h e m a t i c s and of its place in our
techno log ical
2.

culture.

Sch ola rly Modes of K n o w l e d g e , w h ose p urpo se

is to

provide the t e a c h e r - t r a i n e e s with the n e c e s s a r y
bac k g r o u n d for teachin g elementary

school

mathematics.
3.

Profess iona l Us e of K n o w l e d g e , w h i c h provides
t e a c h e r -trainees w i t h an o p p o r t u n i t y
the m a t h e m a t i c s

learned

to t ranslate

in Sch olarly Mo des of

Knowledge into i n s t r uctional st rategi es
children.

the

for

This area p r o vid es the t e a c h e r - t r a i n e e

with an a w a r e n e s s of the i n s t r u c t i o n a l d i m e n s i o n s
to be c o n s i d e r e d in planning
activities.

for rela ted clini cal

7

4.

Human L e a r n i n g , whose basic purpose
the t e a c h e r - t r a i n e e s

is to i n t r o d u c e

to the t hree b a s i c b e h a v i o r a l

areas b r o u g h t into i n t e r a c t i o n in an y p l a n n e d e d u ­
c a t i o n a l ex perience;

tha t

is,

exploring human

capacity

for learning,

understanding environmental

systems,

and enquiring

into the c o g n i t i v e d e v e l o p ­

ment .
5.

Clinical E x p e rience.

The o b j e c t i v e

in this ar ea is

to d e v e l o p an d e x p a n d a p r o s p e c t i v e t e a c h e r ' s
facil i t y in e m p l o y i n g the c l i n i c a l b e h a v i o r s t y l e
into t e a c h i n g m a t h e m a t i c s .

Pre-professional clin­

ical p r o c e d u r e s ar e a n a l y z e d an d p r a c t i c e d t h r o u g h
s i m u l a t e d a n d actual

situ ati ons .

The e x p e r i m e n t a l t e a c h e r e d u c a t i o n p r o g r a m as part
of the T T T P r o j e c t
In 1971-1972,
directed
school

used m a n y

a s p e c t s of the B S T E P mode l.

the m a j o r t h r u s t of the T T T P r o j e c t wa s

toward the d e v e l o p m e n t

teacher who

teach ing

of a ne w kind of e l e m e n t a r y

is b a s i c a l l y w e l l - e d u c a t e d ,

as c l i n i c a l pract ice ,

is an e f f e c t i v e

engages

in

s t u d e n t of

the c a p a c i t i e s and e n v i r o n m e n t a l c h a r a c t e r i s t i c s of h u m a n
learning,
ch ange

and f u n c t i o n s as a r e a s o n a b l e a g e n t of s o c i a l

(22) .

Development
p r o g r a m b e g a n Fall
among th e

of the f i r s t p h a s e of the e x p e r i m e n t a l
term 1971.

Forty

fifty-two who manifested

students chosen

from

a d e s i r e to p a r t i c i p a t e

8

in the program,

c o n s t i t u t e d the first g r o u p of freshmen

involved in the new e l e m e n t a r y

t eache r e ducat ion

curriculum.
Need

Via b l e

teacher

challenge of the
(1) ob jectives

for the Study

e d u c a t i o n p r o grams

future must includ e

to meet the

three aspects:

stated in pe r f o r m a n c e terminology,

t e a c h i n g - l e a r n i n g unit

to imp lement

(3) eva l u a t i v e in s t r u m e n t s

to assess

(2)

these objectives,

the extent to w h i c h

the p r o s p e c t i v e e l e m e n t a r y school t e a chers accomp lish
program objectives.

The

careful d e l i n e a t i o n of

latter

the

two are c o n t i n g e n t upon

the first

C o n s c i o u s of t h e s e

and

(22).

facts,

the teams of sch olars

who w o r k e d c l osely t o g e t h e r to integrat e the p r o g r a m h a v e
rep et itively stres sed

the need

for c o n s t a n t e v a luation

and feedback i n t o the program:

"a c a r e f u l l y designed,

extensive,

and w o r k a b l e e v a l u a t i o n s y s t e m w hich in turn

supports p r o g r a m d e v e l o p m e n t "
The need

(22:11-37).

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

stage of the d e v e l o p m e n t of the e x p e r i m e n t a l pr o g r a m w a s
the basis

for the pr e s e n t study.

9

Purpose of the Study

The gen eral p urpose of this i n v e s t i g a t i o n was the
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 omponent of the
experim ent al e l e m e n t a r y

teache r e d u c a t i o n p r o g r a m at

Michi gan State Unive rsit y.
More sp eci fically,
1.
c omp ete nc ies

this i n v e s t i g a t i o n sought:

To analy ze and e v a l u a t e th ose m a t h e m a t i c a l
sp eci fied by the p rogram to as s ess w h e t h e r they

do in fact m e e t the b a s i c m a t h e m a t i c a l

ne ed of the e l e m e n t a r y

teacher.
2.

To e v a l u a t e

the effe ct of the i n s t r uction aB

prescr ibe d by the m a t h e m a t i c s

comp one nt of the experi m e n t a l

program on the students w h o p a r t i c i p a t e d
to the spe cified c o m p etenci es:

in it,

and to asse s s

in re lat ion

if the students

have a c h i e v e d a degr e e of m a s t e r y over these com petencies.
3.

To e v a luate the bas ic m a t h e m a t i c a l

understanding

of the students w h o p a r t i c i p a t e d in the p r o g r a m p rior to and
after the c o m p l e t i o n of

instruction,

in o rder to assess

ef fectiveness of the p r e s c r i b e d m a t h e m a t i c s
their g enera l m a t h e m a t i c a l
4.

To assess

on the a ttitudes

the

treatment on

knowledge.

the e f f e c t of the e x p e r i m e n t a l p r o g r a m

to war d m a t h e m a t i c s of

the studen ts who

p a r t i c i p a t e d in the program.
5.

To d e t e r m i n e the r e l a t i o n s h i p b e t w e e n s e l ected

variable s and a c h i e v e m e n t

in mat hema tic s.

10

6.

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

attitudes toward m a t h e m a t i c s of p r o s p e c t i v e e l e m e n t a r y
school teachers p a r t i c i p a t i n g in the e x p e r i m e n t a l pro g r a m
with the m a t h e m a t i c a l u n d e r standing and at tit udes of stu ­
dents enro lle d in the regular
7.

teac h e r - e d u c a t i o n program.

To use the results of the i n v e stigation to make

specific recomme n d a t i o n s that may h e l p recti fy any w e a k ­
nesses in the program.

Forma tiv e E v a l u a t i o n and
C u r r i c u l u m Improvement

Forma t ive e va l u a t i o n

involves the c o l l e c t i o n of

appropriate e v i dence during the c o n s t r u c t i o n and try ing ou t
of a new c u r r i c u l u m in such a way that the re vision of the
c u r riculum can be based on this evidence.
tional eva l u a t i o n as in legal

trials,

In all e d u c a ­

the merits of the cas e

rest prima r i l y on the kind and quality of the e v i d e n c e p r e ­
sented.

B ecause for mative e v a l u a t i o n of any new c u r r i c u l u m

focuses on the sta tem ent s of obj e c t i v e s of the program,

the

evidence gathe r e d on these obj e c t i v e s needs to be v a l i d .
Unfortunately,

most p u b l i s h e d or s t a n d a r d i z e d tests do not

meet this criterion,

since the y are d e s i g n e d to faci lit ate

comparison among individuals,

rather th an assessing their

at tainment of s p e c i f i e d c u r r i c u l u m o b j e c t i v e s

(4).

Thus,

de velopment of valid tests that mea s u r e the sp ecific
objectives of the c u r r i c u l u m being e v a l u a t e d

is a m ajor

11

part of any formativ e evaluation.

In this study,

a set

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 are d e v e l o p e d to evaluate
the students'

ac h i e v e m e n t on the m a t h e m a t i c a l com pete ncies

prescribed in the program.

The se m e a s u r e s are based on the

math ema tic al ob jectives spec ifi ed

in the topics

included 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 x p e r i m e n t a l program.
H y p otheses

The

follow ing two hyp o t h e s e s will be tested to

assess the effe ct of the e x p e r i m e n t a l p r o g r a m on the
achievement of the e x p e r i m e n t a l

gro up on the p r e s c r i b e d

mathema tic al competencies:
Al.

There

will

poet-tcat

be

a oignificant

meana

experimental

and

group

the
on

difference

pre-tcot

means

between
of

the

the

the

criterion-referenced

The u n i v a r i a t e hypo t h e s e s

a sso c i a t e d w i t h this

meaaurce.

multivar iat e hypothesis are:
The mean p o s t - t e s t sc ore of the e x p e r i m e n t a l
will be s i g n i f i c a n t l y higher than the mean p r e - t e s t
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 in:
a.
b.
c.
d.
e.
f.
g.
h.
i.

Measurement
Numeration
Sets and Se t Re lations
W h o l e Nu mbers
Fractio ns
De cimals
Rel ati ons an d F unctio ns
Pr o b a b i l i t y and St atistics
M a t h e m a t i c a l Systems.

group
score on

12

A2.

There

will

post-test
on

the

be

no

means

significant
and

the

difference

maetery

criterion-referenced

The uni v a r i a t e hypo t h e s e s

level

between

the

(80 p e r c e n t )

measures.

asso c i a t e d w i t h this

multivar iat e hypo t h e s i s are:
The mean p o s t - t e s t
will be at least equal

score of the e x p e r i m e n t a l group

to the m a s t e r y

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
a.
b.
c.
d.
e.
f.
g.
h.
i.

level

(80 percent)

on

in:

Measurement
Numeration
Sets and Set Re lat ion s
W h o l e N umbers
Frac tio ns
Dec imals
Relatio ns and Fun ctions
P r o b a b i l i t y and Stat is ti cs
M a t h e m a t i c a l Systems.
The

followi ng

assess cha nge s

two hypo t h e s e s w i l l be tes ted to

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

and

attitudes to wa rd a r i t h m e t i c in the e x p e r i m e n t a l group:
Bl.

There

will

basic

mathematical

test

scores

pre-test
B2.

be

There

arithmetic

scores.

of

of

sign ificant

the

difference

understanding
experimental

on

between
group

a

test

the

and

of

post-

their

scores.

will

scores

a

be

a

significant

attitude
the

difference

inventory

experimental

between

group

and

on
the

their

an
post-teat
pre-test

13

The

foll o w i n g

two h y p o t h e s e s w i l l be

compa re the e x p e r i m e n t a l g r o u p and s t u d e n t s

tested to
enrolled

in

the reg ula r t e a c h e r e d u c a t i o n p r o g r a m on b a s i c m a t h e m a t i c a l
understanding
Cl.

The

and a t t i t u d e s

adjusted

mean

experimental
adjusted

proepeative
regular

C2.

poet-teet

group

mean

elementary

mathematical

There

will

adjusted

a

scores
at

of

the

least

equal

to

of a

group

of

oaoree
teaahera

enrolled

program

on

a

in

the

the

teat

of

underotanding.

significant

attitude
scores

difference

‘>ioe-.. o r y

of

post-test

elementary
education

be

be

education

baeic

post-test

will

pogt-teet

teacher

arithmetic

toward arithmetic.

the

experimental

scores

teachers

between

of

enrolled

a
in

group
the

in
the

an
adjusted

group

and

th e

of p r o s p e c t i v e
regular

teacher

program.

Assumptions
The mathematics

c o m p o n e n t of the e x p e r i m e n t a l

pro g r a m is b a s e d on t he se
1.

assumptions:

T h a t the n e e d s of p r e s e r v i c e t e a c h e r
education are better

s e r v e d by p r o f e s s i o n a l i z e d

subject-matter courses
method.

in e l e m e n t a r y

that b l e n d b o t h c o n t e n t and

14

2.

T h a t the c o n t e n t - m e t h o d c ourses sho u l d be taught
in a m a t h e m a t i c a l

l a b o ratory set tin g w h e r e w ell-

p lan ned a c t i v i t i e s u t i l i z i n g m a n i p u l a t i v e m a t e r i a l s
w i l l bet ter

fa cilitate the learning of m a t h e m a t i c s

and the l e a rning of how to teach m a t hemati cs.
3.

Th at the p r e s e r v i c e

teacher should study the

theories of t e a c h i n g and learn ing c o n c u r r e n t l y
with

labo rat ory and clinic al exp e r i e n c e and thus

relate th eor y to practice.
4.

That

this c o m bi ned study a nd exp e r i e n c e

as early as the student's

should begin

freshman year and c o n t i n u e

thr ou ghout his education.
5.

That the study and exp e r i e n c e should
the p r o s p ective teacher has

in te grat e wha t

learned about m a t h e m a t i c s

w i t h what he has learned a b o u t h u m a n i s t i c and b e h a v ­
ioral sciences.
6.

That good m a t h e m a t i c s
tive teachers
instruction,

teach ing b e h a v i o r s by p r o s p e c ­

is fostered by good m a t h e m a t i c s
and that teachers tend to teach as

they are taught.

Li m i t a t i o n s of the Study

The ma jor l i m i t ations of this study w e r e the
following:
1.

W h i l e there are several g oa ls that pe r t a i n to the
for mative e v a l u a t i o n of an e d u c a t i o n a l program,

this

15

study eva luated only the m a t h e m a t i c s

c o m p o n e n t of

the e x periment al t eache r e d u c a t i o n program.
2.

E v a l u a t i o n of the p r o g r a m was confi n e d solely

to

those p r o s p ective e l e m e n t a r y tea chers wh o v o l u n ­
te ered and were selected to p a r t i c i p a t e
first year trial
3.

in the

i m p l e m e n t a t i o n of the program.

The study did not atte mpt to e v a l u a t e

the effect

of the integr ated c o n t e n t - m e t h o d co urse on the t e a c h ­
ing b e h a v i o r of its recip ien ts

in elem e n t a r y school

setting.
4.

The study did not a ttempt to e v a luate the e f f e c t of
the e x perimental p r o g r a m on the m a t h e m a t i c a l c o m p e ­
tency of the school chi ldren w ho w e r e taught by the
e x p e r i m e n t a l p r o g r a m parti cip ant s.

5.

The exte nt to w h i c h

the e v a l u a t i v e ins truments

adequately measured

the e f f e c t s of the int egrated

c o n t e n t - m e t h o d cou r s e and the c l i nica l e x p e r i e n c e
w a s also a limitation.

The i n s t r uments used in this

study ha d the inh erent li m i t a t i o n s of p a per-andp enc i l

tests.1

1As p ointed out by G l e n n o n (53: 395)# far s u p e r i o r to
the p a p e r - a n d - p e n c i l test w o u l d be:
"the study of the b e h a v ­
iors of ea ch per s o n indiv i d u a l l y throu gh con v e r s i n g w i t h h i m
and k eeping ane cdotal r ecords of his p e r f o r m a n c e on the test
i t e m s ."

16

De f i n i t i o n of Terms

This section p r o v i d e s a de fini tion of the major
terms used in this study.
Attitude:

"A learned p r e d i s p o s i t i o n or tendenc y on the part

of an i n d ividual to r esp ond p o s i t i v e l y or neg a t i v e l y
to some object,

situation,

or a nother person"

(36:551) .
C o m p e t ency -Based Tea c h e r E d u c a t i o n P r o g r a m :
p r o g r a m that r e q u i r e s its trainees

"Any training

to de m o n s t r a t e at

a speci f i e d level of com p e t e n c e b eh aviors

that have

been e x p l i c i t l y d e s c r i b e d and prescribed as d e s i r ­
able and e f f e c t i v e p r o f e s s i o n a l be haviors"
Criterion-Referenced M e a s u r e :

"One that is d e l i b e r a t e l y

co n s t r u c t e d to y iel d m e a surement s
interpretabl e
sta ndards"

(97:4).

that are dir ectly

in terms of sp eci f ied p erformance

(13:43).

Curr i c u l u m E v a l u a t i o n :

"Collection,

p r o cessing an d

inter ­

p r e t a t i o n of d a t a pert a i n i n g to an educat ional
progra m"

(34:1) .

Experimental G r o u p

(The) :

A g r o u p of t h i r t y - e i g h t freshmen

el e m e n t a r y e d u c a t i o n majo r s wh o v o l u n t e e r e d

for the

e x p e r i m e n t a l p r o g r a m and p a r t i c i p a t e d in the

first

year trial i m p l e m e n t a t i o n of the m a t h e m a t i c s

c ompo­

nent of the program.

17

Formativ e E v a l u a t i o n :

The use of syst e m a t i c e v a l uation in

the p r o c e s s of c u r r i c u l u m const ruc tio n,

teaching,

and l e a r n i n g for the p u r p o s e of impr oving any of
these p r o c e s s e s
Learni ng U n i t :

(4:117).

The m a t h e m a t i c a l c a p a b i l i t i e s

to be ac quir ed

under a single set of l e a rning conditions.
Mathematics A c h i e v e m e n t :
student

The level of c o m p e t e n c y of the

in reg a r d to the spec i f i e d inst ructional

object ive s of the m a t h e m a t i c s curriculum.
Mathematics C o n t e n t :
tencies of the

D e s c r i p t i o n of the e x p ect ed c o m p e ­
student

Mathematics C u r r i c u l u m :
competen cies an d
achieve

A set of p r e s c r i b e d m a t h e m a t i c a l
the i n s t r u c t i o n a l des ign s

to

these co mpe tencies .

Mathematical U n d e r s t a n d i n g :
student

in m a t h e m a t i c s activities.

Th e level of c o m p e t e n c y of the

in rega r d to the gener al m a t h e m a t i c a l k n o w l ­

edge needed for e l e m e n t a r y school

teaching.

CHAPTER II
REVIEW OF LITERATURE

The re view of l i t e r a t u r e p ertinent to this study
has been o r g a n i z e d u n d e r six categ ori es:
evaluation,

(2) ra ti onal e

education programs,
education,

of m a t h e matics cu rriculum,

curriculum

for the i m p r o vem ent of teach er

(3) d esigns

(4) app r o a c h e s

(1)

for pr ograms of teacher

to t e a c h e r training,
and

(5) conte nt

(6) re searc h on at ti tude s

toward mathema tics .

Curriculum Evaluation

Baker

(41:339)

p ointed out that,

alt hough it is

possible to d i s t i n g u i s h betwe en c u r r i c u l u m e v a l u a t i o n and
cur riculum research,

"it is not very useful

to base the

dis tinction on a rev i e w of lit e r a t u r e deali ng wi th c u r r i c ­
ulum res earch and c u r r i c u l u m e v a l u a t i o n
by authors

. . . the terms used

in their ti tl es and the o p e r a t i o n a l de f i n i t i o n s

assigned to the terms in the ar ticles
W a l b e s s e r and C a r t e r

(71)

is near l y im possible."

r e c o g n i z e d the i m portance

of def ining c u r r i c u l u m by d e v e l o p i n g a s equenced set of
instructional objectives.

Th ey ex p e r i e n c e d p r o blems w i t h

dis crepancy b e t w e e n the s t a t e m e n t of the obj e c t i v e s or the

18

19

propo sed a s s e s s m e n t and th e p e r c e i v e d m e a n i n g
in tention of

the c u r r i c u l u m d e v e l o p e r s .

and

They recommended

pre-evaluation strategies that might help increase
effic ien cy an d c o n t r i b u t i o n of b e h a v i o r a l

the

o b j e c t i v e s to

c u r r i c u l u m d e v e l o p m e n t an d e v a l u a t i o n .
Atkin

(40) , on the o t h e r hand, w a r n e d

tha t

the

rigid a d h e r e n c e to s p e c i f y i n g the b e h a v i o r a l o u t c o m e s of
all i n s t r u c t i o n a l

activities

tend s to d e c r e a s e t h eir

educational

r e l e v a n c e a nd e l i m i n a t e s m a n y o t h e r w o r t h w h i l e

experiences

fr om the c u r r i culum.

Defining Curriculum Evaluation
E v a l u a t i o n is the g a t h e r i n g of i n f o r m a t i o n
purpose of m a k i n g d e c i s i o n s .

Evaluation differs

for the

from b a B i c

resear ch in its o r i e n t a t i o n to a s p e c i f i c p r o g r a m r a t h e r
than to v a r i a b l e s

c o m m o n to m a n y p rograms.

conc ern ed w i t h q u e s t i o n s of u t i l i t i e s

Evaluation

is

th at i n v o l v e v a l u e

and judgment.
Curriculum evaluation requires
process ing and

i n t e r p r e t a t i o n of d a t a p e r t a i n i n g

e d u c a tiona l pr ogram.
(1)

it p r o v i d e s

It s e r v e s two

The

first

important

to an

functions:

a m e a n s of o b t a i n i n g i n f o r m a t i o n that can

be used to i m p r o v e a cours e,
for d e c i s i o n s

the c o l l e c t i o n ,

and

(2)

it p r o v i d e s a b aBis

a b o u t c u r r i c u l u m a d o p t i o n and e f f e c t i v e use.

f u n c t i o n is g e n e r a l l y c a l l e d

formative evaluation

w hile the s e c o n d is r e f e r r e d to as s u m m a t i v e

evaluation

(34).

20

As d e f i n e d by S c r i v e n

(31),

formative evaluation

involves the c o l l e c t i o n of a p p r o p r i a t e e v i d e n c e d u r i n g th e
c o n s t r u c t i o n and t r y i n g ou t of a n e w c u r r i c u l u m in such a
way that r e v i s i o n s of the c u r r i c u l u m can be b a s e d on this
evidence.

B l o o m et a l . (4:117)

as useful not only

regard formative evaluation

for c u r r i c u l u m c o n s t r u c t i o n b u t al so for

instruction a n d st u d e n t learning.
e v a l u a t i o n as:

T h e y d e f i n e this type of

"The us e of s y s t e m a t i c e v a l u a t i o n

in the

process of c u r r i c u l u m c o n s t r u c t i o n ,

teaching and learning

for the p u r p o s e of i m p r o v i n g any o f

the three p r o c e s s e s . "

Dyer

(50) m a i n t a i n s

that f o r m a t i v e e v a l u a t i o n is

very i m p o r t a n t in the d e v e l o p m e n t o f a t e a c h e r e d u c a t i o n
program.

It is not e x p e r i m e n t a l

in an y formal

sense;

cannot tell yo u m u c h a b o u t the u l t i m a t e pay -off;
in fact,

purely descriptive.

a m e a n s of f inding out

in deta i l h o w the ne w m a t e r i a l is

w h a t k i n d of m a t e r i a l

student,

and w h a t c h a n g e s are n e e d e d

a new course,

tries

and w h a t

is w o r k i n g

to fas hion the

he n e e d s to know,

dents are r e a c t i n g
is not"

an d it is,

But it is a b s o l u t e l y v ital as

working,

the e d u c a t o r

it

for w h a t kind of

to m a k e

it better.

i n d i v i d u a l c o m p o n e n t s of

as he goes along,

to the ne w m a t e r i a l s ,

how s t u ­

"w hat is c o n n e c t i n g

(50).

In f o r m a t i v e ev a l u a t i o n ,

As

D y e r continu es:

you do no t w o r r y a b o u t e x p e r i m e n t a l desig n s ,
c ontrol groups, and te st of s t a t i s t i c a l s i g ­
ni fic ance; yo u do w o r r y a bou t th e a d e q u a c y of

21

the w e e k - t o - w e e k and m o n t h - t o - m o n t h feedback,
so that w h e n you bone the c o u r s e down into its
final shape, you w i l l have some assu r a n c e that
it will do the job you inte nd it to do (50:24).
F orm a t i v e e v a l u a t i o n t e c h n i q u e s are e m p l o y e d w h e n
one is interes ted in r e v i e w i n g the c u r r i c u l u m w h i l e it is
still in its d e v e l o p m e n t a l stages.
uation activ iti es m u s t take pla ce

This imp lie s that e v a l ­
in p r e d e t e r m i n e d

stages

in the d e v e l o p m e n t of the c u r r i c u l u m and that strat egi es
must be inclu ded in the a c t i v i t i e s to permit chang es to be
made on the b a s i s of a r e l i a b l e and val id c r i t e r i o n r e f e r ­
ence evidence.

It is also s u g g e s t e d

that a t r i a l - r e v i s i o n

cycle based on p r e a s s i g n e d s tandards be ut ilized
A c c o r d i n g to Bloo m et al.

(4)

(50).

formative eva l u a t i o n

requires a m o r e m i c r o s c o p i c and d i a g n o s t i c a n a l y s i s of the
content.

The i n f o r m a t i o n o b t a i n e d from this

ation is used as a feedb ack

into the system in or de r to

det ermine s u b s e q u e n t a c t i v i t i e s

for the learner.

C o m m e n t i n g on format ive eva l u a t i o n
of E d u c a t ional Obj e c t i v e s

(3),

the taxonomy does not attem pt
methods used by the teacher,
relate themse lve s

B l o o m et al.

in their T a x onomy
indic a t e d that

to cl assify the instructional
or the ways in w h i c h teach ers

to students,

instructional m a t e r i a l s

type of e v a l u ­

or the d i f f e r e n t kinds of

they use;

in fact,

the taxonomy

attempts to c l a s s i f y the i n t e nded b e h avior of the learner,
the w a y in w h i c h the i n d i v iduals m e n t a l l y act or think as
a result of p a r t i c i p a t i n g in some un it of instruction.

It

22

is possi ble that the ac tua l b e h a v i o r of the learn ers after
they have c o m p l e t e d the unit of i n s t r uction ma y dif f e r in
degree as well as in kind

from the inten ded b e h a v i o r

specified by the objecti ves.
In recent years,

many p u b l i c l y

funded m a t h e m a t i c s

c urricu lum groups or pro jects c o n c e n t r a t e d on d e v e l o p i n g
curric ulu m m a t e r i a l s
preparation.

for the e l e m e n t a r y schools and teach er

However,

of these projects
de velopment

there wa s

for syst e m a t i c eva l u a t i o n eit h e r dur i n g

(formative stage)

(summ ati ve) .

little provi s i o n in m o s t

or of the

final p roduct

L o c k a r d (20) surveye d s i x t y -eight

Science and

Mathe mat ics proje c t s and found on ly n i n eteen

(28 percent)

which p ossessed r e s e a r c h e v i d e n c e of success

in a chieving

stated objectives.
C u r r i c u l u m eva l u a t i o n

is d i f f i c u l t

because both the uni q u e and com m o n

to a c c omplish

features of various

curricula m u s t be e v a l u a t e d both o b j e c t i v e l y
outcomes of
judgment)

instruction)

and s u b j e c t i v e l y

(by m e a s u r i n g

(by pers ona l

as to the qua l i t y and a p p r o p r i a t e n e s s of goals.

Cronbach
ization of both
evaluation.

(45)

is more

in clined to favor the u t i l ­

forma t i v e and sum mative,

non-comparative

He feels that in an exp e r i m e n t w h e r e treatments

differ in m a n y respects,

no vali d con c l u s i o n ca n be d r awn

from the fact that the e x p e r i m e n t shows a n umerical
advantage in favor of a new method.

Guba

(96)

pr ese nted

23

an o v e r v i e w of the c u r r i c u l u m e v a l u a t i o n problem s and
p ointed o u t many of the d e ficiencies
in the field of evaluation.
of eva l u a t i o n designs.
National

that c u r r e n t l y e x i s t

He s u g g e s t e d some taxon omy

The sixty- e i g h t h

Yearbook of the

Soci e ty for the Study of E d u c a t i o n

(23) provides

an e x c e l l e n t d i s c u s s i o n of the cha nge s that have b e e n ta king
place an d the m a n y ne ede d c hanges

in c u r r i c u l u m ev aluation.

M e a s u r e m e n t of A c h i e v e m e n t
C o n c e r n e d by the im por ta nce of impr o v i n g the
m e a s u r e m e n t of achievement,

a score of r e s e a r c h e r s have

w r i t t e n e x t e n s i v e l y on the d i s t i n c t i o n b e t w e e n two kinds of
me asures

that are us ed to as se ss subjec t mat t e r proficiency;

that is,

to dete rmi ne the c h a r a c t e r i s t i c s of s tudent p e r ­

formance with respe ct to sp eci f ied standards.

On e is the

relative orderi ng of in div iduals w i t h respe ct to their test
performance.
measure.
"those

Gla s e r

(13) c a l l s it the n o r m - r e f e r e n c e d

It is d efined by P o p h a m and H u s e k

(measures]

ual's p e r f o r m a n c e

which

(28:20)

as:

are u s e d to a s c e r t a i n an i n d i v i d ­

in r e l a t i o n s h i p to the p e r f o rman ce of

other indiv idu als on the same m e a s u r i n g device."
to that d efi ni tion , m o s t

According

s t andardized te sts of a c h i e v e m e n t

or intellec tua l a bility can be clas s i f i e d as n o r m - r e f e r e n c e d
measures.
Th e other type of m e asur ement,
ref e r e n c e d measure,

call ed the criteri on-

d e p e n d s upon an a b s o l u t e standard of

24

qu a l i t y

and it is used

"to a s c e r t a i n an i n d i v i d u a l ’s status

w i t h resp ect to some cr iterion,
(28:20).

i.e.,

p e r f o r m a n c e standard"

W i t h this type of measu r e m e n t ,

not c o m p a r e d w i t h o t h er indiv idu als

the indi v i d u a l is

and his score is not

d e p e n d e n t on those of other studen ts

in the class.

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 m e n t "pr ovides i n f o r m a t i o n as
to the degr e e of comp e t e n c e a t t a i n e d by a p a r t i c u l a r
dent"

(28:8)

stu­

al ong a " c o n t i n u u m of k n o w l e d g e a c q u i s i t i o n

r ang ing fro m no p r o f i c i e n c y at all t o per fec t per form anc e"
(28:7).
It is not always easy to m a k e the d i s t i n c t i o n
b etw een

the two types of measur e m e n t .

that "the d i s t i n c t i o n
for w h i c h

is found by e x a m i n i n g

the test was constructed,

it was con str ucte d,

Gla s e r

(b)

(13:43)

(a)

wr it es

the purpos e

the m a n n e r in which

(c) the s p e c i f i c i t y of the information

yi e l d e d abou t the domain of

instructionally relevant

tasks,

(d) the g e n e r a l i z a b i l i t y of test p e r f o r m a n c e i n f o r m a t i o n to
the domain,

and

(e) the use to be m a d e of the o b t a i n e d test

i n f o r m a t i o n ."
Criterion-referenced measurement
in e d u c a t i o n and has
performance.

is r e l a t i v e l y new

se ld om be en u s e d to ass e s s s t u d e n t ’s

But d e v e l o p m e n t of i n s t r u c t i o n a l techn olog y

and the recent e m p h a s i s on c u r r i c u l u m r e s e a r c h and c u r r i c ­
ulum e v a l u a t i o n have stressed the n e e d

for the kind of

in f o r mation m a d e avai l a b l e by the use of c r iterionref e r e n c e d me asu res.

25

With criterion-referenced
make
uals,

two ty pe s of dec isions:
and

tests,

it is p o s sibl e to

(1) d e c i s i o n s about i n d i v i d ­

(2) d ecisions ab ou t trea tmen ts,

tional programs.

As

to d e c i s i o n s

e.g. , i n s t r u c ­

rega r d i n g individuals,

such tests help determi ne w h e t h e r a studen t had m a s t e r e d
a c r i t e r i o n skill n e c e s s a r y

to c o n t i n u e his program.

This

m e anB that pe r f o r m a n c e s ta ndards mus t be e s t a b l i s h e d prio r
to test c o n s t r u c t i o n and that the pu r p o s e of test ing is to
assess an
ards.

individual's status w i t h re s p e c t to these s t a n d ­

In the case of d e c i s i o n s about treatments,

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 d e s i g n e d to reflect a set of
i n s t r u c t i o n a l o b j e c t i v e s and a d m i n i s t e r e d to the students
af ter they have c o m p l e t e d a sp eci fied i n s t r u c t i o n a l
will p r o v i d e the i n f o r m a t i o n nec es s a r y
r egarding

sequen ce

to reach a d e c i s i o n

the e f f i c a c y of the treatment.

M o s t e d u c a t o r s h a v e used n o r m - r e f e r e n c e d m e a sures
to mak e d e c i s i o n s
systems.

But,

about

i n d i v iduals and

inst ruct ion al

as m e n t i o n e d by the prop o n e n t s of the

c r i t e r i o n - r e f e r e n c e d m e asurem ent,

" n o r m - r e f e r e n c e d mea s ures

we re really d e s i g n e d to spread pe opl e o u t and
suited to that purpose"

(28:22).

c r i t e r i o n - r e f e r e n c e d tests are

. . . are best

On th e other hand,

" s p e c ific ally c o n s t r u c t e d

to suppor t g e n e r a l i z a t i o n s abou t an i n d i v i d u a l ' s p e r f o r m a n c e
r e l a t i v e to a specified d o m a i n of tasks"
sh ould be used,

stresse s Gavi n

(11:62)

(13:42).

Th ey

"to c o n t r o l en try

26

to su ccessive units

in any instruction al

sequence w h e r e

the content is inhere ntl y c u m u l a t i v e and the rigor
progressively greater.”
The use 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 m e n t has
many impl ica tion s
Pop h a m and Husek
1.

for test c o n s t r u c t i o n and evaluation.
(28:17)

h ave m e n t i o n e d the following:

V a r i a b i l i t y is not a n ecessary c o n d i t i o n

a good c r i t e r i o n - r e f e r e n c e d

for

test.

V a r i a b i l i t y is i r r e l ­

evant as the mea n i n g of the score

is only related to the

con ne ction betw een

the items and the c r i t e r i o n and is not

dependent on com p a r i s o n w i t h o t h e r scores.
2.

R e l i a bility is d e s i r a b l e but the aut hors point

out that it is not obvi ous how to assess
ency since the cla ssi cal
variability.
possib le

p r o c e d u r e s are d e p e n d e n t on score

A c c o r d i n g to the authors,

for a c r i t e r i o n - r e f e r e n c e d test

internal c o n s i s t e n c y

internal c o n s i s t ­

index and

it is so me time s
to have a negative

still be a good test.

is needed are new indices and es tim ates a p p r o p r i a t e

What
to

measu re internal c o n s i s t e n c y of c r i t e r i o n - r e f e r e n c e d tests.
3.

Validity

is also a s s e s s e d by p r o c e d u r e s based

on c o rrelations and thus on variability.
of the pr ocedures are useful
not be c o n s i d e r e d d e v a s t a t i n g
general,

Hence,

the results

if they are p o s itive bu t should
if they are negative.

In

validi ty m u s t depe n d u p o n the c o r r e s p o n d e n c e of

the test items w i t h the o b j e c t i v e s to w h i c h the test is

27

referenced.

Thus,

the t e s t items m u s t be c o n s t r u c t e d

for,

or m a t c h e d to, g o als of instruc tion .
4.

Item c o n s t r u c t i o n for 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 m e n t r eq ui res that the items incl ude d

in the t e s t

are an accurate r e f l e c t i o n of the c r i t e r i o n behavior.
use d to mak e dec isions a b o u t individuals,

Wh en

cri t e r i o n -

r e f e r e n c e d tests must be the same or an e q u i v a l e n t fo rm
for all

students.

On th e other hand,

if cri t e r i o n -

r e f e r e n c e d tests are used to e v a l u a t e p r o g r a m s
ma ny tests,
items,

(treatments),

each c o n t a i n i n g d i f f e r e n t c r i t e r i o n - r e f e r e n c e d

c ou ld be constru cte d.
5.

Item analy sis

me nt is d i f f e r e n t
measure men t.

for 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 ­

than i t e m a n a lys is

for n o r m - r e f e r e n c e d

In the c a s e of c r i t e r i o n - r e f e r e n c e d tests,

more c o n c e r n is given to id e n t i f y i n g n e g a t i v e d i s c r i m i ­
nators than n o n - discrimin ators.
dures
(7:71)

for item analy sis

Consequently,

should be

found.

te ste d two m e t h o d s of a n a l y s i s

new p r o c e ­

C o x and V a r g a s

for the e v a l u a t i o n

of items on tests adm ini s t e r e d b o t h as pre- an d post-t est s.
One index was comput ed u s i n g the c o m m o n u p p e r minus
group s tech n i q u e whil e the

second

lower

index w a s c o m p u t e d by

su b t r a c t i n g the p e r c e n t a g e of p u p i l s wh o p a s s e d the items
on the p r e -test

from the p e r c e n t a g e w h o p a s s e d the i t e m on

the post-test.

In the

first case,

the i n dex p r o v i d e d

i n f o r matio n on h ow we ll each i t e m d i s c r i m i n a t e d b e t w e e n

28

the groups, w h i l e

in the se cond case the index p r o v i d e d

d i s c r i m i n a t i o n i n f o r m a t i o n b e t w e e n pre- and p o s t - t e s t
groups,

in dicating items use f u l

The autho rs of the study

for p r e - t e s t di agnosis.

found that the pre- and p o s t - t e s t

m e t h o d of item a n a lysis

should be c ons i d e r e d

r e f e r e n c e d m e asurement,

where

in c r i terion-

score v a r i a b i l i t y

is not the

concern.
6.

R e p o r t i n g and i n t e r p r e t a t i o n of an indivi d u a l ' s

pe r f o r m a n c e on a c r i t e r i o n - r e f e r e n c e d test does not requir e
the same g r o u p - r e l a t i v e d e s c r i p t o r s

as when p e r c e n t i l e

ran king or s t a n d a r d scor es are used to i nterpret re sults
in n o r m - r e f e r e n c e d tests.
"on-off"

Sc or es o b t a i n e d are e s s e n t i a l l y

in n a t u r e and we are i n t e r e s t e d in k n o w i n g

indivi dua l has m a s t e r e d the c r i t e r i o n or not.

if the

It is also

p o s s i b l e to repo r t the d e g r e e of the stude n t ' s p e r f o r m a n c e
and

to dete r m i n e how far away he is from the crit er ion.

Such reports on the d e g r e e of

"less-than-criterion"

per­

forma nce e x c l u s i v e l y d e p e n d on the use m a d e of the data.
If 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 are use d for the e v a l u a ­
tion of treatments,

different procedures

are possi ble :

(1) re po rt the n u m b e r of i n d i v iduals w h o a c h i e v e d the pree s t a b l i s h e d cr iterion,
sta ti stics such as
deviations,

(3)

(2) us e t raditional d e s c r i p t i v e

" p e r c entage corr ect ," m e a n s and stand a r d

if a c r i t e r i o n

level has bee n set,

report

the prop o r t i o n of the g roup w h i c h r e a c h e d that level and

29

repor t the p r o p o r t i o n and de gre e of the "be tte r-thancriterion'* perform anc es.

Popham an d Husek

(28) bel i e v e

that the best cour se of ac tio n w o u l d be to us e as m a n y
r e p o r t i n g p r o c e d u r e s as pos s ible in order to pe rmi t a
be tter

i n t e r p r e t a t i o n of the test results.
R a t i o n a l e for the Impr ove ment of T eacher
£<3ucaiion P r o grams at the
Ele m e n t a r y Le vel

For some years,

c o ntinuing m o v e m e n t toward r e f o r m of

m a t h e m a t i c s c u r r i c u l a at the secon d a r y level has been e v i ­
dent in the U n i t e d States and in m a n y c o u n t r i e s of the world.
Such c hanges have s t r e s s e d the i m p ortance of the p r i m a r y
stage in a child 's

learning.

R e f o r m is no w d i r e c t e d

m o d i f i c a t i o n of basic teachin g in the p r i m a r y
o r d e r to prepar e
of

to the

school in

the child a d e q u a t e l y for the new p a t t e r n s

further ed ucation.

T h e s e c hanges

in the c ontent of c u r ­

ricula have b e e n a c c o m p a n i e d by e x p e r i m e n t s

in the d e v e l o p ­

men t of new teach i n g m e t h o d s and in a new c o n c e p t i o n of
p r e s e r v i c e and i n - service e d u c a t i o n
e l e m e n t a r y school

for p r o s p e c t i v e

teachers.

As the d e m a n d s of an incre as in gly c o m p l e x s ocie ty
for q uali ty e d u c a t i o n c o n t i n u e to mount,
t eac her a c c e l e r a t e w i l d l y and th ere

d e m a n d s u p o n the

is g e n e r a l agree m e n t

from all segmen ts of so ciety that teachers and their e d u ­
ca tio n are the prin c i p a l object b e h i n d any ef fort m a d e a n y ­
w here

for the u l t imate i m p r o v e m e n t of e d u c a t i o n a l systems.

30

A c c o r d i n g to B r o u s s e a u
be b e t t e r p r e p a r e d

(42:265):

for two reasons:

"tea chers must

(1) they ar e

w i t h an e n l i g h t e n e d g e n e r a t i o n of ch ildren who,
the

impact of te levision,

faced

du e to

have a c c u m u l a t e d m u c h know l e d g e

upon ent ra n c e in to t o d a y ’s e l e m e n t a r y schools,

an d

(2)

they

are not a d e q u a t e l y p r e p a r e d to e x p l o i t this ac c u m u l a t e d
r e s e r v o i r of k n o w l ed ge."
The r e q u i r e m e n t

to r etrain ele m e n t a r y school

t e a c h e r s in m a t h e m a t i c s prese n t s m a n y problems.
p r o p o r t i o n of e l e m e n t a r y

school t e a c h e r s have ha d

or no m a t h e m a t i c a l train i n g

(21:108)

A large
little

and they have not

s t u d i e d m a t h e m a t i c s b e y o n d the ag e of 14.

Some have had

s e c o n d a r y t r a i n i n g thro u gh r o t e - l e a r n i n g of sol utions to
c e r t a i n kinds of pr oblems and m a t h e m a t i c i a n s a n d / o r m a t h e ­
m a t i c s e d u c a t o r s agree that such train ing does not result
in a true k n o w l e d g e of m a t h e m a t i c s .
e x p o s e d to m a t h e m a t i c s
manne r.

These

Others h a v e been

ca r r i e d ou t in a hi gh ly logical

teach ers ma y u n d e r s t a n d the struct ura l

p r o p e r t i e s of m a t h e m a t i c s but they will have to learn ho w
to u s e the i n d u c t i v e m e t h o d in the te achin g of m a t h e m a t i c s
to y o u n g children.
C o n s i d e r i n g such pro blems,
for su ccessful

t eacher e d u c a t i o n ?

w h a t are the c r i t e r i a
How muc h m a t h e m a t i c s

s h o u l d an e l e m e n t a r y sc hool t e a c h e r know?

How m u c h p s y ­

c h o l o g i c a l k n o w l e d g e of the d e v e l o p m e n t of chi ldren,

an d

of the m ethods of th inki ng sho u l d he be f a m iliar with?

31

S p e c i a l i s t s in the e d u c a t i o n of m a t h e m a t i c s
for the e l e m e n t a r y sch ool

(6,

16,

the follow ing c h a r a c t e r i s t i c s

21,

46,

are b asic

59,

64)

teachers

agree that

to a suc c e s s f u l

t eacher e d u c a t i o n program:
1.

T e a c h e r s sh ould have a c learer p ictur e of the
objectives

2.

T h e y m u s t have a high deg r e e of m a s t e r y of the
mathematics

3.

for a c o n t e m p o r a r y m a t h e m a t i c s program.

they need to teach.

They m u s t a l s o have a thorough m a s t e r y of the
m a t h e m a t i c s t h e y need to give p e r s p e c t i v e

to their

teaching.
4.

They m u s t be k n o w l e d g e a b l e a b o u t t e a c h i n g steps and
instructional materials

5.

They m u s t be c o m p e t e n t

involved.
in d e v e l o p i n g in the e l e m e n ­

tary school c l a s s r o o m m a t h e m a t i c a l c o n cepts
on a s o p h i s t i c a t e d
6.

level.

They must k n o w eno ugh about h o w c h i l d r e n
mathematics
acti v i t i e s

lea rned

learn

to select and d e s i g n a p p r o p r i a t e
for children.

A d d r e s s i n g the N a t i o n a l C o u n c i l of T e a c h e r s of
Mathematics
(16:8)

at a f o r u m on T e a c h e r Educa tio n,

spoke of the c h a l l e n g e

faced by

J a m e B Gray

teac her e d u c a t o r s

to "provi de th e ne w te a c h e r w i t h as m u c h of a s cience of
methodology

as it can.

They

(teacher ed ucators)

lenged to at t e m p t to get the future m a t h e m a t i c s

are c h a l ­
teach er aB

32

an artist,

as one w h o has an i n d e p e n d e n t c o n f i d e n c e in his

u n d e r s t a n d i n g of m a t h e m a t i c s and of his s t u d e n t s so that he
c a n best a rrange w h a t is done
a c h i e v e m e n t of o b j e c t i v e s . "

in the c l a s s r o o m to the
He added:

It is onl y in the s e c u r i t y of high c o m p e t e n c e
that a teache r feels suffic i e n t l y free to h i m ­
self p r o p o r t i o n this m a t h e m a t i c s to the need of
his studen ts w i t h o u t feel ing b o und by the p a r ­
ticul ar p a c k a g i n g of the textbook, his teachers'
note or other p r e pared m a t e r i a l s w h i c h do not
e xactly fit the p a r t i c u l a r s ituation of his
s t u d e n t s . In the security of truly u n d e r s t a n d ­
ing w h a t he is about, the teac he r can c o n s i d e r
his situation, his students* need an d his m a t h e ­
ma t i c a l obj e c t i v e s an d go abou t his own str ategies
for the a c c o m p l i s h m e n t s of these o b j e c t i v e s (16:4).
For Lebl anc

(59:606),

the ba sic c h a r a c t e r i s t i c of

the n e w l y - t r a i n e d e l e m e n t a r y school t eacher will be "his
abi l i t y an d c o m p e t e n c e in m a k i n g w i s e d e c i s i o n s
progr a m s an d o r g a n i z a t i o n s

con c e r n i n g

in te aching m a t h e m a t i c s . "

This

implies no t only the d i f f e r e n t c o m p e t e n c i e s menti o n e d
earlier,

bu t also k n o w l e d g e in b e h a v i o r a l

te rms of the

goals of a c o n t e m p o r a r y m a t h e m a t i c s p r o g r a m and the seque nce
in whic h they s e e m e a s i e s t and most a p p r o p r i a t e to be
attained.
ment

In addition,

k n o w l e d g e about

tests and m e a s u r e ­

in m a t h e m a t i c s w i l l h e l p the t eacher d e s i g n e v a l u ations

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

learning,

an im por tant

c o m p o n e n t of e d u c a ­

tional programs.
Th e

International

Study G r o u p

Le a r n i n g has for m u l a t e d some q u e s t i o n s

for M a t h e m a t i c s
that wi ll help

33

d e t e r m i n e some cr i teria

for the t r a i n i n g of p r o s p e c t i v e

teache rs of m a t h e m a t i c s at the e l e m e n t a r y level
1.

If the c onten t to be include d

{21:114-115).

in e l e m e n t a r y school

m a t h e m a t i c s is the one s u g g e s t e d by the Camb r i d g e
C o n f e r e n c e of 1963,
asked,

then the que stions m u s t be

How well the teacher is c o nversan t with

these topics? and,
home w i t h

Can he foll o w and be q u i t e at

such c ourses as the UICSM C o u r s e

I or

S u p p e s 1 Logic for E l e m e n t a r y Schools?
2.

A n o t h e r que stion w o u l d be:

How well is the teach er

p r e p a r e d to help d evelop c r e a t i v e m a t h e m a t i c a l
enquiry

in his students?

tea ch ers
avenues

A c c o r d i n g to the authors,

"should ha ve at t h e i r finge rti ps many
through the e n c o u r a g e m e n t of w h i c h c h i ldren

can be shown at q u ite an e a r l y stage th at mat h e m a t i c s
is an induct ive and creat ive enterpr ise "
3.

It is al so r e l evant
p r e p a r e d to o b t a i n

to ask:
insight

How well

(21:115).

is the teacher

into how c h i l d r e n

learn?

A go od know ledge of the liter atur e of educators,
child p s y c h o l o g i s t s and r e s e a r c h m a t h e m a t i c i a n s
a must

4.

is

to a c quire c o m p r e h e n s i o n on i nsight into ho w

ch i l d r e n

learn.

Finally,

the q u e s t i o n sho u l d be asked:

ext e n t are t e a chers able to u n d erstan d

To what
the pu rposes

and m ethods of m a t h e m a t i c a l educ ation and to w h a t

34

e x t e n t can they explain t h e s e purpo s e s and m e t h o d s
to o t h e r people:
parents ,

their o w n pupils,

and a d m i n i s t r a t o r s ?

othe r teachers,

This suggests an

a w a r e n e s s of the kinds of a p p l i c a t i o n s that are
b e i n g ma de of m a t h e m a t i c s

in the p r e s e n t s o c i e t y

today.
T e a c h e r e d u c a t o r s have a l s o been c o n c e r n e d w i t h the
n e g a t i v e a t t i t u d e s t o w a r d m a t h e m a t i c s of m a n y pr 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 a n d they have

r e p e a t e d l y stres sed the

importance of d e v e l o p i n g a m a t h e m a t i c s c u r r i c u l u m for t e a c h ­
er e d u c a t i o n

that w o u l d get peop l e

to r espond to it,

to a t t e n d to m a thematics,

to v a l u e it, to e njoy and p a r t i c i p a t e in

their m a t h e m a t i c a l e d u c a t i o n e x p e r i e n c e s in s t e a d of m e r e l y
s u b mitting

to or f i g h t i n g them.

b e l i e v e tha t

interest

d o m a i n mu s t p a r allel

M o s t t e a c h e r ed uc ator s

in the o b j e c t i v e s of

the affec t i v e

intere st in t he o b j e c t i v e s of the

c o g n i t i v e domain.

D e s i g n s for P r o g r a m s of
Te ache r E d u c a t i o n

M a n y m o d e l s a n d programs b a s e d on some or all the
c r i t e r i a m e n t i o n e d in the p r e v i o u s

section hav e be en p r o ­

pose d for the e d u c a t i o n of e l e m e n t a r y s c h o o l teach ers

in

genera l and

the e d u c a t i o n of e l e m e n t a r y s c h o o l teac her s

mathematics

in part icu lar .

in

Some o f the l a t t e r are familiar

35

and h a v e b e e n
the UICSM,

fu n c t i o n i n g for a few years now:

etc.

Late in the 196D's,

Offi c e of E d u c a t i o n p r o p o s e d
educa tio n"

that w o u l d

on the subject.

to fund

the SMSG,

the U n i t e d States
"models

for teache r

i n c o r p o r a t e the m o s t recent

findings

Ten mod e l s w e r e c h o s e n to be funded,

among

them the M i c h i g a n Sta te Mode l p r o p o s e d by M i c h i g a n State
University

(the B S T E P ) .

S.

C. T. C l a r k e

(6) has li ste d a num b e r of factors

to be c o n s i d e r e d in the p r e p a r a t i o n of a t eacher educa t i o n
p r o g r a m an d he has e v a l u a t e d
States O f f i c e of E d u c a t i o n
w i t h these

factors.

the m o d e l s

funded by the Uni t e d

in terms of the w a y they d e alt

Cla r k e states that d e s i g n s

for teacher

e d u c a t i o n m u s t deal w i t h t e a ching in terms of presage,
process,

and product

factors.

Presage Factors
Presage

factor s r epresent d e c i s i o n s m a d e prior to

the d e v e l o p m e n t of a p r o g r a m and w h i c h shape the d i r e c t i o n
of the program.
of lead

They

in terms of

include:

time,

context,

and b o u n d a r i e s

g r a t i n g g e n e r a l educati on,

cybe rna ti on,

ex tent

in te rms of i n t e ­

s ub ject matter ,

and related

disciplines.
D e c i s i o n s a b out the c o n t e x t
being p r e p a r e d must be made
of t e a c h e r education.
p r o grams d e v e l o p e d

for w h i c h t eac he rs are

in a d v a n c e of p l a n n i n g a p r o g r a m

M o s t of the m o d e l

teac her e d u c a t i o n

in the last half of the 1960's are b a sed

36

up on p r e d i c t i o n s of w h a t society and educ a t i o n wi ll be like
in 1980.

The exte n t of

lead varies,

however.

Mo st of the p r o g r a m s also c ontai n c y b e r n e t i c selfc o r r e c t i n g de v i c e s for per i o d i c a l e x a m i n a t i o n and u p d a t i n g
of the p r o g r a m as well as the p r e p a r a t i o n of s e l f - r e n e w i n g
teachers cap a b l e of sh a p i n g the c h a n g e s that "s eem c ertain
in the future w o r l d of educa tio n."
In the mo dels e v a l u a t e d by Clarke,

there seems to

be a general te ndency for the r e p r e s e n t a t i v e s of the i n s t i ­
tuti ons to dete r m i n e context,

cybernation,

at the same time pro f e s s i n g the n e e d
m a n y orga nizations,
There

institutions,

lead, w h i l e

for the i n v o l v e m e n t of

an d agencies.

is a g reement that g e n e r a l edu cati on,

m a t t e r to be tau g h t to pupils,
discipli nes ,

and

su b j e c t

and comman d of related

are wi th in the b o u n d a r i e s of t eacher education.

A T e a c h e r E d u c a t i o n P r o g r a m can be vi ewed as

the 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, w h i l e the

subject m a t t e r and gen e r a l

e d u c a t i o n p o r tions are regar ded as

"givens."

the Standards E v a l u a t i v e

According

C r i t e r i a of Te a c h e r E d u c a t i o n

(6:124):
[General education! should in c l u d e the studies
m o s t w i d e l y g e n e r a l i z a b l e to life and f urther
learning. . . . T h e gener al s t udies c o m p o n e n t
for p r o s p e c t i v e t eac he rs r e q u i r e s that from
o n e - t h i r d to o n e -half time b e d e v o t e d to
studies in the s ymbolic s of in for mati on,
b a s i c p h y s i c a l and b e h a v i o r a l sciences, and
humanities.

to

37

Aa to the p r o f e s s i o n a l component,

it cov ers all r e q uirements

that are justi f i e d b y the w o r k of the s p e c i f i c v o c a t i o n of
teaching.

B. O.

Smith

says

(6:124):

the subject m a t t e r p r e p a r a t i o n of the teach er
sh oul d consis t of two i n t e r r e l a t e d parts:
first, c ommand of the c o n t e n t of the d i s c i p l i n e s
c o n s t i t u t i n g his t e a c h i n g field and of the s u b ­
ject mat t e r to be taught; and second, co m m a n d
of k n o w l e d g e about knowledge.
Th e M i c h i g a n State M o d e l

is an o u t s t a n d i n g e x a m p l e

of h o w to deal w i t h the impo r t a n t m a t t e r of bound ar ies.
five m a j o r ar eas of this p r o g r a m are:
cation,

s c holarly m o d e s of kn owledge,

knowledge,

human

learning,

general

The

liberal e d u ­

p r o f e s s i o n a l use of

and c l i nical studies.

The p r o ­

g r a m wa s d e v e l o p e d by an i n t e r d i s c i p l i n a r y t e a m and its
c ont inued d i r e c t i o n w a s to be r e p r e s e n t a t i v e of the v a r i o u s
interests.
complete

A c c o r d i n g to Smith,

no o ther m o d e l has su c h a

t reatm ent of r e l a t i o n s h i p w i t h g e n e r a l e d u c a t i o n

and a c a d e m i c di sc iplines.

P rocess Fac to rs
The p rocess
i ndividualization,
s upp ort systems,

factors

inc lud e d i m e n sions ,

ex ten t of

graduated conceptualization-practice,

and t a s k - c e n t e r e d c urr ic ulum .

In the model p r o g r a m s there is a m o v e m e n t a w a y
the tr a d i t i o n a l d i m e n s i o n s
p e r f o r m a n c e modules,

(time,

that is,

credits,

courses)

from

to war d

t e a c h i n g tasks which c a n be

m a s t e r e d in few or m o r e hour s of i n s t r u c t i o n - p r a c t i c e , and

38

w h o s e end p roduct
Clarke,

is t e a c h i n g behavior.

the M i c h i g a n State M o d e l

A c c o r d i n g to

is the m o s t c o m pletely

d e v e l o p e d e x a m p l e a m o n g ne w m o d e l s of tea ch er ed uca tion
p r o g r a m s and
fo rmat

includes over 2,7 00 modules.

The standard

for th ese m o d u l e s in clu des ob jec tives,

experience,

setting,

materials,

level,

hours,

pr erequisites,
and evaluation.

The m a j o r trend in t e a c h e r e d u c a t i o n as e x e m p l i f i e d
by the m o d e l s
r eco g n i z e d
programs,

the

is i n d i v i d u a l i z a t i o n even if most progra ms
insti t u t i o n a l b a r r i e r s to i n d i v i dualized

such as time r e q uired

cre d i t require men ts.
graduated exercises
as si mulation,
teaching.

for a d e g r e e and course or

The m o d e l s p r o p o s e d ha ve stresse d
leading up to p r a c tice t e a c h i n g such

a n a l y s i s of teaching,

tutoring,

and m i c r o ­

Th e M i c h i g a n State M o d e l pla ces c o nsider able

e m p h a s i s on the d e v e l o p m e n t of c l i nical b e h a v i o r w i t h the
se que nce st arti ng d u r i n g

the first two yea rs w i t h tuto rial

e x p e r i e n c e s w i t h chi ldren,
seminar,

anal y t i c a l

microteaching,

c o n t i n u i n g wi th a care e r d e c i s i o n

study of teaching u sing si mulation and

th r o u g h tea m teaching,

internship,

and

teache r spec ial ization.
The mode l s var y gr e a t l y
ag e m e n t systems.

T h e y all

in their t reatment of m a n ­

face the p r o b l e m s of r e c o r d i n g

and s tudent a c c o u n t i n g cr e a t e d by the p r o c e s s

factors and

by the m u l t i p l e e n t r a n c e s and e x its p r o v i d e d for selection.

39

In the mo del s
of Education,

funded by the Uni t e d States O f f i c e

there is an emph asi s on task analysis,

specification,

required behaviors,

d e v e l o p these behaviors,

tr eatme nts design ed

This pro g r a m of e x p e r iences

for teacher cand i d a t e s stresse s the

p e r f o rmance c r i t e r i a
The M i c h i g a n

style of teachers,
(a) desc ribing,

starts

including:

(b) analyzing;

(a) hypothesi zing ,
(a) treating,

formu lat ion

State M o d e l

to

and a s sess ment of results in terms

of the o r i gi nal task analysis.
d e s igned

tas k

idea that

is the base of most models.
from the cli nical b e h a v i o r

(1)

the ref l e c t i n g phase:

(2)

the propo s i n g phase:

(b) prescribing;

and

(3)

the doing phase:

(b) seeki ng e v i dence on consequences.

Produ ct Factors
The produ ct
rated

factors refer to the features i n c o r p o ­

in a t eacher e d u c a t i o n p r o g r a m to e v a l u a t e the p r o g r a m

and the t eac her b e h a v i o r s produced.

Most of

the mod els

studied have str esse d the need for e v a l u a t i o n but,
to Smith,

accor d i n g

"designs to e v a l u a t e teach er behaviors we re not,

on the whole,

well-developed,

e duc a t i o n p r o g r a m

with the e x c e p t i o n of one

(the M i c h i g a n Stat e Model )"

(6:153).

40

M y t h s About T e a c h e r E f f e c t i v e n e s s

The q u e s t i o n of t e a c h e r effe cti ven ess,
of m e a s u r i n g it,

the p r o b l e m

and the p r o b l e m of p r e d i c t i n g it are

e x t r e m e l y important.

In an y edu c a t i o n a l

sy st e m a vast

number of d e c i s i o n s are m a d e whic h req u i r e some k n o w l e d g e
about tea cher e f f e c t i v e n e s s .
chang es

Among

them,

d e c i s i o n s about

in the c u r r i c u l u m sho u l d be based,

info rmation about

in part,

on

the e f f e c t i v e n e s s of the t e a c h e r s who

wi ll be ca lled on to i m p l e m e n t the chang es.
Becaus e of the i m p o r t a n c e of thi s matt e r
(93), d uri n g the cour se of a rather e x t e n s i v e

the SMSG

five year

longitu din al s t u d y of m a t h e m a t i c s a c h i e v e m e n t w h i c h started
in the fall of 1962,

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

i n f o r mation ab out a large n u m b e r of t e a chers and comp l e t e d
an analys is of s o m e of t h e s e data to find more a b o u t teacher
e f f e c t i v e n e s s b a s e d on st u d e n t achievem ent .
significant,

and in most cases,

R e s u l t s showed

large vari a t i o n s

in teacher

e f f e c tiveness but

the v a r i a t i o n did no t seem to be c o r r e ­

lated w i t h any of

the e x t e n s i v e i n f o r m a t i o n a b o u t

age,

sex,

t e a c h i n g experience,

that m i n i m a l l y required

teachers:

am oun t of tr aining beyond

for the job,

re cen t i n s e r v i c e

training,

a t t i t u d e s toward mathem a t i c s ,

teaching,

a t t i t u d e s toward

students.

a tti tudes

toward

In all cases,

regres­

sion analysi s sho w e d that this am ount of i n f o r m a t i o n about
the teach ers di d not a c c o u n t

for more than a s mall

fraction

41

of the v a r i a n c e in the t e a c h e r e f f e c t i v e n e s s scores,
most cases

less than 10 percent.

The bel ief that m a t h e m a t i c a l ability,
ligence,

in

is not sh ar ed e q u a l l y am ong

like i n t e l ­

i n d i v iduals still

in flu ences most m a t h e m a t i c s c u r r i c u l u m d e s p i t e the fact
that it has b e e n c h a l l e n g e d by m a n y r e s e a r c h m a t h ematicians.
The SMSG c o n d u c t e d an e x p e r i m e n t to test C a r r o l l ' s h y p o t h ­
e s i s 1 that all studen ts c o u l d be b r o u g h t to the same level
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,

am ount of i n s t r u c t i o n n e e d e d

to b r i n g a s tudent to a p a r ­

tic ular level of a c h i e v e m e n t w o u l d var y
student.

b u t the

from s tudent to

R esults of the study p r o v i d e d e v i d e n c e in favor

of this h y p o t h e s i s

(108).

M o r e recently,

some r e s e a rcher s h a v e taken the

p o s i t i o n that it is the teaching,

not the teacher,

is the key to the learni ng of students.

That

that

is, no t what

teachers

like but w h a t they do in i n t e r a c t i n g w i t h

student s

(36,

th eir

38).

Approaches

to T eacher T r a i n i n g

M a n y m e t h o d s m a y be used for the i m p l e m e n t a t i o n
of a suc c e s s f u l t eache r e d u c a t i o n program.
departments

Mathematics

in a num b e r of colleges a n d u n i v e r s i t i e s have

‘John Carroll, A M o d e l for Sc hool L e a r n i n g , Vol.
T e a chers C o l l e g e Record, 1963, pp. 725-733.

64,

42

i n s tituted special cours es
school teachers;

in m a t h e m a t i c s

for elemen tar y

a num ber of c o l l e g e s and univer s i t i e s

offer a subject- m a t t e r m a jor for ele m e n t a r y school teachers
in preparat ion ;

and several

schools have e x p e r i m e n t e d w i t h

laboratory cou rse s in tea chi ng

for un der gra dua tes .

Fr om its b e g i n n i n g to the present,

t eac her ed uca tion

for e l e m e n t a r y school m a t h e m a t i c s has p r o g r e s s e d
ra ther

formal ize d approach to a t t e n t i o n to m o r e p r o fe ssional

p r o blems t hrough demonstr atio ns,
ings,

l a b oratory work,

(62:434),

field work,

and p a r t i c i p a t i o n

school m a t h e m a t i c s classes.

bes t

from a

But,

read­

in e l emen tary

a ccording to Muelle r

"strong i n d ication as to whi ch

is still lacking:

projects,

type of cour se is

sepa rat e m e t h o d s and cont ent courses,

c o m b i n e d c o n t e n t - m e t h o d s course,

CAI course,

remedi al course,

c our s e w i t h or wi t h o u t discu s s i o n s . "
Leblanc

(59) be lieves

that p r o s p e c t i v e m a t h e m a t i c s

teacher s need both a good conten t course and a wellstr u c t u r e d m e t h o d s co ur se
that the con t e n t

in math ema ti cs.

He po ints out

taught should be in closer a l l iance w i t h

the c o n t e n t that teachers will be teaching c h i l d r e n and
that the cour s e needs to be c a r e f u l l y
of p e r f o r m a n c e objectives.

fa shi one d in terms

As to the m e t h o d s course,

sho u l d prepa re the teache r to:
a)
b)

be able to list some s e q u e n c e of lear nin g
e x p r e s s e d in p e r f o r m a n c e objectives;
be abl e to identify some "need -to -kn ow"
conce p t s and skills, or "nee d-t o-kn ow"
obj e c t i v e s as o ppos ed to " n i c e - t o - k n o w " ;

it

43

c)

be able to use m a t h e m a t i c a l lab oratories
or l e a r n i n g - r e s o u r c e centers an d know the
reaso ns for m a t h e m a t i c s resou rce centers
(59:607) .
W h i l e c o n s i d e r i n g se parate c o u r s e s

and one on teaching)

an a p p r o p r i a t e m e a n s

goals set up for t eacher prepara tio n,

(one on c ontent
for a c h i e v i n g

P h i llips

(64)

the

lists

three a d v a n t a g e s to the c o m b i n e d m e t h o d s - c o n t e n t c o u r s e ov er
two se parate courses:

(1)

the p r o s p e c t i v e teac her

m a t h e m a t i c a l c o n c e p t s on the abstract
addition a function al knowledge,
learns the te aching steps,

and

learns

level and o b t a i n s in

(2) the p r o s p e c t i v e

teacher

(3) the cou rse o f f e r s e f f i ­

ciency in learning.
Lit t l e r e s e a r c h is a vailab le as

to the e x t e n t of

the e f f e c t of the c o m b i n e d c o n t e n t - m e t h o d s

co urs e on the

c ogn i t i v e and a f f e c t i v e b e h a v i o r s of the learner.

Ph illips

c ond u c t e d a r e s ea rch study at the U n i v e r s i t y of Illin ois
(Urbana)

in 196 4-1965 with 73 p r o s p e c t i v e e l e m e n t a r y

school

teachers e n r o l l e d in the first re quire d m a t h e m a t i c s course.
Students w e r e pla c e d

in t hree groups:

two grou ps w e r e

taught a m a t h e m a t i c s co n t e n t co urse w h i l e a c o m b i n e d - c o n t e n t
teaching a p p roach w a s used w i t h the t h i r d group.

S t u dents

were t e s t e d and c o m p a r e d on o p e r a t i o n a l skill in a r i t h m e t i c
and algebra,

m e a n i n g and u n d e r s t a n d i n g

v o c a b u l a r y kno wledge.

Results

in arithm eti c,

indicat ed

and

that s t u d e n t s

enrolled in the c o m b i n e d c o n t e n t - t e a c h i n g cou rse h a d hi gher
mean s c o r e on all

three te sts

than s t u d e n t s in the o ther two

44

sections.

Phillips

relat ed his findi ngs to thos e of anoth er

study at the U n i v e r s i t y of Ill inois w h i c h sho w e d that 94
p e r c e n t of

for t y - n i n e e x p e r i e n c e d

combined content-teaching course
by c o r r e s p o n d e n c e

te ac he rs c o m p l e t i n g a
in e l e m e n t a r y m a t h e m a t i c s

favored the c o m b i n e d c o n t e n t - t e a c h i n g ove r

the s e p a r a t e - c o u r s e a p p r o a c h

(64).

A r e s e a r c h study by Ma x Bell and ass o c i a t e s

(103)

on an a c t i v i t 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 s course
for p r e s e r v i c e tea chers w a s c o n d u c t e d dur i n g
Preliminary
1.

r esults

i ndicated

the following:

l e a rning w i t h m a n i p u l a t i v e m a t e r i a l s
d e s i r e to use,
changes

1972.

in creases

incr e a s e s a b i l i t y to use,

and

teacher's b e h a v i o r w i t h re s p e c t to the

use of m a n i p u l a t i v e m a t e r i a l s

for the teac hin g

of m a t h e m a t i c a l concepts;
2.

learning with manipulative materials
desire

i ncrea ses

the

to teach and ori e n t act ual t e a c h i n g b e h a v i o r

in a l e a r n e r - f o c u s e d way;
3.

the a c t i v i t y - o r i e n t e d c o m b i n e d c o u r s e had a
s i g n i f i c a n t p o s i t i v e eff e c t on the a t t it ude of
the p r e s e r v i c e t r a i n e e tow a r d t e a c h i n g mat hema tic s;

4.

the p r e s e r v i c e tr ainees sho w e d su b s t a n t i a l gain s
understanding elementary mathematics.
s h o w e d a sizeable i n c r e a s e
self-image:
could

They also

in their m a t h e m a t i c a l

after the course,

learn mathem a t i c s .

in

they b e l i e v e d

they

45

A n a l y s i s of the i n t e r v i e w data gathe red by Bell et
al.

indi c a t e d that u t i l i z a t i o n of manipu l a t i v e m a t e r i a l s

the c o m b i n e d cour se acts as
mathem a t i c s .

"enablers"

for train ees

in

learning

M a n i p u l a t i v e m a t e r i a l s ena ble the tra in ee to

"play around" w i t h the concepts#
that is entailed.

to actually see the p rocess

T h e y e n a b l e the train ee to d i s c o v e r a

con c e p t for himself.

For some trainees#

the i nsight r e ­

c e i v e d th r o u g h the us e of physi c a l m aterials d i f f e r s in
ki n d

from the one r e c e i v e d

in abstract,

verbal teaching.

Insigh t from phy si cal m a t e r i a l s is a more powerful,

believ­

ab le kind of i nsight for t hese trainees.
P a s t cri t i c i s m s of m e t h o d s cou rs es both by teac her s
and n o n - t e a c h e r s c e n t e r e d a r o u n d two points of view:
they are to o t h e o r etica l and un re alis tic ,
too s u p e r ficial and insulting to o ne's
e v e r vali d these c r i t i c i s m s ma y be,
that the real p r o b l e m lies
o f t e n s t r e s s e d by m e t h o d s

in the

and

(2)

they are

intelligence.

Zahovic

(1)

How­

(72) maint a i n s

fact that "the topics

cour ses are not teachi ng but

rat h e r the plann i n g and p r e p a r a t i o n that take p l a c e prior
to teaching;

in short,

c u r r i c u l u m concerns.

. . . They are

largely v o i d of m a t t e r s d e a l i n g w i t h teacher b e h a v i o r in
the i n t e r a c t i v e c l a s s r o o m s i t u a t i o n w h e r e teachers co nfro nt
learne rs
(72:198).
enrolled

in an ef fort to e f f e c t learn er b e h a v i o r a l change"
A c c o r d i n g to this author,
in a m e t h o d s c o u r s e sho uld

p r o s p ec tive teach ers
learn:

46

• what k i n d s of q u e s t i o n s

to ask learners;

• what k i n d s of d i r e c t i o n s to provide;
• what clue s or p r o m p t i n g s to give;
• how m u c h s t r u c tur ing to do for the learner;
• what kind s of feedback to provide;
• how to lif t levels of thought;
• how to e x t e n d and use learners'

ideas;

• what kin ds of p r a i s e to use;
• how to terminat e d i s c u s s i o n of a t o pic and ma ke
tra n s i t i o n to a new one;
• how and w h e n to e m p l o y c o n v e r g e n t memory,
ing,

associate,

classify­

or a f f i r m a t i v e - d e n y i n g questions.

The m e t h o d s course

"should deal not only w i t h w h a t

b e h a v i o r s or act s teachers sh ould emp l o y in the in t e r a c t i v e
s i t u a t i o n but a l s o w i t h the timing of a p a r t i c u l a r act and
the sequence or p attern in w h i c h it sh ould be used in o rder
to provid e m a x i m u m service to learners"

(72:198).

Zahovic

concludes:
The need for the inclusion of these a spects of
teaching in me t h o d s c o u r s e is clear.
Teaching
met hods c o u r s e s must ind e e d b eco m e t e a c h i n g
me thod s c ourses and no t just c u r r i c u l u m
courses (what they are n o w ) .
One wa y of i mproving the m e t h o d s cour s e
by Kali k

is sugg e s t e d

(57) w h o argues t h a t the t r a d i t i o n a l o n e - s e m e s t e r

m e t h o d s co urs e w i t h o u t an a c c o m p a n y i n g c l a s s r o o m e x p e r i e n c e
is d e f i c i e n t in terms of time and r e a l i t y to pre pare

the

47

st udent a d e q u a t e l y for the m y r i a d p r o b l e m s he w i l l
a b e g i n n i n g teacher.

Ideally,

of m e t h o d s and fo un da tion s

face as

in th e auth or' s view,

instructors

"a team

should be a s s i g n e d to

a g r o u p of t w e n t y to tw e n t y - f i v e students,

w i t h w h o m they

sh ould wo rk w i t h i n a n e a r b y pu bli c sch o o l sett ing over a
three-year period"

(57:262).

This n o t i o n of a ful ly
the p r o s p e c t i v e

fie l d - c e n t e r e d p r o g r a m w here

teacher in his t r a i n i n g is pla c e d e arly in

the posit i o n of having to a n s w e r the w h y s of his studen ts
is sh ared by m a n y

teacher educators.

a si m i l a r point of view b y
s tudent t e a ching

saying that

(69)

exp res ses

"an important end of

is to a s s i s t the s t u d e n t te acher to b e c o m e

a st u d e n t of h i s own te ach ing "
sees the need fo r a:

(69:374).

For so doing,

" s t r o n g c l i n i c a l c om ponent

pr o f e s s i o n a l e d u c a t i o n of
of training w h i c h

Traver s

teachers,

he

in the

t h a t is that e lement

iB p r o b l e m - c e n t e r e d and gi ves t r a ining

in finding so lut i o n s w i t h i n the c o n t e x t of actu al

situations”

(69:375).
The p h i l o s o p h y of

Zo lta n D i e n e s on the s ubject has

been very i n f l u ential in s h a p i n g the m o d e r n p r o g r a m s of
mathematics

for p r o s p e c t i v e e l e m e n t a r y school teachers.

He wrote:
A teach er w i l l teach as he was t a u g h t himself.
. . . If he w a s taught in school and even in
teac h e r ' s c ollege t h r o u g h lectures, he w i l l
te nd to le c t u r e to the children.
In o th er
words, he w i l l tend t o e xplain rath e r than set
up s i t u a t i o n s through w h i c h the c h i ldren can
be led to understand.

48

If we w i s h t e a chers to be able to set up
concrete p r o b l e m sit u a t i o n s that the c h i l d r e n
c a n m a n i p ulate, then they m u s t also learn to
set up such c o n c r e t e s i t u a t i o n s for t h emselves
and to m a n i p u l a t e t h e m themselves.
T h e y must
feel in t h e i r own skins what it is like to
start from s cratch and learn something.
Many d e m o n s t r a t i o n c las ses w i t h c h i l d r e n
(should be i n c l u d e d in t eacher e d u c a t i o n p r o ­
gram) and, f ollo w i n g those, m a n y si tuations in
w h i c h the t e a chers the m s e l v e s ca n b e g i n to
han dle gr oup s of children.
Any p r i n c i p l e s ar r i v e d at should be learned
by the t r a inees the m s e l v e s as the res u l t of
their e x p e r i e n c e w i t h m a t e r i a l s and w i t h c h i l ­
dren and a m o n g themselves.
Demonstration
sessions or w o r k s h o p l a b o r a t o r y se ssions should
be fo llowe d by s e m i n a r - t y p e d i s c u s s i o n s b etwe en
the t ea cher e d u c a t o r and the t eacher traine es
(46:268).
This n o t i o n n egates the usua l lecturing,
and testing

found in m a n y

formal u n i v e r s i t y cours es and

r e p l a c e s it w i t h an acti v e
sharing,

ev alu ating ,

reciting,

i n v o l vement

and redoing)

(discussing,

doing,

as a m e t h o d of learning.

A l t h o u g h the t eacher has a so me what d i f f e r e n t role
in a la boratory

setti ng from that in a m o r e

c l a s s r o o m situation,
program.

he is still

the key to a success ful

He m u s t select or d e v i s e w o r t h w h i l e a c tivities

tha t will be a p p r o p r i a t e
period,

traditional

he acts

for his class.

as a g u i d e or cou nselor.

Duri n g a labor ato ry
A f t e r the activity,

he m u s t e v a l u a t e and rec o r d pupil progress.
L a b o r a t o r y ac t i v i t i e s ma y be u s e d in three ways:
se p a r a t e d

from,

i n t e g r a t e d into,

r e g u l a r instr u c t i o n a l program.

an d c o r r e l a t e d w i t h the
Experimental

stud ies of

m a t h e m a t i c s l a b o r a t o r y have be en d o n e by a few invest iga tors .

49

Vance

(87) e v a l u a t e d a m a t h e m a t i c s lab o r a t o r y s e p a r a t e d

from the instruc tio nal p r o g r a m and he
rea ct ion was more

favora ble to the l a b oratory s etting than

to class setting.
mat ics

found that student

Wilkinson

l a b oratory inte g r a t e d

(100)

found that the m a t h e ­

into the regula r p r o g r a m

appear ed 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 and
low in telligence.

Wasylyk

(99)

c o m pared studen ts taught

in the t r a d i tio nal m a n n e r w i t h st udents
integr ate d a p p r o a c h

(mathematics labo r a t o r y i n t egr ated

into the regu lar program)
ment of students

taught by the

and he i n d i c a t e d that the a c h i e v e ­

in the labo r a t o r y g r o u p s wa s s i g n i f i c a n t l y

higher than the a c h i e v e m e n t of the c o n t r o l g r o u p ta ugh t in
a t e a c h e r - d i r e c t e d setting.
found that p e r f o r m a n c e of

On the o t h e r hand,

Johnson

student s tau g h t e x c l u s i v e l y by the

activity a p p roach was i n f e r i o r to that of student s
textbook - b a s e d or a c t i v i t y - e n r i c h e d

in struction.

analys is of r e s e a r c h d o n e on m a t h e m a t i c s
and Ki eren

(70)

ma t h e m a t i c a l

concl u d e d that:

ideas

(1)

(2)

ful i n s t r uction appea rs to w o r k as w e l l

the

(4)

if not better.

argume nts

(58:231):

and

in

other m e a n i n g ­

for use of

m a n i p u l a t i v e a c t i v i t i e s and p l a y - l i k e acti v i t i e s
Ki er en says

Vance

for lo w-a bility students,

m a x i m i z i n g a c h i e v e m e n t on cognitive variables,

learning,

In their

student s can learn

(3) ther e is limi ted e v i d e n c e of a t t i t u d e change,

S u m m a r i z i n g the t h e o r e t i c a l

rece i v i n g

laboratories,

from la boratory settings,

a p p roach is p a r t i c u l a r l y e f f e c t i v e

mat i c s

(80)

in m a t h e ­

50

The y [man ipulative activities] h a v e a
funda men tal p o s i t i o n in the s e q u e n c e of
e x p a n d e d l e a rning act i v i t i e s bo th on a
mac r o - and m i c r o - i n s t r u c t i o n a l basis; they
can p ro vide an i n f o r m a t i o n - s e e k i n g , no n ­
a u t h o r i t a r i a n env ironment; they sh oul d b e s t
inc lude a var i e t y of c o n cr ete r eferents for
a concept; they can c o n t r i b u t e a readiness
fo und ation for la tter ideas.

M a t h e m a t i c a l C o m p e t e n c i e s of E l e men tary
School T e a c h e r s
W h a t are the m a t h e m a t i c a l
e l e m e n t a r y school teac her s?

competencies

In 1961,

U n d e r g r a d u a t e P r o g r a m in M a t h e m a t i c s
recommendations

nee ded by

the C o m m i t t e e on the
(CUPM) m a d e

s p e cific

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

tary scho ol teachers.

In 1966,

the C a m b r i d g e C o n f e r e n c e on

Teache r T r a i n i n g a d v a n c e d b o l d r e c o m m e n d a t i o n s

for the m a t h e ­

ma tics c ontent to be in cluded in e l e m e n t a r y t e a c h e r ed uca tion
programs,

these r e c o m m e n d a t i o n s e x c e e d i n g by

p r e v i o u s l y by the CUPM.

Si nce then,

far those m a d e

other g u i d e l i n e s ha ve

been suggested and r e s e a r c h c o n d u c t e d to e v a l u a t e the c o n ­
tent of m a t h e m a t i c s c u r r i c u l u m for e l e m e n t a r y
ers i n d icate that

school

teach­

in g ener al t e a c h e r p r e p a r a t i o n in the

Unite d St ates did not meet the m i n i m u m r e q u i r e m e n t s of
CUPM

the

(52).
In 1972,

the C o m m i t t e e on G u i d e l i n e s of the C o m m i s ­

sion on P r e - S e r v i c e T e a c h e r E d u c a t i o n of the N C T M a n a l y z e d
g ui d e l i n e s p r e p a r e d by the CUPM,
the A m e r i c a n A s s o c i a t i o n

the C a m b r i d g e Co n f e r e n c e ,

for the A d v a n c e m e n t of S cience

51

(AAAS), the A s s o c i a t e d O r g a n i z a t i o n s of T e a c h e r E d u c a t i o n
( A O T E ) , and o t h e r groups.
analysis,

Based o n the findi ngs of the

the C o m m i t t e e me mbers p r e p a r e d ne w gui d e l i n e s

the p r e p a r a t i o n of e l e m e n t a r y sch o o l teachers
T h e s e g u i d e l i n e s are b a s e d on the p r e mises
tial for teache rs

for

in ma thematics.

that it is e s s e n ­

to k n o w more than they are e x p ecte d to

teach and to be able to learn m o r e than t h e y already know.
Spec ifically,

the C o m m i t t e e re c o m m e n d e d the

m i n i m a l k n o w l e d g e and c o m p e t e n c i e s
school t e a c h e r
1.

fo llo win g as

needed by the e l e m e n t a r y

(94:24).

T e a c h e r s of e a r l y c h i l d h o o d and p r i m a r y gr ade s
(ages 4-8) should:
a.

Be able to us e and e x p l a i n b a s e ten n u m e r a t i o n
system.

b.

Be ab le to d i s t i n g u i s h b etween r a t ional
(meaningful) cou nt ing and rote counting.

c.

Be able to recog n i z e stages in the c o n s e r v a t i o n
of number an d q u a n t i t y in a c t i v i t i e s of children.

d.

Be able to p e r f o r m the four b a s i c o p e r a t i o n s
w i t h who le num ber s an d with p o s i t i v e r ational s
w i t h r e a s o n a b l e sp eed and accuracy.

e.

Be able to explain, at a p p r o p r i a t e levels,
wh y o p e r a t i o n s are p e r f o r m e d as they are,
and n u m e r a l s p r o c e s s e d as they are.

f.

Be able to use equality, gr e a t e r than,
less than relat ions co rrectly.

g.

Be able to relate the nu mbe r line to w h o l e
n u m b e r s an d p o s i t i v e ra tional numbers.

h.

Be able to relate the nu mber line to the c oncept
of m e a s u r e an d d e s c r i b e and i l l u s t r a t e b a sic
c o n c e p t s of m e a s u r i n g such q u a n t i t i e s as weight,
volume, etc.

and

52

2.

i.

Be able to extra ct concepts of two- an d
th r e e - d i m e n s i o n a l geometry fro m the real
w o r l d of the child, and be a ble to d iscuss
the p r o p e r t i e s of simple g e o m e t r i c figures
such as line, line segment, triangle,
qua dri lateral, circle, p e r p e n d i c u l a r and
pa r allel lines, pyramid, cube, sphere, etc.

j.

Be able to use a pro tractor,
str aig ht edge.

k.

Be able to use the met ric s y s t e m of w e i g h t s
and me asures, and be able to e s t imate such
m e a s u r e m e n t s in metr i c u n i t s bef ore a c t ua lly
measuring.

1.

Be able to create and i n t e r p r e t simple bar
and line graphs on two d i m e n s i o n a l c o o rdinate
system s and understand the na ture of scale
changes.

m.

Be able to use a cal c u l a t o r to help so lve
problems.

n.

Be able to use all of the abov e c o m p e t e n c i e s
(a-m) to help create, recognize, and solve
p r o b l e m s w h i c h are real to adults and children.
(To "solve problems" in th is conte xt includes
re c o g n i t i o n of problem s w h i c h have no solution
and a b i l i t y to esti mat e th e ex pected magni t u d e
of the so lution of a problem.)

o.

Be able to d iscuss the history , ph ilosophy,
nature and cu ltural signif i c a n c e of mat hem ati cs,
both g e n e r a l l y and specifically.

compass

and

Te achers of up per elementary a n d mi ddle school
gra d e s (ages 8-12) should:
a.

Have all c o m petencies

li sted above.

b.

Be able to name large and small n u m b e r s and
cre at e their own physi cal e x a m p l e s of a p p r o x ­
imati ons for such numbers (e.g., one m i l l i o n is
a p p r o x i m a t e l y the number of m inutes in two years;
one b i l l i o n is a p p r o x i m a t e l y the n u m b e r of
seconds in 32 years, etc.) and d i s t i n g u i s h
b e t w e e n infinit y and such n umbers as a
googl epl ex.

53

c.

Be able to p roduce reas ona ble ,
and l ogic al arg ume nts (proofs)
m a t h e m a t i c a l facts.

cons ist ent ,
for e l e m e n t a r y

d.

Be able to p e r f o r m the four b a s i c o p e r a t i o n s
w i t h p o s i t i v e and n e g a t i v e r a t i o n a l num bers
using d e c i m a l n o t a t i o n an d f r actional n o t a t i o n
and explain , at a p p r o p r i a t e levels, w h y the
o p e r a t i o n s are p e r f o r m e d as they are.

e.

Be able to de v e l o p ne w alg o r i t h m s for o p e r a t i o n s
and be ab le to test the e f f e c t i v e n e s s and
c o r r e c t n e s s of algorithms.

f.

Be able to solve p r a c t i c a l and th e o r e t i c a l
p r o blems in two and three d i m e n s i o n a l g e o m e t r y
r e l a t i n g to co ngruence, p a r a l l e l and p e r p e n d i c ­
ular lines, similarity, symmetry, incidence,
areas, vo lumes, circles, spheres, po lygons, and
polyhedrons.

g.

Be able to
use the m e t h o d s of p r o b a b i l i t y and
sta t i s t i c s to solve s i m p l e p r o b l e m s p e r t a i n i n g
to m e a s u r e s of central t e n d e n c y an d disp ers ion ,
e x p ectation , pre dic tio n, and r e p o r t i n g of data.

h.

Be able to
grap h fun ctions and r e l a t i o n s r elated
to p o l y n o m i a l s and to m a k e a p p r o p r i a t e s e l e c t i o n
and us e of such relat ions in the s o l ution of
p r a c t i c a l problems.

i.

Be able to w r i t e flow char t s for simple m a t h e ­
m a t i c a l o p e r a t i o n s and o t h e r act ivities.

j.

Be ab le to
recognize,
problems.

use q u a n t i t a t i v e skills to help
create and s o lve a p p r o p r i a t e

54

M a t h e m a t i c a l C o m p e t e n c i e s of E l e m e n t a r y
--------------------------S c h o o l ^ R l i a r e n ------------------------

The m a t h e m a t i c a l p r e p a r a t i o n of the e l e m e n t a r y
school teach ers m u s t be h i g h l y c o r r e l a t e d w i t h that of
e l e m e n t a r y school pu p i l s as the C o m m i t t e e on G u i d e l i n e s
suggested.

The m a t h e m a t i c a l c o n t e n t of the c u r r i c u l u m for

e l e m e n t a r y sc hool studen ts

sho uld be a sub s e t of the c ontent

of the m a t h e m a t i c s c u r r i c u l u m for p r o s p e c t i v e e l e m e n t a r y
school teachers.
What are the m a t h e m a t i c a l c o m p e t e n c i e s that sh ou ld
be a c q u i r e d by the e l e m e n t a r y sc hool ch ild ?

A mong the m o s t

known studi es m a d e in this c o u n t r y in rece n t years c o n c e r n ­
ing w h a t m a t h e m a t i c s s h o u l d be at the e l e m e n t a r y
the S t r a n d ’s R e p o r t p u b l i s h e d in 1968.
statement of goals and o b j e c t i v e s
program,

follo win g

for the m a t h e m a t i c s

K-8 w a s p r o p o s e d by tho se w h o c o n t r i b u t e d to

the Str a nd's R e p o r t
1.

Th e

level is

(105:34).

Nu m b e r s and O p e r a t i o n s
To use e f f e c t i v e l y the f u n d a m e n t a l o p e r a t i o n s
of arith met ic, c o m p u t i n g w i t h f r a c t i o n s and
w i t h dec imals; to u n d e r s t a n d and u t i l i z e the
p r o p e r t i e s of the o p e r a tions, and the p r o p ­
erties of o r d e r and a b s o l u t e value; and to
u n d e r s t a n d the s t r u c t u r e of the s e v e r a l n u m ­
ber s ystems and the s p e c i a l p r o p e r t i e s of
each.
T o read and u n d e r s t a n d m a t h e m a t i c a l
sent e n c e s invol v i n g operations, exp onents,
and letters, an d to f o r m u l a t e an d use Buch
sen ten ces in the a n a l y s i s of m a t h e m a t i c a l
problems.

55

2.

Geo metry
To rec ognize and us e com m o n g e o m e t r i c
concepts and conf igurations; to ut i l i z e
compass and st rai g h t edge for simple
constructions; and to und e r s t a n d and to
construct simple d e d u c t i v e proofs.
To use
the e l e men tary q u a n t i t a t i v e g e o m e t r i c notions,
such as mea s u r e of angle, area and volume; to
util ize the c o n c e p t s of s i m ilarity and con­
gruence in a p p l i c a t i o n s such as p l a n s and maps;
and to u tilize the c o o r d i n a t e plane.

3.

Measurement
To make measureme nts; to u n d e r s t a n d the n o ­
tion of unit of m e a s u r e m e n t , and to use and
interpret various units; to u n d e r s t a n d the
degree of a c c uracy of an a p p r o x i m a t e m e a ­
surement; to e s t i m a t e m e a s u r e m e n t s and the
results of simple c a l c u l a t i o n s involv ing
measur eme nts ; and to c o n c e i v e and use forma
of me a s u r e m e n t as functions.

4.

Applications
To analyz e c o n c r e t e p r o blems by u s i n g an
appro priate m a t h e m a t i c a l model; to employ
graphs, scale drawings, sentences, formulae,
co m p u t a t i o n s and r e a s o n i n g in s t u dying the
m a t h e m a t i c s of such a model; to in ter pre t
m a t h e m a t i c a l c o n s e q u e n c e s in c o n c r e t e terms;
and to e xamine the c o n crete res ult s of such
an analy sis in terms of r e a s o n a b l e n e s s and
accuracy.

5.

Sta tis tic s and P r o b a b i l i t y
To c o n s t r u c t and read o r d i n a r y graphs.
To
coll ect and o r g a n i z e d a t a by m eans of graphs
and tables; to i nterpret data u sin g concepts
d e s c r i b i n g c e n t r a l tendency, such as mean,
m e d i a n and mode; and to u n d e r s t a n d s t a t i s ­
tical v a r i a n c e as a m e a s u r e of c entra l
tendency.
To underst and , at a s i m p l e level,
the idea of sampling, and to int e r p r e t and
predict from d a t a samples.
To und e r s t a n d
rud i m e n t a r y no t i o n s of p r o b a b i l i t y theory
and of ch ance events.

56

6.

Sets
To u n d e r s t a n d and use routi n e l y the b asic
set c o n c e p t s , notations, and op erations.

7.

F u n c t i o n s and Graphs
To us e the coo r d i n a t e p lane to d i s p l a y
r e l a t i o n s and to o r g a n i z e data; to r e c o g ­
nize and u ti lize the c oncept of function,
or fu n c t i o n a l relation; and to use fu nct ion s
and the usual f u n ctiona l n o t a t i o n in analysis
and p r o b l e m solving.

8.

Logical Thinking
To und ers tand, to ap preciate, and to use
p r e c i s e statements; to u n d e r s t a n d and use
c o r r e c t l y the simple logi cal c o n n e c t i v e s
suc h as:
"and," "if-then," etc.; to d i s ­
ti nguish, c o n c e p t u a l l y and in operations,
b e t w e e n the "for some" and "for all"
q u a ntifiers ; and to foll o w an d to c o n s t r u c t
simp l e d e d u c t i v e arguments.

9.

P r o b l e m Solvin g
To d e v i s e and apply s t r a t e g i e s for analysis
and s o l u t i o n of problems, and to use e s t i ­
m a t i o n and a p p r o x i m a t i o n to v e r i f y the
r e a s o n a b l e n e s s of the outcome.

It is o bvious that the re is h i g h a g r e e m e n t b e t w e e n
the c o ntent p r o p o s e d by the C o m m i t t e e on G u i d e l i n e s

for the

p r e p a r a t i o n of ele m e n t a r y school t e a c h e r s and the S t r a n d 's
Report.

However,

so far,

no p r o g r a m has b e e n im p l e m e n t e d

w h i c h i n c o r p o r a t e s the C o m m i t t e e ' s r e c o m m e n d a t i o n s .

57

Re s e a r c h on A t t i t u d e s To w ard
Mathematics

It has be en m a i n t a i n e d r e p e a t e d l y by p r o f e s s i o n a l s
inv ol ved in the d e v e l o p m e n t of m o d e r n m a t h e m a t i c a l p r o g r a m s
that students'

a t t i t u d e s to war d m a t h e m a t i c s w o u l d improve

g r e a t l y w i t h change B in the c u r r i c u l u m and in the m e t h o d s
of t e a c h i n g the " n e w ” mathem a t i c s .
A l t h o u g h many

i n v e s t i g a t i o n s have b e e n c o n d u c t e d in

the last fifteen y ears
have

to d e t e r m i n e w h e t h e r m o d e r n c u r r i c u l a

fostered m o r e p o s i t i v e a t t i t u d e s

toward the subject,

the e v i d e n c e to s upport this c l a i m is still meager.
section

sum marizes the

toward m a thematics:
tudes,

This

f ollowing r e s e a r c h on attit u d e s

(1)

t e chniques use d to m e a s u r e a t t i ­

(2) at titu des and p e r s o n a l i t y c h a r a c t e r i s t i c s ,

teachers'

a ttit udes t o w a r d m a t h e m a t i c s ,

ach iev eme nt,

and

(5) a t t i t u d e s and

(4) attit u d e s

(3)
and

the new m a t h e m a t i c s

curricula.

T e c h n i q u e s Use d to M e a s u r e A t t i t u d e s
A numb er of tec h n i q u e s have b e e n e m p l o y e d to m e a s u r e
a t t i t u d e s 1 toward mathem a t i c s :
interviews,

questionnaires,

b e h a v i o r a l o b s e rvat ions,

rank o r d e r i n g of sc hoo l subjects,

2A t t i t u d e is d e f i n e d by A i k e n (36:551) as a "lear ned
p r e d i s p o s i t i o n or t e n d e n c y on the p a r t of an i n d ivid ual to
r e s p o n d p o s i t i v e l y or n e g a t i v e l y to some object, situation,
concept, or a n o t h e r per son ."

58

a t t i t u d e scales,

sen ten ce com pletions,

p i c t u r e pr eferences,

c ontent anal ysi s of essays,

and e v e n appar a t u s

physiological

Among

sta tes

(36).

indicati ng

these methods,

the most

p o p u l a r have been the a t t i t u d e scales d e v e l o p e d by Thur s t o n e
and Likert and the s e m a n t i c - d i f f e r e n t i a l techniques.
In vi ew of the fact that the types of m e a s u r i n g
i n s t r u m e n t s e m p l o y e d in th e r e s earch should af fec t to some
d e g r e e the i n t e r p r e t a t i o n of r e s u l t s , a few i n v e s tigators
have questioned
ni ques used.

the r e l a t i v e me rits

of the m e a s u r i n g t e c h ­

M o r r i s e t t and V i n s o n h a l e r

(109)

stated a few

y ears ag o that there we re no v a lid m e a s u r e s of atti tudes
towar d m a t h e m a t i c s .

Aiken

(36) c o n c l u d e d from a rev iew of

re s earch that r e l i a b i l i t y an d v a l i d i t y of the attit ude
scales v a r y s o m e w h a t w i t h grade levels,

b e i n g ge nerally m o r e

r e l iable and v a l i d in hi gh school and college.
reasons he o f f e r e d to e x p l a i n this

fact was

One of the

the m a n y p r o b ­

lems of r e a d i b i l i t y and i n t e r p r e t a b i l i t y of self-report
i n v e n tories e n c o u n t e r e d in the lower grades.

Anttonen

(7 3)

m e n t i o n e d the same p r o b l e m in his docto r a l d i s s e r t a t i o n and
he poi n t e d out the need for improving the re a d a b i l i t y of
a t t itude m e a s u r e m e n t s at the e l e m e n t a r y scho o l level.
R omb erg
validity,

(66)

i n d i c a t e d that many p r o b l e m s of

inte rna l c o n s i s t e n c y and score stab i l i t y result

fro m an o p e r a t i o n a l d e f i n i t i o n of a ttit u d e s
paper-and-pencil

tests.

from scores on

He stressed the ne ed for a

59

th eore tical

fo rmu l ation w h i c h w o u l d c o n c e i v e of a t t i t u d e s

as "a set of m o d e r a t o r varia b l e s

that affe c t

the s u b j e c t ' s

respons e to m a t h e m a t i c a l s i t u a t i o n s in obs e r v a b l e and
pre d i c t a b l e ways"
using a single,
matics, w h i c h

(66:481).

R o m b e r g also point ed out

that

gl obal m easure of a t t i t u d e s toward m a t h e ­

is w h a t most i n v e s t i g a t o r s do,

is not r e a l ­

istic "si nce there is probably a set of p r e d i s p o s i t i o n s or
feelings th at vary f r o m c o m p u t a t i o n to p r o b l e m - s o l v i n g ,
etc."

(66:481).

S imilar r e c o m m e n d a t i o n s w e r e also m a d e

by Aiken and Moss and Kagan

(36,

61).

A t t i t u d e s and Personal ity
Characteristics
M a n y studies have been d o n e to inv estigate r e l a t i o n ­
ships b e t w e e n attitudes toward m a t h e m a t i c s and a n u m b e r of
p e r s o n a l i t y and soci al

factors,

toward scho o l work in general,
to learn,
parental

namely anxiety,
achieve men t,

a t t itude of one's peers,
influences,

of interests,

low.

general

abilit y

socio e c o n o m i c status,

sex d i fferences,

masculinity-feminity

etc.

T h e corre l a t i o n s in m o s t
ge nerally

attitude

In m o s t cases,

relatin g p e r s o n a l i t y v a r i a b l e s

studies w e r e found to be

the finding s of st u d i e s
to m a t h e m a t i c s a t t i t u d e s

indicate t h a t ind ivi duals w i t h m o r e p o s i t i v e a t t i t u d e s

tend

to have b e t t e r perso n a l and s o c i a l a d j u s t m e n t

38,

39).

However,

it is n e c e s s a r y

va riables ar e also a f f e c t e d b y

to r e m e m b e r
family,

(36,

37,

that p e r s o n a l i t y

sch o o l and society.

60

P are nts do hav e some effe c t on c h i l d r e n ' s a ttitud es
toward m a t h e m a t i c s

(38,

74,

m igh t be m o r e i nfl ue ntia l
language de ve lopm ent .
to be si gnif icant
(38,

43,

60,

101)

but it a p p e a r s

in m o r e verbal

Socioeconomic

that they

subjects,

such as

status does not see m

in d e v e l o p i n g at ti tudes

to war d m a t h e m a t i c s

81).

According

to A i k e n

(38:231),

a t t i t u d e s toward

m a t h e m a t i c s are " p o s i tively r e lated to both ve rbal and
q u a n t i t a t i v e abi lity and w i t h a m a s c u l i n e - i n t e r e s t pat tern."
It w o u l d
only

seem that these attitudes and a b i l i t i e s are not

learned-response

te ndencies d e t e r m i n e d by social and

cu ltu ral e x p e r i e n c e s but are d e p e n d e n t on a ge n e t i c a l l y
d e t e r m i n e d t e m p e r a m e n t a l and abi li ty base.

Teachers' A t t i t u d e s and
E f f e c t i v e n e s s TowarcT
Mathematics
M o s t e d u c a t o r s view t e a c h e r ’s a t t i t u d e s and e f f e c ­
tivene ss
at tit u d e s

in m a t h e m a t i c s as the prime d e t e r m i n e r s of students*
and p e r f o r m a n c e

out by Banks

in the subject.

As was p o i n t e d

(2:16-17):

The teac her w h o feels insecure, w h o d r e a d s and
d i s l i k e s the subject, for w h o m a r i t h m e t i c is
la r g e l y rote m a n i p u l a t i o n , devo i d of u n d e r ­
standing, can n o t avoi d transm i t t i n g his feelings
to the children. . . . O n the other hand, the
t e a c h e r who has conf idence, unde rst and ing ,
in t e r e s t and e n t h u s i a s m for a r i t h m e t i c has gone
a lo ng wa y t o w a r d in suring success.

61

Da ta c o n c e r n i n g the re lat ionships b e t w e e n teacher
att itudes and stud ent a tti tudes supp ort B a n k *9 assertion.
Torrance et al.

(112)

grade m a t h e m a t i c s

stu died 127 sixth throug h

twelfth

te achers and he found that the tea ch er

e f f e ctiveness had a p o s i t i v e eff ect on stu de nt attitudes.
In a s t u d y c o n c e r n i n g
algebra,

Garner

(77)

teachers*

Pes kin

(84)

at tit udes

the s ubject and students'

atti­

c o m p a r e d the attitude and u n d e rstanding

of t e a chers and st udents

in nine junior high schools.

c orr el a t i o n s b e t w e e n teachers'

and students'

the c o r r e l a t i o n s b e t w e e n teachers'
attitudes.

The

u n d e r s tandings

of a l g e b r a and g e o m e t r y w e r e s i g n i f i c a n t l y positive,

students'

toward

found si g n i f i c a n t r elations betwee n

teac her 's a t t itud e toward
tudes.

and pupils*

as were

u n d e r s t a n d i n g scores and

The relat i o n s h i p s of teach er u n d e r ­

st anding and a t t i t u d e to student a c h i e v e m e n t and attitude
were no t so clea r cut.
a t t itude and a "high"

T e a chers w i t h a "middle of the road"
u n d e r s t a n d i n g had stud ent s achieve

d i f f e r e n t l y a c c o r d i n g to the m a t h e m a t i c a l

topics.

Corre­

lations be t w e e n teache r u n d e r s t a n d i n g and studevt attitud e
and a c h i e v e m e n t w e r e also a f f e c t e d d i f f e r e n t l y by students
h av i n g ver y high or very

low levels of achievement.

Wh at are the reasons w h y t e a c h e r s and pr osp ective
te ache rs

like or d i s l i k e ar i t h m e t i c ?

D u t t o n and others have

c o n d u c t e d e x t e n s i v e stud ies on the subje ct
89),

(32,

principally with prospective elementary

48,

49, 65,

teachers.

62

Reasons g i v e n for liki n g the s ubject were:
its pr actical applicat ion ,
specific skills,

its ex actness,

and solving pro blems.

a r i thmetic gave reas ons such as:
work,

long problems,

a p p r e c i a t i o n of

Th ose w h o disli ked

w o r d problems,

lack of un de rstanding,

lack of t eacher enth usi asm,
Dreger an d Aike n

its challenge,

bor i n g

p o o r teaching,

failure or fear of

failure.

(47) e s t i m a t e d that a p p r o x i m a t e l y o ne-third

of pr o s p e c t i v e e l e m e n t a r y school t e a chers and p erhaps of
college stude nts in gen er al have u n f a v o r a b l e attit u d e s
toward arithmetic.

Rey s and D elon

(65) r e p orted

that the

m a j ority of the p r o s p e c t i v e e l e m e n t a r y school t e a chers in
their study d e v e l o p e d their attit u d e toward a r i t h m e t i c
duri ng the seventh to ni nth grades.
I nv e s t i g a t i o n s conducted by Dutton,
and Delon, and Gee

(48,

49,

65,

78)

Purcell,

Reys

to find r e l a t i o n s h i p s

betwee n the at tit u des and a c h i e v e m e n t s of p r o s p e c t i v e
teachers

in t e a c h e r - t r a i n i n g c ourse s

at titudes

toward m a t h e m a t i c s

indi ca te d that:

(1)

i m p rov ed signi f i c a n t l y after

the st udent s had c o m p l e t e d the c o u r s e

(65,

78,

86),

(2)

p ost - t e s t scores on an ari t hmetic c o m p r e h e n s i o n test were
s i g n ificantly hig h e r than p r e - t e s t

scores

(48,

(3) n o n 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 changes
and c hanges

in u n d e r s t a n d i n g of m a t h e m a t i c s

In general,

78,

86),

in at ti tudes

(48,

49,

86).

the results of these i n v e s t i g a t i o n s

indicate that impro v i n g teacher a t t i t u d e t o w a r d ma t h e m a t i c s

63

can res u lt in m o r e p o s i t i v e a t t itude and b e t t e r
u n d e r s t a n d i n g on the part of students.

Proper training

is a l s o very li kely to i mprove at ti tudes of p r o s p e c t i v e
e l e m e n t a r y sc hool teach ers to ward m a thematics.

A t t i t u d e s and A c h i e v e m e n t
Res ults of i n v e s t i g a t i o n s r e l ating a t t i t u d e s to
a c h i e v e m e n t in m a t h e m a t i c s ar e often co n t r a d i c t o r y .
(36) b e l i e v e s that this is due

to the fa ct that the m a j ority

of th ese i n v e s t i g a t i o n s ha ve employed e x p e r i m e n t a l
that w e r e

Harrington

(79), at the U n i v e r s i t y of

studied 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 s

m a t h e m a t i c s and grad e o b t a i n e d
course.

des ign s

inad e q u a t e for a n s w e r i n g the q u e s t i o n s pose d by

the investi gat ors .
Florida,

A iken

toward

in a f r e s h m a n m a t h e m a t i c s

He r e p o r t e d a s t a t i s t i c a l l y i n s i g n i f i c a n t r e l a ­

ti onship b e t w e e n a t t itude a n d pe rform anc e,

a l t h o u g h he

found that the s e l e c t i o n of an elect ive c o u r s e in m a t h e ­
mat i c s wa s s i g n i f i c a n t l y r e l a t e d to atti tude.

Whitnell

(88),

in a study done to d e t e r m i n e 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
c ollege students,
me nt w e r e ability,
ba ckground.

found that the best p r e d i c t o r s of a c h i e v e ­
at titude,

In hi s stud y of

an d high scho o l m a t h e m a t i c a l
160 st udent s e n r o l l e d in three

d i f f e r e n t s e c t i o n s of an u p p e r d i v i s i o n m e t h o d s

co urs e

d ealing w i t h the t e a dhing of ari thmetic,

(49)

Dutt o n

repo rte d that s t u d e n t a t t i t u d e s toward a r i t h m e t i c reflected
a g rowing a p p r e c i a t i o n of th e subject as they incre a s e d

64

their u n d e r s t a n d i n g of ari thmetic.

T h e g eneral at titude

of abou t 75 p e r c e n t of the studen ts towa r d a r i t h m e t i c was
q u ite f a v o r a b l e - - v a r y i n g from 6.0 to 9.5
scale items ranged
favorable).

(value of the

from a low of 1 . 5 - d islike to 10.5-ve ry

T h e lo wes t 2 5 p e r c e n t of the stud ents in this

st udy held u n f a v o r a b l e a ttitudes tow a r d arithmetic.
Litwiller

(82)

i n v e s t i g a t e d the ch ang e

of p r o s p e c t i v e e l e m e n t a r y teachers r e s u l t i n g
in "method."

in atti t u d e s

from a change

Her samp le c o n s i s t e d of 145 studen tB e n r o l l e d

in a c o n t e n t c o u r s e at Indian a Uni versity,

n i n e t y - f i v e of

w h o m w e r e in the e x p e r i m e n t a l gro up an d fifty in the contr ol
group.

Resul ts

ind icated

the following:

(1)

the a t t i t u d e s

of the e x p e r i m e n t a l gr oup c h a n g e d s i g n i f i c a n t l y re lative to
the att it u d e s of the con tr ol group,

(2) there was a s i g n i f i ­

ca nt d i f f e r e n c e b e t w e e n the a c h i e v e m e n t
students
group,

sc ore s of th ose

in the e x p e r i m e n t a l g r o u p r e l a t i v e to the co ntr ol

and

(3) there was a 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

the p ost- test a t t i t u d e sco re and a c h i e v e m e n t and SAT m a t h e ­
m atical

scores,

re spe ctivel y.

In s o m e w h a t d i f f e r e n t studies,
(47)

Dreger and A i k e n

found that sco r e s on an inv ent ory of a n x i e t y was

s ign i f i c a n t l y r e l a t e d to the

final g r a d e of 704 freshmen

e n r olled in a m a t h e m a t i c s course.
same i n v e s t i g a t o r s

In a n o t h e r study,

(39) r e p o r t e d that sc ore s on the

M a t h e m a t i c a l A t t i t u d e Scale c o n t r i b u t e d s i g n i f i c a n t l y

the

65

to the p r e d i c t i o n of final gra d e s in a m a t h e m a t i c s course
for s i x t y -seven c ollege women,

bu t not for the sixt y men

subjects of the same study.
A n in ter nation al study reporte d b y Huston
compare

(17)

to

the ma t h e m a t i c s a c h i e v e m e n t of s econdary stude nts

in twelve co unt rie s pr ovided dat a c o n c e r n i n g the r e l a t i o n
of a ttitudes and intere sts to m a t h e m a t i c s
Three of the five at titude

achievement.

scales a d m i n i s t e r e d were:

measur es of at tit ude s toward m a t h e m a t i c s

as a process,

attitude s ab out the d i f f i c u l t i e s of learning m athema tic s,
and a ttitudes about the p l a c e of m a t h e m a t i c s

in society.

C o r r e l a t i o n a l resu lts of this i n v e s t i g a t i o n were:

signif­

icant n e g a t i v e r a n k - o r d e r c o r r e l a t i o n s be t w e e n m e a n m a t h e ­
matics ac h i e v e m e n t and m e a n scores acr o s s cou ntries on the
attit ude scales;

small c o rrelations b e t w e e n a c h i e v e m e n t and

attitude w i t h i n countries;

m o d erate to h i g h correlations

between a c h i e v e m e n t and in teres t m e a s u r e s within countries.

Attit u d e s and the N e w M a t h e m a t i c s
Curr i c u l a
R e s earch d e s i g n s used to c ompare at tit u des and
a c h i e v e m e n t of student s e n r o l l e d in new m a t h e m a t i c s prog ram s
wit h t hose of stude nts e n r o l l e d in tr a d i t i o n a l p r o g r a m s
resulted

in findi ngs that are not consistent.

Inv e s t i g a t o r s

who have compar ed SMSG and trad itiona l curricula
tary and

junior h i g h school

(55,

83,

85,

90,

101)

in e l e m e n ­
found that,

66

gen erally,

the mean m a t h e m a t i c s a t t i t u d e sco r e s of students

ta ugh t by S M S G c u r r i c u l u m was no t s i g n i f i c a n t l y g r e a t e r than
the mean a t t i t u d e sc or e s of s t u dents taught m a t h e m a t i c s by
the t r a d i tional curri culu m.
conventional

As to ac hiev eme nt,

scores on

s t a n d a r d i z e d tests te nde d to favo r traditi ona l

pr o g r a m s w h i l e scores on more s p e c i a l i z e d t ests favored the
S M S G curriculum.
Simil ar d e s i g n s wer e u s e d to c o m p a r e othe r m a t h e ­
m a t i c s prog ram s w i t h the tr a d i t i o n a l progr ams .
c o m pared the colleg e m a t h e m a t i c s

Comley

(75)

a c h i e v e m e n t and at titu des

of students e n r o l l e d in the U n i v e r s i t y of Ill inois Commi t t e e
on Sdhool M a t h e m a t i c s

(UICSM)

program with

those of students

w h o had tr adit io nal h i g h - s c h o o l math e matics.

There w e r e

few

di f f e r e n c e s betwe en the two g r o u p s in c o l l e g e mathem a t i c a l
achievement

after th e c r i t e r i o n sc ore s of the U I C S M and non-

U I C S M gr oup s were a d j u s t e d on a nu mbe r of varia ble s,
U I C S M stude nts had s i g n i f i c a n t l y more

but the

favor a b l e m a t h e m a t i c s

a tti t u d e s tha n the n o n - U I C S M group.
Y asu i

(91)

c o m p a r e d a g r o u p of s t u d e n t s e xpose d to

a modern-mathematics

p r o g r a m w i t h a group not e x p o s e d to

m o d e r n m athem ati cs.

After adjustment

ences,

the i n v e s t i g a t o r

for i ndi v i d u a l d i f f e r ­

found that w h i l e the d i f f e r e n c e

b e t w e e n the mean sco r e s of the tw o gr oup s on an in ventory
of att itudes toward m a t h e m a t i c s was not si gnificant,

atti­

tude scores w e r e 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 achievement

67

in both groups.
experimental

Ryan

"modern"

(30)

c o m p a r e d the eff ects of three

programs

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

on the a t t i t u d e s and interests d e v e l o p e d in n i n t h - g r a d e
pupils.

Th e resu lts

sho wed that the e x p e r i m e n t a l p r o g r a m s

had little d i f f e r e n t i a l effe c t on a t t i t u d e s and interests.
I n v e s t i g a t i n g p r o g r a m - c e n t e r e d vs.
ing of

fi rst-year algebra*

Devi n e

teacher-centered teach­
(76)

con c l u d e d

an a verage or abov e av e r a g e t each er is available*
a c h i e v e m e n t is o b t a i n e d in a c o n v e ntiona l,
c l a s s r o o m ap proach and a t t i t u d e s

that w h e n
gr e a t e r

teacher-centered

to war d m a t h e m a t i c s are not

a f f e c t e d signif ica ntl y.
In his e v a l u a t i o n of the r e s e a r c h done on atti t u d e s
to ward mat hem a t i c s ,
cautious

Aiken

(38)

i n d i c a t e d that one

in i n t e r p r e t i n g the results of

on the subject.

Fo r one thing,

sho uld be

the i n v e s t i g a t i o n s

a v a i l a b l e subj ect s w e r e not

al ways a s s i g n e d at r a n d o m to the two types of cu rricula.
is q u ite p o s sible that s t u d e n t s

in s pecial p r o g r a m s were

i nit ially a t t r a c t e d to or s e l ected
of thei r p o s itive a t t i t u d e s

It

for the p r o g r a m be c a u s e

toward mat hematics.

Osborn

(83)

s u g g e s t e d that m o d e r n c u r r i c u l a are m o r e abstract and
d e m a n d i n g than the t r a d i t i o n a l c u r r i c u l u m with the result
that s t u dents e n r o l l e d in m o d e r n m a t h e m a t i c s p r o g r a m s

fail

to c h a n g e their a t t itu de tow a r d m a t h e m a t i c s or b e c o m e mo re
n e g ative as the p r o g r a m develops.

68

For Aiken, w h o e x p lored e x t e n s i v e l y the research
on attitudes toward mathemati cs,
rather than the curriculum,
influential v a r i a b l e as

it seems that "the teacher,

still a ppears to be the more

far as attit u d e s are concerned"

(36:581).
Summa ry
This cha pter wa s c o n c e r n e d w i t h a search of the
most recent and perti n e n t lite r a t u r e on c u r r i c u l u m e v a l ­
uation,

teac her e d u c a t i o n p r o g r a m s

teachers,

for e l e m e n t a r y school

appr o a c h e s to teach er training, c o n t e n t of m a t h e ­

matic s c u r r i c u l u m and research on a ttitu des

toward and

ach iev ement in mathematics.
The incre a s e d c onc ern of m a t h e m a t i c s educ ators
from all over the w o r l d w i t h the n e c e s s i t y of improving
ma t h e matics e d u c a t i o n has b e e n very i n f l u ential in the
d e v e l opment of new tea ch er e d u c a t i o n p r o g r a m s and in the
impro vem ent of m e t h o d s of evaluation.
Since teacher s and their e d u c a t i o n are the p r i n c i p a l
substance b e h i n d any ef for t m a d e
ment of e d u c a t i o n a l systems,

for the u l t imate i m p r o v e ­

e d u c a t o r s hav e d evoted a g r eat

deal of time to the i m p r o vement of teac he r e d u c a t i o n p r o ­
grams, d e v e l o p i n g c r i t e r i a for the t r a ining of p r o s p e c t i v e
teachers of m a t h e m a t i c s at b o t h the p rimar y and the s e c o n d ­
ary level.

Chang es

in the c o n t e n t of c u r r i c u l a have been

69

a c c o m p a n i e d by e x p e r i m e n t s
teachi ng methods,

in the d e v e l o p m e n t of new

and by n e w resear ch on the m a t h e m a t i c a l

compe t e n c i e s n e e d e d by e l e m e n t a r y scho o l teacherB and
e l e m e n t a r y school children.
C u r r i c u l u m r e s e a r c h and e v a l u a t i o n has conti n u e d
to progress.

W h i l e sununative e v a l u a t i o n is still rega rded

as an ad equate and n e c e s s a r y meth od to make d e c i s i o n s about
c u r r i c u l u m a d o p t i o n and e ffective use,

fo rmative e v a l u a t i o n

techniques are c o n s i d e r e d more and m o r e imp ortant by mos t
c u r r i c u l u m spe ci alis ts d u r i n g the d e v e l o p m e n t of a teacher
e d u c a t i o n p r o g r a m and a l s o

for i n s t r u c t i o n and studen t

learning.
C o n c e r n e d by the

imp o rtance of im pr oving the

m e a s u r e m e n t of achievement,

a score of researchers have

prop ose d a more e x t e n s i v e u b c of c r i t e r i o n - r e f e r e n c e d
measu r e s

in the as s e s s m e n t of the d e g r e e of comp e t e n c e

at tai ned by a p a r t i c u l a r

student.

This type of m e a s u r e m e n t

is r ela t i v e l y new in e d u c a t i o n but the d e v e l o p m e n t of
in s t ruction al

technology and the r e c e n t e m p h a s i s on c u r ­

r i c u l u m rese arch and c u r r i c u l u m e v a l u a t i o n have s t r essed
the ne ed

for the kind of i n f o r m a t i o n ma de a v a i l a b l e by the

use of c r i t e r i o n - r e f e r e n c e d measures.
A t t i t u d e s towa rd m a t h e m a t i c s , an impo r t a n t e lement
in the success of m o d e r n m a t h e m a t i c s e d u c a t i o n programs,
have al so b e e n i n v e s t i g a t e d e x t e n s i v e l y in r e l a t i o n to

70

p e r s o n a l i t y c h a r a c teristics,
e ffect ive nes s,

students'

teac her's atti t u d e s and

a c h i e v e m e n t and the new m a t h e m a t i c s

curricula.
M a t e r i a l s d i s c u s s e d in the r e v i e w of l i t e r a t u r e
were used for the d e v e l o p m e n t of a d e q u a t e p r o c e d u r e s to
be

followed in the

fo rmative e v a l u a t i o n of the m a t h e m a t i c s

comp one nt of the M i c h i g a n St ate U n i v e r s i t y e x p e r i m e n t a l
t eacher e d u c a t i o n program.
features of the study.

Chapter

III d e s c r i b e s the

CHAPTER III
D E S C R I P T I V E FEA TUR ES OF THE STUD Y

The

fo rma tive eva l u a t i o n of the m a t h e m a t i c s

c omp onent of the e x p e r i m e n t a l ele m e n t a r y teach er e d u c a t i o n
p r o g r a m at M i c h i g a n Sta te U n i v e r s i t y was c o n d u c t e d du ring
the a c a d e m i c year 1971-1972.

This chapte r s u m mari zes the

d i f f e r e n t pr oc e d u r e s w h i c h w e r e

follo wed in c a r r y i n g out

the pr e s e n t evaluat ion .
Students

in the Study

The students in the e x p er imental p r o g r a m were those
freshm en e l e m e n t a r y e d u c a t i o n maj o r s w h o v o l u n t e e r e d and
wer e s e l e c t e d to p a r t i c i p a t e in the program.
fif ty-two e n t ering
were selected.
d rop ped

fresh men volunteered,

forty of w h i c h

At the b e g i n n i n g of the year,

from the program;

thus,

Initially,

two student s

the remaining thirty - e i g h t

students c o m p r i s e d the g r oup of p r o s p e c t i v e ele m e n t a r y
teach ers w h o p a r t i c i p a t e d in the first cou r s e of the
m a t h e m a t i c s c u r r i c u l u m of the program.
In a d d i t i o n to these students,
students w e r e u t i l i z e d in this study:

71

othe r grou ps of

72

1.

A r e p r e s e n t a t i v e sa mple of three

was used for c o m p a r i s o n purposes.
were

These

fresh men grou p s

"comparison grou ps"

selected by the TT T p roject eva l u a t i o n t e a m on the

basi s of a study of U n i v e r s i t y

records of n u m b e r of fresh m e n

wi th d e c lared m a j o r s in v arious d i s c i plines
"c ompa rison groups"
d e c l a r e d maj o r s

ch ose n w e r e

in:

(98).

first term f r e s h m e n w i t h

(1) E l e m e n t a r y Education,

m a t i c s and S e c o n d a r y Edu ca tion,

The t h r e e

and

(2) M a t h e ­

(3) Mathema tic s.

Th e

in ves tigator wa s able to u t i l i z e the three c o m p a r i s o n g r o u p s
to assess w h e t h e r the s t u dents w h o v o l u n t e e r e d and were
a c c e p t e d in the e x p e r i m e n t a l p r o g r a m were init i a l l y d i f ­
ferent in their c ognitive and a f f e c t i v e b e h a v i o r s toward
mathematics

from ot her

fr esh men studen ts wh o sha red the

same p r o f e s s i o n a l

int erests

showed a s p e cific

in terest in ma thematics.

2.

and a g r o u p of f r e s h m e n wh o

A g r o u p of p r o s p e c t i v e ele m e n t a r y

school

teache rs e n r o l l e d in the r egular teache r e d u c a t i o n p r o g r a m
at M i c h i g a n State U n i v e r s i t y was a l s o used for this i n v e s ­
tigation.

T h e s e students had al r e a d y c o m p l e t e d the m a t h e ­

m at i c s c ontent cou r se

(Mathematics

201)

in the m ethods co urse

(Education 325E)

and w e r e e n r olled
at the same time

that the e x p e r i m e n t a l g r o u p was inv olv ed in th e in te g r a t e d
content-methods
substantially

course.

T h e s e re g u l a r s t u d e n t s differ

from the e x p e r i m e n t a l group a n d

p ari son groups"

the "com ­

in that they are se c o n d - y e a r college

73

students

(or h i g h e r ) .

T h e regular stu dents w e r e used for

two differe nt purposes:

(1) to e v a l u a t e the r e l i a b i l i t y

of the a c h i e v e m e n t tests d e v e l o p e d in this study,

and

(2)

to assess t h e i 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 and attitude s
toward a r i t h m e t i c afte r the c o m p l e t i o n of the m e t h o d s
cou rse and to c ompare t h e s e results w i t h those of the
experim ent al group.

Th e

students in these gro u p s are

referr ed to in the study as the "r e g u l a r m e t h o d s co urse
(325E)

student s."
E v a l u a t i o n of the E x p e r i m e n t a l P r o g r a m

D e s c r iption of the M a t h e m a t i c s
Component of the E x p e r i m e n t a l
T eac her E d u c a t i o n P r o g r a m
At M i c h i g a n S t a t e Un iversity,

undergraduate elemen­

tary e d u c a t i o n majors are require d to c o m p l e t e a s e q uence of
two m a t h e m a t i c s e d u c a t i o n courses.
the Dep a r t m e n t of Mat hema tic s,

The

first,

o f f e r e d by

is a f o u r - q u a r t e r hour c o n ­

tent course e n t i t l e d A r i t h m e t i c for E l e m e n t a r y T e a c h e r s
(Math.

201).

teachers s p e n d

Dur ing thi s course,

prospective

elementary

three h o u r s 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.

Thi s c o u r s e is a p r e ­

re quisite to th e seco n d requi red c o u r s e in m e t h o d s of
teaching e l e m e n t a r y m a t h e m a t i c s

(Education 325E)

w h i c h is

o ffe red by th e D e p a r t m e n t of E l e m e n t a r y Scho o l Educ ati on.
T hese two c o u r s e s are u sually taken d u r i n g the s o p h o m o r e or
junior year.

74

T h e total e x p e r i e n c e of the p r e s e r v i c e e l e mentary
school t eacher e n r o l l e d in the re g u l a r p r o g r a m in math ematics
ed ucation thus consists of

forty

class hours of con te nt and

thirty c lass hours of methods.
The m a t h e m a t i c s component of the exp e rimental
teacher e d u c a t i o n program,

on the other hand,

offers two

co mbi ned m a t h e m a t i c s c o n t e n t - m e t h o d s cou rses w h i c h integrate
the m a t h e m a t i c a l c o n cepts

i n t roduced to the students w i t h

the me t h o d s to teac h these concepts
children.

to e l e m e n t a r y school

Eac h course is a c c o m p a n i e d by a clinical e x p e ­

rience in w h i c h the student s a c t u a l l y p r a c t i c e teach in g the
concepts c o v e r e d in the c o u r s e to e l e m e n t a r y school children.
The
during the
year.

first of these two c ourses

1 b to be offered

freshman yea r an d the second,

d u r i n g the junior

T h e maj or topics c ove red in the first course are:

(1) Measurem ent ,

(2) Set Theory,

(4) Who le Nu m b e r System,

(3) N u m e r a t i o n Systems,

(5) Rational N u m b e r System,

I ntr odu ct ion to R elations and Functions,
and Statistics.

and

(6)

(7) Probability

The sec o n d co urs e e m p h a s i z e s the ar eas of

the Real Num b e r System,

Alge bra,

and Geometry.

U p o n c o m p l e t i o n of these two courses,

the c omp osite

m a t h e m a t i c a l e x p e r i e n c e of the p r e s e r v i c e el ementar y school
teacher e n r o l l e d in the e x p e r i m e n t a l p r o g r a m woul d consi st
of 160 c l a s s ho ur s of c o n t e n t - m e t h o d s and 80 clinical
hours.

school

75

Duri ng the 19 71 -197 2 a c a demic year,

the firs t course

was de signed and tried w i t h the first g roup of p r o s p e c t i v e
elementary teachers who p a r t i c i p a t e d
program.

in the exper i m e n t a l

It is this p a r t i c u l a r co urs e that is the ob jec tive

of the formative e v a l u a t i o n d o n e by this investigator.

A s s e s s m e n t of the M a t h e m a t i c s
C o m p o n e n t of the E x p e r i m e n t a l
T e a c h e r E d u c a t i o n frrogram:
D e v e l o p m e n t of a C r i t e r i o n R e f erenced List
In o r d e r to c o n s t r u c t a scor e c a r d of m a t h e m a t i c a l
to pics sugge s t e d for 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,

the i n v e s t i g a t o r sought the advice of m a t h e m a t i c s

educators at the Univer sity , w h o r e c o m mended a t h o roug h
review of the related l i t e r a t u r e such as the r e p o r t of the
Committee on the U n d e r g r a d u a t e P r o g r a m in M a t h e m a t i c s
the Camb r i d g e C o n f e r e n c e R e p o r t "Goals

(CUPM),

for M a t h e m a t i c s

E duc ation of Elem e n t a r y S c h o o l Teachers,"

the S t r a n d 1s

R e p o r t , the p u b l i c a t i o n s of the N a t i o n a l Cou nci l of Teachers
of Ma th emat ics ,

and v arious t extbooks

mati c s e d u c a t i o n for e l e m e n t a r y
The list of top ics

in the f i e l d of m a t h e ­

school teachers.

su gge sted

in the p u b l i c a t i o n s

re vie wed ser ved as a b a s i s upon w h i c h to assess w h e t h e r the
c o n t e n t i n c luded in the e x p e r i m e n t a l pro g r a m at M i c h i g a n
State U n i v e r s i t y is s u f f i c i e n t

in m e e t i n g the n e e d of the

p r o s p e c t i v e teac her in the field of mat hema tic s.

76

The r e v i e w of literat ure r e v e a l e d a highly related
survey study c o n d u c t e d b y Hicks a n d P e r r o d i n

{54) w h i c h

provi ded a sound b a s e for the s e l e c t i o n of topics a p p r o ­
priate

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 of

e l e m e n t a r y sc hoo l teachers.

Fo ur

types of sources w e r e

i n t e n sively r e v i e w e d by the a u t h o r s
data.
1.

to p r o v i d e the n e c e s s a r y

T h e y were:
R e v i e w of fo rty -six s e l e c t e d r e s e a r c h studies
p o i n t i n g out the m a t h e m a t i c a l c o m p e t e n c i e s or
w e a k n e s s e s of e l e m e n t a r y sch o o l teachers.

2.

R e v i e w of th i r t y - t w o sets of r e c o m m e n d a t i o n s of
m a t h e m a t i c s e ducators an d n a t i o n a l l y - r e c o g n i z e d
a d v i s o r y gr oup s or o r ganiz ations.

3.

P a g e - b y - p a g e analysis of s ixteen r ece n t text b o o k s
designed

for c ollege c o u r s e s

in m a t h e m a t i c s

for

e l e m e n t a r y sch ool 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 ser i e s or teacher 's
guid es

for gra d e s K-7 p u b l i s h e d

A composite

list of m a t h e m a t i c a l

since

1962.

to pic s

from the

above s ources w a s then compiled b y Hicks and P e r r o d i n
and a s y s t e m of rating these t o p i c s wa s devis ed.
which a p p e a r e d at least once in th e c o m p o s i t e
c a t e g o r i z e d as Level

I.

(54)

Topics

list w e r e

To be c a t e g o r i z e d as Level

topics had to m e e t one of the f o l l o w i n g co nditions:

II,

77

•

app 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 of 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 or a d v i s o r y groups;

•

appear in

at leaBt eight

of the sixteen college

textbooks in mathematics

for elementary school

teachers;
• a p p e a r in at le ast six of the ele v e n a r ithmetic
se rie s or te ach e r ' s gui d e s
Finally,

for gra d e s K-7.

to be c l a s s i f i e d as Level

at least two of the four c r i t e r i a

III,

a to pic had to meet

listed a b ove for Level

II

topics.
A total of n i n e t y - e i g h t
four s ources
two,

topics w e r e

(nineteen in source one,

e i g h t y - f o u r in source three,

four).

Of these topics,

fif t y - f o u r in source

and s e v e n t y - n i n e in source

f ifty-one w e r e c a t e g o r i z e d as Level

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 evel
T a b l e 1 shows the topics
the sourc es

located in the

III.

in level three along with

in w h i c h they appeared.

It is o bvious

from this

table that the last three sour ces are in close a g r e e m e n t on
what

sh ou ld be included

in some m a n n e r

c u r r i c u l u m of the e l e m e n t a r y

in the m a t h e m a t i c s

school teacher.

The rel a t i v e l y

low p e r c e n t a g e in the first source d o e s not in dicate d i s ­
a gre ement w i t h the o t h e r sources;

it o n l y

indic ate s

the lack

of e x p e r i m e n t a l r e s e a r c h done on the s e l e c t i o n of m a t h e m a t i c l
topics

for 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.

78

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.
10.
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 Numeration
Systems K 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 s> 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
Total
% of Total No. of Topics

Source 1

X

X

Source 2

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

13
37

Source 4

X
X
X
X

X

X
X
X
X

Source 3

28
80

X
X
X
X
X
X
X
X
X
X

X
X
X
X

X
X
X

31
89

32
91

79

Re vie w of

the s ources u t i l i z e d

in the Hicks and

Pe r r o d i n survey sugg e s t e d that almo s t all of

the sources

r e c o m m e n d e d to the i n vestigat or by the m a t h e m a t i c s educa t o r s
at M i c h i g a n State U n i v e r s i t y w e r e included.
the v a l i d i t y of this list,

T o further test

the i n v e s t i g a t o r revie wed p u b ­

licat i ons of s i m i l a r s ources

for the years

1968-1972.

Th e

topicB suggested in these sources are very c o nsisten t w i t h
the li st d e s c r i b e d

above ex cep t in the field of G e o m e t r y

and in the field of Logic.
An a l y s i s of the c o n t e n t of
elementary mathematics

for teac her s

revealed that c o o r d i n a t e ge ometry
were not

inclu ded

Perrodin

(54).

five t e x t b o o k s
(5, 18,

24,

for

26, 27)

an d m a t h e m a t i c a l logic

in the list d e v e l o p e d by H i c k s and

Th e A r i t h m e t i c T e a c h e r , a p u b l i c a t i o n of

the N a t i o n a l C o u n c i l of T e a chers of M a t h e m a t i c s ,

a n n ually

p u b l i s h e s a su m m a r y of r e s e a r c h and artic les on m a t h e m a t i c s
education conducted
ing year.
1969,

in the Uni ted St ate s d u r i n g the p r e c e d ­

R e v i e w of these s u m m a r i e s

1970,

and

for the years 1968,

1971 a g a i n p o i n t e d out that m o s t r e s earch

done on the c o n t e n t was in topics n o ted in the Level III
list as d efined by Hicks and Perrod in.
of research,

one by Shah

(67)

Howeve r,

on the a p p l i c a b i l i t y of t e a c h ­

ing g e o m e t r y to e l e m e n t a r y school children,
O' B r i e n and S h a p i r o
learn m a t h e m a t i c a l

(63)
logic.

two p i e c e s

th e other by

conf i r m e d c h i l d r e n ' s ability

to

R e s e a r c h c o n d u c t e d by Suppes

80

(111)

at S t a n f o r d U n i v e r s i t y in t e a c h i n g l o gic to e l e m e n t a r y

schoo l c h i l d r e n has no t as ye t p r o v i d e d con c l u s i v e ev ide nce
to the c h i l d r e n ' s a bility to learn and com p r e h e n d m a t h e ­
m a t i c a l logic.
the

B ased on this r e v i e w of r e c e n t

literature,

i n v e s t i g a t o r c o n c l u d e d that only the t o p i c "Coo rd inat e

Geome t r y " m et the q u a l i f i c a t i o n s of the L e v e l
by Hicks and Perrodin,

III p r e s c r i b e d

and t h e r e f o r e d e c i d e d to include it

as the t h i r t y - s i x t h topic in the cri teria list.

The Int e g r a t e d C o n t e n t - M e t h o d s
C ou r s e in Ma t h e m a t i c s

D e s i g n i n g the C u r r i c u l u m of
the Int e g r a t e d C o n t e n t - M e t h o d s
C o u r s e in M a t h e m a t i c s
A m o n g the o b j e c t i v e s of the TT T p r o j e c t is the
d e v e l o p m e n t of a c o m p e t e n c y - b a s e d e l e m e n t a r y teac her e d u c a ­
ti on p r o g r a m that i n c o r p o r a t e s aspect s of th e Mode l P r o g r a m
BSTEP.
In the m a t h e m a t i c s c o m p o n e n t of the program,
competency-based

tea che r e d u c a t i o n p r o g r a m means

p r o g r a m that requires
s pec ified

its train e e s

level of compete nce ,

h a v e been e x p l i c i t l y

a trai nin g

to demonst rat e,

at a

m a t h e m a t i c a l beha v i o r s that

speci f i e d as e f f e c t i v e p r o f e s s i o n a l

behaviors.
These competencies

(knowledge,

skill,

and behaviors)

are d e t e r m i n e d by the p r o g r a m d e v elopers as s p e c i f i c s t a t e ­
m e n t s of c o m p e t e n c i e s

ne ede d by the future e l e m e n t a r y school

81

teacher for m a t h e m a t i c s instruction.

It is upon these

statements that the i n s t r ucti onal m a t e r i a l s and design s
were d e v e l o p e d and implemented.
The task of d e t e r m i n i n g wh at m a t h e m a t i c a l c o m p e ­
tencies

{knowledge,

skill,

and behaviors)

should be incl ude d

in the p r o g r a m to make it effec t i v e in prod u c i n g c ompetent
teachers was c a r r i e d by an interdisc i p l i n a r y team of p r o ­
fessional pe ople

involved in education.

of two faculty me m b e r s
three faculty m e m b e r s
Education,
teachers

The tea m c ons isted

fr om the Dep a r t m e n t of Mathema tic s,
from the De partme nt of E l e m e n t a r y

four doct ora l students,

and

four el emen tary

from the school w h e r e the student s had their

clinical ex perience.

The

faculty m embers all had speci fic

interest and b a c k g r o u n d in ma t h e m a t i c s e ducation and have
had recent e x p e r i e n c e w o r k i n g with elem e n t a r y or junior
high schools.
The team wo r ked c l o s e l y together on deve l o p i n g
m a thematical exp e r i e n c e s

for the first year trial i m p l e m e n ­

tation of the e x p e r i m e n t a l program.

The p roduct of their

work consi s t e d of a series of nine learn ing units,
dev oted to one of nine m a t h e m a t i c a l
necessar y

Measurement
N u m e r a t i o n Systems
Sets and Set Rel ati ons

each one

topics d e e m e d via b l e and

for the future ele m e n t a r y school teacher.

topics were:
1.
2.
3.

the

The

82

4.
5.
6.
7.
8.
9.

The Whole N u m b e r System
Fractions
Decima ls
Relation s and Fu nctions
Probability and Sta tistics
Mathematical Systems.

The se learning units w e r e all d e s i g n e d in a c c o r d a n c e w i t h
gu idelines propo sed by the B S TEP and all h a v e c e r t a i n
features in common:
1.
2.

3.
4.
5.
6.
7.

A s s e s s m e n t tests
Goals and O b j e c t i v e s
a.
Requi red Act i v i t i e s
b.
Op tional A c t i v i t i e s
Str ate gie s to achieve the o b jectives
Instr u c t i o n a l design
Instr u c t i o n a l de sign to be used w i t h c h i l d r e n
at the e l e m e n t a r y level
Instructional feedback
Comments.

A complete

file on e a c h of the nine topi cs used in the

integrated co n t e n t - m e t h o d s c o u r s e was p r e p a r e d for eac h
of the students e n r o l l e d in the e x p e r i m e n t a l program.
(A speci men of suc h a file is found in A p p e n d i x F.)

E x p e rimenting w i t h the Integ rat ed
C o n t ent-Methods C o u r s e :
Procedure Follow ed
The i n tegrat ed content - m e t h o d s c o u r s e was
in a Michi g a n State Uni v e r s i t y m a t h e m a t i c s
laboratory was a large room e q u ipped w i t h
tables,

a u d i o - v i s u a l materi als ,

laboratory.

as well as n u m e r o u s shelves

at tri b u t e blocks,

Madison Proj ect s h o e b o x kits,

The

spacious w o r k

and cabine ts loaded w i t h m a n i p u l a t i v e m a t erials,
Cuisenaire rods,

cond u c t e d

geoboards,

b a l a n c e beams,

e.g.,

Dien e s blocks,
mirro r c a r d s .

83

The

s t u d e n t s w o r k i n g in th e l a b oratory al so had at thei r

disposal

a sizea ble c o l l e c t i o n of e l e m e n t a r y school m a t h e ­

matics textb o o k s

and n u m e r o u s copi es of Th e A r i t h m e t i c and

M a t h e m a t i c s T e a c h e r as w e l l as p u b l i c a t i o n s of many m a t h e ­
matics e d u c a t i o n projects.
T h e basi c m o t i v a t i o n
a mathematics
not o n l y

lab o r a t o r y

for c o n d u c t i n g the cour s e

in

is that such a s e t t i n g e n h a n c e s

the l e a rning of m a t h e m a t i c s

and m e t h o d s of teachi ng

m a t h e m a t i c s but a l s o p r e p a r e s the p r o s p e c t i v e t each er in
using m a n i p u l a t i v e m a t e r i a l s when t e a ching e l e m e n t a r y pupils.
T h e e x p e r i m e n t a l g r o u p met
days a week,

in the

M o n d a y th r o u g h T h u r s d a y

the S p r i n g term of 1972.

labo r a t o r y

from 3 to 5 p.m.

Each da y of class,

four or

four
during
five

in structors involved in the p r o g r a m w ere a v a i l a b l e to assist
the students;
ing unit)
was as

and every week,

was covered.

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

The weekly

topic

{learn­

sched u l e for the c o u r s e

follows:

Monday
3:00-3:30

A p r e - t e s t s p e c i f i c a l l y c o n s t r u c t e d to assess
the s tudent b e h a v i o r 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 compet e n c i e s for that w e e k was
a d m i n i s t e r e d (see A p p e n d i x A ) .

3:30-4:00

F i l e s for th at w e e k lea rning unit w e r e
d i s t r i b u t e d to eac h student.
The files
(deve lop ed ea rli er by the u n i t de signers)
c o n t a i n e d the goals and o b j e c t i v e s and a
d e s c r i p t i o n of the a c t i v i t i e s for each of
t h e s e objectives.

4:00-5:00

S t u d e n t s d i v i d e d into grou p s of four students
w o r k e d on the p r e s c r i b e d a c t i v i t i e s util i z i n g
the m a n i p u l a t i v e m a t e r i a l s a v a i l a b l e to
m a s t e r the o b j e c t i v e s set for that week.

84

TuesdayW edn e s d a y
3:00-5:00

Stude nts cont inued w o r k i n g on the ac tivities
pre s c r i b e d in the learn ing un it files.
When
one objec t i v e was completed, the student
d e m o n s t r a t e d his m a s t e r y over that obje c t i v e
to one of the inst ructor s and had it checked
on his file.
When the instructors felt that some or all
the studen ts we re hav i n g d i f f i c u l t i e s c o m ­
pre h e n d i n g certai n concepts, a short lecture,
o ften util i z i n g m a n i p u l a t i v e mat erials, was
condu c t e d by one of the instructors.

T h u rsday
3 :00-3:30

A p ost-tes t (parallel form of the pre-test)
on the conte nt l earned on the prev iou s week
w a s a d m i n i s t e r e d (see A p p e n d i x A ) . The p o s t ­
test on any p a r t i c u l a r topic wa s given only
after the students ha d gone thro ugh one we ek
of in str u ction in the la borat ory and one wee k
of self -teaching the pupi ls in the e l e ment ary
school (see the sched u l e on the following
page).
Note:
The only e x c e p t i o n was for the last
topic M a t h e m a t i c a l Systems, w h i c h was not
ta ught in the e l e m e n t a r y school.
In this
case, the post - t e s t wa s giv en at the end of
the w e e k of study in th e MSU laboratory.

3:30-5:00

E a c h g r o u p of four studen ts w o r k e d w i t h an
ins t r u c t o r or one of the e l e m e n t a r y teachers
p resent that day on d e s i g n i n g a lesson w h ich
w o u l d in cor porate ass essment, goals and
objectives, strategies, and evaluation.
Th is lesson was ta ug ht the n e x t w e e k to the
p up i l B of the e l e m e n t a r y sch ool by the four
student s of each group.

Monday
through
Thursda y
8 : 00 - 12:00

E v e r y day, nine or ten students went to the
c h o s e n elem e n t a r y school in the Lan si ng
are a for clinic al exp erience.

Clinical Experience
St udents

im ple men ted the i n s t r u c t i o n a l designs

devel o p e d at the u n i v e r s i t y l ab o r a t o r y w i t h
the e l e m e n t a r y school m a t h e m a t i c s

the c h i ldren at

laboratory.

INTEGRATED CONTENT-METHODS COURSE SCHEDULE
SPRING TERM 1972
Week
Beginning

Unit Covered
in the Lab.

Pre-Test

April 3, 1972

Measurement

Measurement

April 10, 1972

Numeration

April 17, 1972

Unit Taught
m School

Post-Test

Numeration

Measurement

Measurement

Sets and Set
Relations

Sets and Set
Relations

Numeration

Numeration

April 24, 1972

Whole Numbers

Whole Numbers

Sets and Set
Relations

Sets and Set
Relations

May 1, 1972

Fractions

Fractions

Whole Numbers

Whole Numbers

May 8, 1972

Decimals

Decimals

Fractions

Fractions

May 15, 1972

Relations and
Functions

Relations and
Functions

Decimals

Decimals

May 22, 1972

Statistics and
Probability

Statistics and
Probability

Relations and
Functions

Relations and
Functions

May 29, 1972

Mathematical
Systems

Mathematical
Systems

Statistics and
Probability

Statistics and
Probability
and
Mathematical
Systems

86

E a c h stu d e n t spent one
every w e e k

full m o r n i n g

in the ele m e n t a r y school.

(four hours)

T h r e e h o u r s were

spent w o r k i n g in the c l a s s r o o m w i t h the t eacher and the
re main ing hou r w a s
of t e a c h i n g

split into two parts:

four to six p u p i l s the m a t h e m a t i c a l concepts

designed at the u n i v e r s i t y
week;

(1) o n e - h a l f h o u r

lab o r a t o r y d u r i n g the p r e vious

(2) o n e - h a l f hour of m e e t i n g w i t h

the d o c toral stude nt

or f ac ulty m e m b e r w h o o b s e r v e d the c l i nical p r a c t i c e to
excha nge c o m m e n t s

and r e c e i v e

or des i g n UBed in teach ing

feedb ack o n the m e t h o d s

the pupils.

Th e c l i n i c a l e x p e r i e n c e p r o v i d e d

the p r o s p e c t i v e

ele m e n t a r y t e a c h e r with:
1.

T h e o p p o r t u n i t y to rel ate theo r y
ap p l y i n g

to pra ctice,

by

the k n o w l e d g e gain e d at the u n iversity

to actual

teach ing

sit u a t i o n s at the e l e m e n t a r y

school.
2.

T h e o p p o r t u n i t y to o b s e r v e d i f f e r e n t classes,
teachers,

3.

and t e a ching methods.

T h e o p p o r t u n i t y to ini tiate his t e a c h i n g e x p e r i e n c e
by w o r k i n g with a small g r o u p of children,
benefiting

thus

from c l o s e r ind i v i d u a l r e l a t i o n s and

m i n i m i z e d pr oblems of d i s c i p l i n e and control.

4.

T h e o p p o r t u n i t y to r e c e i v e i m m e d i a t e

f e e dback on

th e m e t h o d s of t e a c h i n g u t i l i z e d from e x p e r i e n c e d
in-service

teachers or faculty members.

87

A s s e s s i n g the C o n t e n t of the
Int e g r a t e d Content-rietho3ii
Cou r s e
i nstru men ts

select ed or d e v e l o p e d for use in the

c o l l e c t i o n of data were:
1.

Nine c r i t e r i o n - r e f e r e n c e d a c h i e vement m e a sure s to
asse ss m a t h e m a t i c a l c o m p e t e n c i e s on presc ribe d
ob jec tives

2.

(two para lle l

forms).

Te st 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

(two

p a r allel forms).
3.

R evised form of the D u t t o n A t t i t u d e Inventory,
F o r m C.

4.

A t t i t u d e s S ca l e s towa r d d i f f e r e n t aspects of
m a t h e m a t i c s d e v e l o p e d by the Internation al Study
of A c h i e v e m e n t

in Ma the mat ics .

D e v e l o p m e n t of C r i t e r i o n - R e f e r e n c e d
Achievement Measures
U n d e r l y i n g the conc ept of ac h i e v e m e n t m e a s u r e m e n t
is the not ion of a c o n t i n u u m of k n o w l e d g e acquisition
rangin g from no p r o f i c i e n c y at all to perfec t performance.
A stud e n t ' s a c h i e v e m e n t

level

falls at some point in this

c o n t i n u u m as ind i c a t e d by the be hav i o r s d i s p l a y e d during
testing.

The deg r e e to w h i c h his a c h i e v e m e n t res embles

des i r e d p e r f o r m a n c e at any s p e c i f i e d level is as sessed by
criterion-referenced measures
(28).

The term "criter ion, "

of a c h i e v e m e n t or p r o f i c i e n c y
w h e n used in this way,

does not

88

n e c e s sarily refer to final e n d - o f - c o u r s e beh avior.
Cr iterion

level can be,

and i n f o r m a t i v e e v a l u a t i o n should

be, e s t a b l i s h e d at c e r t a i n pointB

in i n s t r uction when it

is n e c e s s a r y to obta i n i n f o r mation as to the a d e q uacy of
a stu dent's p e r f o rmance w i t h re s p e c t

to some spe cified

standard a n d to know w h e t h e r l e a r n i n g is p r o m o t e d by the
p r e s e n t a t i o n of the seque n c e of m a t h e m a t i c a l

learnin g units.

A fairly s t r a i g h t f o r w a r d m e t h o d can be employe d to
test the e f f e c t i v e n e s s of the p r o p o s e d cur ric ulum .

This

consists in de t e r m i n i n g and a d m i n i s t e r i n g test s w h i c h have
been specif i c a l l y c o n s t r u c t e d to y i e l d i n f o r m a t i o n on the
a c h i e v e m e n t of the s t u d e n t s on e a c h learning un it w i t h i n the
curriculum.

The d a t a f r o m such t e s t s are the n analyze d to

reveal the eff ect.of the c u r r i c u l u m on the stu dents w h o have
been e x p o s e d to the i n s t r u c t i o n iden t i f i e d by the learning
units

(12).
In or der to ass e s s the e f f e c t i v e n e s s

of the m a t h e ­

matics c o m p o n e n t of the e x p e r i m e n t a l pr o g r a m on the p r o s p e c ­
tive e l e m e n t a r y te achers p a r t i c i p a t i n g in the program,

it

was t h e r e f o r e n e c e s s a r y to d e v e l o p a series of criteri onre fer enced 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
preser vic e

teacher c o u l d or coul d not ex hib it

implied by the p r e s c r i b e d o b j e c t i v e s

the c o mpetency

in each learni ng unit.

It was a l s o ess enti al to d evelop two e q u i v a l e n t

forms for

each tes t in order to ass ess the e n t e r i n g b e h a v i o r s and the

89

terminal b e h a v i o r s of the p r e s e r v i c e t e a c h e r toward

the

p r e s c r i b e d o b j e c t i v e s wit h i n e a c h l e a r n i n g unit.
D e v e l o p m e n t of the Test
In s t r u m e n t s
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 gain d e e p e r
in sigh t on the m e t h o d o l o g y of c o n s t r u c t i n g good tests.
M u c h of the theo r y of a c h i e v e m e n t t e s t i n g was o u t l i n e d
in the m i l e s t o n e volume E d u c a t i o n a l M e a s u r e m e n t
Lind quist,

1951),

(ed.

in which L i n d q u i s t rec o m m e n d s the

by
follow­

ing steps in th e p r e p a r ation of an e d u c a t i o n a l a c h i e v e m e n t
test:

(1) p l a n n i n g the test,

(2) w r i t i n g

the test items,

(3) tr ying out the test form a n d a s s e m b l i n g the finis h e d
test a f te r

tryout,

(4) p r e p a r i n g the d i r e c t i o n s

i steri ng and B c o r i n g the test,

an d

for a d m i n ­

(5) r e p r o ducing

the test

(19:119).
In this study,

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

test

for the s e q u e n c e of nine l e a r n i n g units that make up the
m a t h e m a t i c s c u r r i c u l u m for the f i r s t - y e a r trial i m p l e m e n ­
t at i o n of the e x p e r i m e n t a l

t e a c h e r e d u c a t i o n p r o g r a m under

investigation:
1.

I d e n t i f y i n g the ob jectives
T h e s p e c i f i c state men ts

t h a t the test is to measure.
of m a t h e m a t i c a l o b j e c t i v e s

for e a c h l e a r n i n g unit as s p e c i f i e d by the p r o g r a m
developers were

ident ifi ed a nd listed.

These o b j e c t i v e s

90

are 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

that served

as the c r i t e r i o n - r e f e r e n c e for the a c h i e v e m e n t tests.

2.

D e v e l o p i n g t he test instrument.
L i n d q u i s t made s p e c i f i c s u g g e stions
items

for a c h i e v e m e n t t esti ng

(19).

for w r i t i n g

The follo w i n g w e r e

includ ed a m o n g the list of s u g g e s t i o n s he made:
a.

Expre ss th e items as c l e a r l y as p o s si ble

b.

Choo se w o r d s that have p r e c i s e m e a n i n g w h e r e v e r
possible

c.

Av oid c o m p l e x or a wkward w o r d a r r a n g e m e n t s

d.

Include al l q u a l i f i c a t i o n s n e e d e d to pr o v i d e a

e.

r e a s o n a b l e basis

for r e s p o n s e s election

Avoid i r r e l e v a n t

inaccuracies

in any part of the

items
f.

Avoid i r r e l e v a n t clues to the c o r r e c t resp o n s e s

g.

Avoid i r r e l e v a n t

sour ces of d i f f i c u l t i e s

There a r e many forms of test
such as essay,

true-false,

m u l t i p l e choice.

items in g e n e r a l use,

s h o r t answer,

ma tc hin g,

M o s t t esting a u t h o r i t i e s

and

indicate that

the need for test item s to be o b j e c t i v e l y and e f f i c i e n t l y
s c o r e d can b e best a t t a i n e d t h r o u g h the u t i l i z a t i o n of
m u l t i p l e - c h o i c e type of items.
format,

The m u l t i p l e - c h o i c e

in w h i c h the answ e r c h o i c e s are s u p p l i e d and

the student w o u l d c h o o s e the b e s t or c o r r e c t answ e r

is

91

the m o s t g e n e r a l l y app l i c a b l e
a c h i e v e m e n t tests

(19) .

for m a t h e m a t i c s

The m u l t i p l e - c h o i c e type of

items w e r e used e x t e n s i v e l y in test d e v e l o p m e n t
this study;

however,

for

the invest i g a t o r rec o g n i z e d that,

in m a n y m a t h e m a t i c a l situa tion s,
(rather than re c o g n i z i n g it)

p r o d u c i n g the ans wer

is an e s s e n t i a l part of

the abili ty b e in g tested.
Among the specific p r i n c i p l e s suggested by N o l l

(25)

for the c o n s t r u c t i o n of m u l t i p l e - c h o i c e type test items
were:
a.

All o ptions should be p o s s i b l e and p l a u s i b l e answers

b.

Irrele van t g r a m m a t i c a l clue s sho uld be a voided

c.

The s t e m should not be loaded d o w n w i t h

i r r elevant

ma ter i a l s
d.

3.

The nu mber of ch o i c e s sh oul d be at least four.

P r e p a r i n g the test items.
O n c e the m a t h e m a t i c a l 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 ­
ing unit had b e e n spec i f i e d and the plans
had b e e n deter min ed,

for the test

the p r e p a r a t i o n of a supply of test

e x e r c i s e s that c o n f o r m e d to the s p e c i f i c a t i o n s
above was

initiated.

listed

Th e i n v e s t i g a t o r u t i lized the

foll o w i n g sourc es to ass i s t in the c o n s t r u c t i o n and
s e l e c t i o n of test items:
a.

Test e x e r c i s e s

from p r e vious m a t h e m a t i c s co n t e n t

courses at M i c h i g a n Stat e U n i v e r s i t y

201)

(Mathematics

92

b.

Chapter exercises
mathematics

c.

fro m recent te xt book s on

for e l e m e n t a r y teach e r B

T e s t s d e v e l o p e d by r e c e n t s tudies

in m a t h e m a t i c s

education.

Most

test items,

however,

i n vestigato r w i t h a s s i s t a n c e

were d e v e l o p e d by the
from m a t h e m a t i c s

at M i c h i g a n St ate Uni ver sity.

When

su ffi cient number of items had been

educators

it was felt that a
f o r m u l a t e d for e a c h

sp ecif ied m a t h e m a t i c a l o b j e c t i v e w i t h i n the l e a rning
units,

a t t e n t i o n w a s turned

the b e s t

to the p r o b l e m of

items w h i c h can be a s s e m b l e d

lent forms.

were prepared:

E a c h test w a s

the

to c o n t a i n ten

i t e m s 1 (an item c o u l d c o n t a i n one o r more

questions).
for e a c h

tw o e q u i v a l e n t tests

on e was to serve as a pre-test;

as a post-test.

or e l e v e n

into t w o e q u i v a ­

For each of the nine m a t h e m a t i c a l topics

com p r i s i n g the f i rst year content,

second,

sele c t i n g

A f t e r the poo l of test

learning unit,

t h e y w e r e r e v i e w e d by

uals s e n s i t i v e to com mon e d i t o r i a l
exercises.

items w e r e w r i t t e n
individ­

s h o r t c o m i n g s of test

Th e i tem s we re a l s o c h e c k e d for m a t h e m a t i c a l

c o r r e c t n e s s and p r e c i s i o n of s t a t e m e n t by i n d e p e n d e n t
m a t h e m a t i c a l educa tor s,

1E x c e p t for the test for M a t h e m a t i c a l Syste ms, in
w h i c h five items w e r e in cluded d u e to t h e l i m i t a t i o n s of
i n s t r u c t i o n a l per iod and t esting time.

93

For each l e a r n i n g u n i t , two forms of the test w e r e
assembled

f r o m the test

items and w e r e reviewed by the

g r o u p of e d u c a t o r s and teacher s w h o d e v e l o p e d and
d e s i g n e d the a c t i v i t i e s
unit.

for that p a r t i c u l a r

They w e r e a s k e d to re vie w the test

d e t e r m i n e w h e t h e r each
the o b j e c t i v e

learning

items and

item was a vali d a sse s s m e n t of

it p u r p o r t e d to evaluate.

C o m m e n t s and

s u g g e stions m a d e by the unit d e s i g n e r s were u t i lized

in

r e v i s i n g and

It

for r e p l a c i n g items on the tes t forms.

mu s t be n o ted

that in s electing the items for the final

forms of the final tests,

an a t t e m p t wa s m a d e to sa mple

the b e h a v i o r s under c e r t a i n object ive s.

For example,

the i n s t r u c t i o n a l o b j e c t i v e was sta t e d as:
. . . be a b l e to add two numerals
ten,"

"the student

in base o t h e r than

it w o u l d hav e b e e n i m p r a ctical to inclu de items

that a s s e s s e d the s t u d e n t a bility to add n u m e r a l s
bases
sam ple

if

2, 3, 4,

5,

...

etc.;

in

it w a s more a p p r o p r i a t e to

two or three b a s e s and w r i t e items

for the b a ses

s e l e c t e d .2

4.

P r e p a r i n g the d i r e c t i o n s

for a d m i n i s t e r i n g and sco rin g

the tests.
S inc e the

test c o n s t r u c t o r a d m i n i s t e r e d the tests,

it w a s not n e c e s s a r y to p r e p a r e d e t a i l e d d i r e c t i o n s

for

2A c c o r d i n g to Ra jar atnam, C ronbach, and Glas e r (110),
it is p o s s i b l e to g e n e r a l i z e from such s e l e c t i o n and still
attain p r e d i c t i v e v a l i d i t y for the c u r r i c u l a r obje cti ve.

94

the test examiner.
however,

On the front p a g e of eac h test,

dir e c t i o n s w e r e g i v e n for ta king the test

(due to the nat ure of the test items, most w e r e
explanatory) .

The time allo c a t e d

self-

for testing was

a p p r o x i m a t e l y 30 mi n u t e s but the stu de nts w h o needed
more time were al way s a llowed to c o m plete t h e i r test.
In scoring the test,
question,
answer;

if an item conta i n e d only one

ten poi nts w e r e a llowed

if an item cont a i n e d

for the c o r r e c t

two qu estions,

poi nts w e r e g i ven for each c orrect answer,
p art ial credit wa s a llowed

then five
etc.

No

for incom ple te answers.

The score of a studen t on each test was d e t e r m i n e d in
percentages.

5.

A d m i n i s t e r i n g the tests.
A f t e r the two parallel

forms of the tests

learning unit w e r e d e t e r mined,

for each

copies w e r e p r e p a r e d for

use w i t h the e x p e r i m e n t a l group.

F or each learning unit,

the p r e - t e s t wa s a d m i n i s t e r e d prior

to instruction,

and

the p o s t - t e s t wa s a d m i n i s t e r e d one w e e k af ter i n s t r u c ­
tion.

Al l test ing w a s supervised by the investigator.

When a student wa s abse nt d u r i n g the p r e - t e s t period,
he w a s asked to take the test b e f o r e st arting on the
activities

for that unit.

95

Evaluation of 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 vemen t Me asu res
D e s c r i b e d above are the steps

leading to the

d e v e l opmen t of the set of c r i t e r i o n - r e f e r e n c e d m a t h e m a t i c s
achi evement tests used in this study.

It is up on the

soundness and a p p r o p r i a t e n e s s of t h e s e p r o c e d u r e s that the
c l a i m of v a l i d i t y of the in str uments must p r i m a r i l y rest.
However,

s t a t i s tical e v i d e n c e

is c e n t r a l to e s t a b l i s h i n g

the re l i a b i l i t y of the m e a s u r e s and doe s ha v e
ment ary v a l u e

some s u p p l e ­

in a t t e s t i n g to their validity.

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 m e a s u r e s are
val idated p r i m a r i l y in t e r m s of the a d e q u a c y w i t h w h i c h
they re pre sen t

the criter ion ;

approaches are best s u i t e d

therefore,

to such tests

content validity
(28:29).

The

inherent m e t h o d by w h i c h the set of tests w e r e d e v e l o p e d
assu red c o n t e n t validity,

since the test items,

in the

judgment of the team of m a t h e m a t i c s e d u c a t o r s w h o d e v e l o p e d
and d e s igned the learni ng units,

di d

in fact r eflect the

specific o b j e c t i v e s w i t h i n the m a t h e m a t i c a l c o n t e n t of that
unit.
R e l i a b i l i t y .— S i n c e each te s t is c o n s t r u c t e d to
assess the i n s t r u c t i o n a l o b j e c t i v e s w i t h i n a specified
topic,

it is neces s a r y to e s t i m a t e th e r e l i a b i l i t y of e a c h

test inde p e n d e n t l y

(28:28).

S t u d e n t s in t h re e
course

(Education 325E)

sections of the r e g u l a r m e t h o d s

w e r e made a v a i l a b l e to test the

96

r e l i a bility of the pre- and p o s t - c r i t e r i o n m e a s u r e
a c h i e v e m e n t tests.
20 stude nt s

T h e r e wer e r e s p e c t i v e l y

19,

17,

and

in these c lasses and the inves t i g a t o r was

a l l o w e d a p p r o x i m a t e l y one hour and a half for t esting
purposes.

This

implied

that e a c h

one pre- and on e p o s t - t e s t
Therefore,
of tests

student cou ld co mp lete

in the set period,

but not more.

the i n v e s t i g a t o r ha d the ch oic e of g i v i n g one set

(pre- and post-test)

to a sample of six students,

or to sample the test items and give the s ampled tests to a
la rger number of students.*
After consultation with
able

in m e a s u r e m e n t theory,

select a 5-it e m sample
tests

it w a s d e c i d e d to ran dom ly

from each of the nine pre- and p o s t ­

(about 50 p e r c e n t ) .

par t i c u l a r

faculty m e m b e r s k n o w l e d g e ­

When th e s e l e c t i o n of these

items was completed,

thre e t h i r t y - i t e m tests

w e r e assembled:
1.

Test

I contained

five items

from pre- and five items

from p o s t - t e s t s on Measureme nt,

N u m e r ation,

and Set s

and Set R e l a t i o n s .
2.

Test
items

II c o n t a i n e d

five items from pre- and

from pos t - t e s t s on W h o l e Numbers,

five

Fractions,

and Decimals.

*Cook and S t u f f l e b e a m (44) and o t h e r s d e m o n s t r a t e d
e m p i r i c a l l y t h a t gro up p e r f o r m a n c e is m o r e e f f i c i e n t l y
m e a s u r e d u s i n g small subse ts of items d i s t r i b u t e d among
large num be rs of s t u d e n t s than v i c e versa.

97

3.

Test III contai ned

five items

f r o m pre- and five

items from p o s t - t e s t s on R elations and Functions,
Pr o b a b i l i t y and Stat ist ics ,

and M a t h e m a t i c a l

Systems.

Copies of these tests we r e r a n d o m l y d i s t r i b u t e d
to the

56 students

in the t hree sectio ns of E d u c a t i o n

(Methods of T e a ching E l e m e n t a r y School M a t h e m a t i c s ) .
on the s t a t i stical results of
estim ate s

for each

Est ima tes
sampled

B ased

reliability

test w e r e obtained.
of the r e l i a bility of each of the item-

tests w e r e c a l c u l a t e d using the Hoyt Reliability

Co effi c i e n t s
techn i que
statist ics

(19:570)

t h r o u g h an a n a lysis of v a r i a n c e

(see A p p e n d i x Q ) .

Tables

26-4 3 cont ain the

for the analysis of va riance

The S p e a r m a n - B r o w n

for each test.

formula w as app lie d to the Hoyt R e l i ­

ability coeffic ient s
Table 2

these teBts,

325E

to o b t a i n the total

(see A p p e n d i x M)

show s

test reliability.

the re s u l t s o b t a i n e d

for

each test from the s t a t i s t i c a l pro c e d u r e s d e s c r i b e d above.
The re liability c o e f f i c i e n t s
from a low of 0.77
high o f

for the tests va ried

for the p o s t - t e s t on M e a s u r e m e n t to a

0.93 for the p r e - t e s t on R e l a t i o n s and Functions.

These 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 acce p t a b l e
c r i t e r i o n - r e f e r e n c e d test

(12).

for a

98

Table 2
Reliability Coefficients for Pre- and Post-CriterionReferenced Achievement Tests

Pre-Test
(1)
(2)

Tests

Post-TeBt
(1)
(2)

Measurement

.6669

.8144

.6317

.7743

Numeration Systems

.7981

.8877

.7831

.8784

Sets and Set Relations

.8038

.8912

.7920

.8839

Whole Numbers

.6945

.8197

.7161

.8346

Fractions

.7212

.8380

.6840

.8124

Decimals

.7680

.8688

.7552

.8605

Relations and Functions

.8702

.9306

.8376

.9116

Probability and Statistics

.8412

.9138

.8041

.8914

Mathematical Systems

.8470

.8470

.8244

.8244

£

(1) Hoyt Reliability coefficients obtained from 50 percent
item-sampled test.
b

(2) Reliability coefficients of total test after applying the
Spearman-Brown formula to Hoyt Reliability coefficients.

R
R
R

tt

-

2R _
st
1 + R
st

= Reliability of total test.
St

= Reliability of sampled test.

E q u i v a l e n c y of the two forms of pre-

and p o s t -

tests .— The p r o b l e m of p r e p a r i n g t ruly e q u i v a l e n t forms
of a tes t is#

a c c o r d i n g to T h o r n d i k e

(19:575):

in the logic and p r a c t i c e of test c o n s truction.

"a p r o b l e m
. . . The

best g u a r a n t e e of e q u i v a l e n c e for two test forms w o u l d seem
to be that a c o m p l e t e a n d d e t aile d set of s p e c i f i c a t i o n s

for

99

the test be prepared in advance of any final test
construction."

a further check on the degree of eq uivalency

wa s made by examining cor re lati on coe ffic ien ts between the
two test forms.

Table 3 shows the Pea rson-mom ent c o r r e l a ­

tion coeffi cien ts obtai ned

from the test results of the

students enrolled in the regul ar m ethods course, Educ ati on
325E

(Methods of Teac hin g Elem enta ry School Mathematics).
T h e cor relation co eff icie nts between pre- and

post-test scores varied from a low of
Measurement to a high of
Systems.

It was noted,

of Measurement,
(see Table

the

.65 for the test on

.90 for the test of Ma them atical
however,

that except for the test

lowest co r r e l a t i o n coeffi cie nt was

.77

3).

Table 3
Correlation Coefficients Between Pre- and Post-Test Scores of
the Students in Regular Methods Course (Education 325E)
on Item-Sampled Criterion-Referenced Achievement

N

Correlation Coefficients

Measurement

19

.6452

Numeration

19

.8035

Sets and Set Relations

19

.0173

Whole Numbers

17

.7944

Fractions

17

.7781

Decimals

17

.B232

Relations and Functions

20

.0223

Statistics and Probability

20

.8615

Mathematical Systems

20

.9026

Tests

100

Selec tio n of a Test of Mathe m a t i c a l
Under s t a n d i n g
This phase of the st udy began by searching
w e l l - d o c u m e n t e d in stru ment
standing.

for a

for m e a s u r i n g m a t h e m a t i c a l u n d e r ­

It was h o p e d to find a standar dize d ins t r u m e n t

that w o u l d test the m a t h e m a t i c a l topics covered in the
recommended conten t
teachers.

for prospective ele m e n t a r y school

It was a l s o hoped to find a test w i t h two e q u i v ­

alent forms to m i n i m i z e

the testing effect.

search of the lit e r a t u r e on the subject,

A f t e r careful

the i nves t i g a t o r

came ac ros s an instru ment d e s i g n e d by M i l d r e d J.

Dosse tt as

part of her d o c t o r a l d i s s e r t a t i o n at M i c h i g a n S t a t e U n i v e r ­
sity in 1964. “

The test was d e e m e d m o s t a p p r o p r i a t e for the

purpose of this i n v e s t i g a t i o n since the test it ems covered
m a t hematical topics r e c o m m e n d e d by p r o f e s s i o n a l an d advi sor y
groups

in m a t h e m a t i c s education.

Per mission was g ranted by

the auth o r to use the test for the pr e s e n t study.
D ossett's

in strument entitl ed

matical U n d e r s t a n d i n g "

"Test of B a s i c M a t h e ­

had a reliability c o e f f i c i e n t of 0.87

obtained by c o r r e l a t i n g the scores made by 50 c o l l e g e s t u ­
dents on the two e q u i v a l e n t

forms of the test.

Equivalency

of the two forms was d e t e r m i n e d by us ing a t- tes t sug gest ed
by McNemar.

The t - v a l u e o b t a i n e d indi cated no significant

“M i l d r e d J. Dossett, "An A n a l y s i s of the E f f e c t i v e ­
ness of the W o r k s h o p as an In -Servi ce M e a n s for M a t h e m a t i c a l
Understanding of E l e m e n t a r y Scho o l Teach e r s " (unpublished
doctoral d issertation, M i c h i g a n State Uni versity, 1964).

101

d i f f e r e n c e s b etween the scores on the two forms of the test
w h e n a d m i n i s t e r e d to the 50 c o l l e g e students.
Se lection of an A t t i t u d e Inventory
The

"Ar ith metic A t t i t u d e Inventory,"

an a t t i t u d e

scale d e v e l o p e d by W i l b u r Dutt o n at the U n i v e r s i t y of
California,

w a s used in this study

(48).

For this scale,

Dutton u t i l i z e d a t echnique p e r f e c t e d by Thurstone and
C hav e

(48) .

statements

He first sele cted a large number of w r i t t e n

reg arding a t t i t u d e s towa rd ari t h m e t i c o b t a i n e d

from papers of six h u n d r e d u n i v e r s i t y students over a period
of five years.

The s t a tements w e r e sorted by judges using a

scale of o n e to eleven
favorable).

(extremely u n f a v o r a b l e to ex tr emely

Th e p r o p o r t i o n of judges w h o pla ced each

statement in the d i f f e r e n t cate g o r i e s c o n s t ituted the basic
dat a for c o m p u t i n g the scale val ueB of the statements.
instrument wa s
dents.

The

used w i t h over two h u n d r e d e i g h t y - n i n e s t u­

A rel i a b i l i t y of

retest p r o c e d u r e s

.94 was o b t a i n e d through test-

(49).

O n the att itud e

instrument,

the fifteen items have

values that rang e from 1.0 to 10.5 r e p r e s e n t i n g extr e m e l y
negative to e x t r e m e l y p o s i t i v e at titudes.

The i n dividual

score is the aver age sca le value of the statem ent s w h i c h
the individual

checked.

102

C o n s t r u c t i o n of S tudent Q u e s t i o n n a i r e
The

i n v e s t i g a t o r sought to obtain f r o m the students

in the e x p e r i m e n t a l g r o u p r e a c t i o n s to the e x p e r i m e n t a l
p roc e d u r e s of the m a t h e m a t i c s curriculum.

An eleven-item

q u e s t i o n n a i r e was d e v e l o p e d for the purpos e of de t e r m i n i n g
w h a t the student s t h o u g h t or felt at the e n d of the school
yea r toward par t i c u l a r a spects of the m a t h e m a t i c s c u r r i c u l u m
that they h a d e n c o u n t e r e d dur i n g the first y e a r trial i m p l e ­
m e n t a t i o n of the e x p e r i m e n t a l program.
of the q u e s t i o n n a i r e w e r e

disagree).

agree,

first ten items

sta t e m e n t s each r e l a t i n g to one

par t i c u l a r asp e ct of the p r o g r a m w i t h
(strongly agree,

The

undeci ded,

five s c a l e d - r e s p o n s e s

disagree,

and st rongl y

The e l e v e n t h q u e s t i o n wa s a free re sponse q u e s ­

tion that elicit ed the students'

recommendations

of im pro v i n g the i n t e g r a t e d c o n t e n t - m e t h o d s
of the q u e s t i o n n a i r e is included

S t a t i s t i c a l Pr ocedu res
A nalys is of Data

course.

A copy

in A p p e n d i x E.

for the

To ana ly ze the d a t a c o l l e c t e d dur i n g
inves t i g a t o r selected s everal
purposes of clar i f y i n g

for m ethods

the study,

s t a t i sti cal p r o c e d u r e s

the

for

some a s p e c t s of the s t u dy and to

test the hypo t h e s e s s t a t e d

in C h a p t e r I.

A o n e - w a y m u l t i v a r i a t e a n a lysis of v a r i 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

(35:332)

anal yzi ng the data r e l e v a n t to th e

was s e l e c t e d for use in

testing

for si gnificant

103

d i f f e r e n c e s b e t w e e n the p o s t - t e s t sco r e s of the e x p e r i m e n t a l
grou p on th e ni ne c r i t e r i o n - r e f e r e n c e d tests and
pre-test scores

(Hypothesis A l ) , (2)

(1)

their

the m a s t e r y level of

80 percent or high e r c o m p l e t i o n of the p o s t - t e s t

items

(Hypothesis A 2 ) .
To test the h y p o t h e s e s rel a t e d to the ef fec t of the
m a t h e m a t i c s 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 s t a n d ­
ing

(Hypothesis Bl)

esis B2),

and a ttitudes to war d a r i t h m e t i c

the a n a l y s i s w as done in tw o parts.

The p r e - t e s t

scores w e r e s u b t r a c t e d fr om the p o s t - t e s t sc ore s
individual,

r e s u l t i n g in d i f f e r e n c e s scores.

ences w e r e a n a l y z e d u sing t-t est

(Hypoth­

for each

These d i f f e r ­

for the s i g n i f i c a n c e of

difference between correlated means

(?) .

In o r d e r to ass e s s the r e l a t i v e p e r f o r m a n c e of the
ex p e r i m e n t a l g r o u p as c o m p a r e d w i t h a g r o u p of p r o s p e c t i v e
ele m e n t a r y t e a chers in the re gular teac her e d u c a t i o n p r o ­
gram on the b as ic 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 itud es tow ard a r i t h m e t i c

(Hypothesis Cl)

(Hypothesis C 2 ) , the

analysis of c o v a r i a n c e t e c h n i q u e w a s u t i l i z e d as sug gested
by Winer

(35:753).

The r e s p e c t i v e p r e - t e s t scores w e r e

cov ariate measures.
To study the d e g r e e of r e l a t i o n s h i p be t w e e n
criteria u n d e r inve stigation,

the p r o d u c t - m o m e n t c o r r e l a t i o n

co e fficients b e t w e e n all v a r i a b l e s m e a s u r e d
were o b t a i n e d

selec ted

for the e x p e r i m e n t a l group.

in this study
Th e r esu l t i n g

104

c o r r e latio n m a t r i x

(Appendix J) w a s u t i l i z e d to test for

signi fic ant c o r r e l a t i o n s be t w e e n se lecte d variables.
To comp are the enterin g c ognitive and affectiv e
be haviors to war d m a t h e m a t i c s of the e x p e r i m e n t a l g r oup w i t h
those of o t h e r f r e s h m e n groups,

the D unnett t-t e s t

(35) wa s

used to d e t e r m i n e w h e t h e r s i g n i fic ant d i f f e r e n c e s ex i s t e d
bet ween the e x p e r i m e n t a l gr oup and each of the freshmen
co mpa rison groups on tests of m a t h e m a t i c s and a r i t h m e t i c
achievement,

and tests of a t t i t u d e toward d i f f e r e n t aspec ts

of m a t h e m a t i c s and m a t h e m a t i c s

le arning in general.

Significance Le vel C h o s e n
Th e

5 p e r c e n t level of a c c e p t a n c e or reje c t i o n of

statistical h y p o t h e s e s being i n v e s t i g a t e d w a s

sele cted as

be ing s u f f i c i e n t l y rigor o u s for the c o n d i t i o n s 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 hund red that the o b s e r v e d d i f f e r e n c e c o u l d be attr i b u t e d
to chance,

the r e s e a r c h hypo t h e s i s was accepted;

if the

observ ed d i f f e r e n c e wa s of such m a g n i t u d e t h a t it m i ght
ar ise mo re than five times
op eration of cha n c e
rejected.

factor,

in on e hu n d r e d t h r o u g h the
the r e s e a r c h h y p o t h e s i s was

105

Su m m a r y

T h i s c h a p t e r d e s c r i b e d the m a t h e m a t i c s compo n e n t
of a n e w e x p e r i m e n t a l te a c h e r educa t i o n p r o g r a m at M i c h i g a n
State U n i v e r s i t y an d the p r o c e d u r e s

followe d for its

assessment.
The forma t i v e e v a l u a t i o n of the m a t h e m a t i c s c o m p o ­
nent of the experi m e n t a l e l e m e n t a r y teacher e d u c a t i o n p r o ­
gram at M i c h i g a n Stat e U n i v e r s i t y took p lace d ur i n g the
academic year 1971-1972.

The t h i r t y - e i g h t st udents

forming

the fi rst g rou p of p r o s p e c t i v e ele m e n t a r y t e a chers w h o
pa rtic i p a t e d in th e e x p e r i m e n t a l p r o g r a m w e r e u t i l i z e d
this ev aluation.

for

In a d d i t i o n , samp les of o t h e r student

groups w e r e used for c o m p a r i s o n purposes.
Th e f o l l o w i n g steps w e r e

followed for the e v a l u a t i o n

of the e x p e r i m e n t a l program:
1.

A s s e s s m e n t of the m a t h e m a t i c s c o m p o n e n t of the
p r o g r a m b y mea ns of a c r i t e r i o n - r e f e r e n c e d list
developed

a c c o r d i n g to top i c s s u g g e s t e d by s p e c i a l ­

ists

the p r e p a r a t i o n of e l e m e n t a r y school

for

teachers.
2.

P a r t i c i p a t i o n in the d e v e l o p m e n t of an i n t egrated
c o n t e n t - m e t h o d s c o u r s e in m a t h e m a t i c s w i t h an
i n t e r d i s c i p l i n a r y t e a m of p r o f e s s i o n a l people
i nvolv ed in education.

106

3.

I m p l e m e n t a t i o n of the i n t e g r a t e d c o n t e n t - m e t h o d s
c o u r s e w i t h the t h i r t y - e i g h t stude nts p a r t i c i p a t i n g
in th e e x p e r i m e n t a l program.

4.

A s s e s s m e n t of the c o n t e n t of the int egrated cont ent m e t h o d s c o u r s e by means of the f o l l o w i n g instruments:
a.

Nine criterion-referenced achievement measures
to as sess m a t h e m a t i c a l c o m p e t e n c i e s on p r e ­
s c r i b e d obj ect ive s.

b.

T e s t of B a s i c M a t h e m a t i c a l Unde r s t a n d i n g .

c.

Attitude

Inventory an d A t t i t u d e Scales.

The d e v e l o p m e n t and u se of the test i n s t r uments
the s t a t i s t i c a l p r o c e d u r e s used
were described

for the a n a l y s i s of dat a

in the last s e c t i o n of thi s chapter.

Re s u l t s o b t a i n e d

from the d i f f e r e n t analyses

their i n t e r p r e t a t i o n a r e d i s c u s s e d in the
chapter.

as w e l l as

following

and

CHAPTER IV
A N A L Y S I S OF D A T A AN D R E S U L T S

Th is chap ter present s a summary of the data
co llected d u r i n g this inve stig ati on,

the a n a l y s i s of data,

and results b a s e d on this analysis.

It c o n s i s t s of seven

sections:

(1) analy sis of the m a t h e m a t i c a l

learning units,

content of the

(2) c o m p a r i s o n of the e x p e r i m e n t a l g r o u p and

other freshm an groups on c o g n i t i v e and a f f e c t i v e beha viors
toward mathemati cs,

(3) e v a l u a t i o n of the experi m e n t a l g roup

pe rformance on 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 me as ures ,
(4) effect of the e x p e r i m e n t a l p r o g r a m on th e basic m a t h e ­
matic al u n d e r s t a n d i n g s and a t t i t u d e s tow a r d ma thematics,

(5)

co mpa rison of the e x p e r i m e n t a l g r o u p with a regular e l e m e n ­
tary teacher e d u c a t i o n group on m a t h e m a t i c a l u n derstandings
and att itudes

toward m a thematics,

(6) c o r r e l a t i o n data,

and

(7) e v a l u a t i o n of s t u d e n t r e a c t i o n to the m a t h e m a t i c s c o m p o ­
nent of the e x p e r i m e n t a l

program.

conclude C h a p t e r IV.

107

A s u m m a r y of results

108

A n a l y s i s of the M a t h e m a t i c a l C o n t e n t
ol: the L e a r n i n g O n i t B

Most fu n d a m e n t a l
is the

to the use of

formative e v a l u a t i o n

s ele ction of a unit of learning.

W i t h i n a c o u r s e or

e d u c a t i o n p r o g r a m there are p a r t s or d i v i s i o n s whic h h a v e a
sep arable e xistence such that the y can,
purposes,
parts.

at least for ana lytic

be c o n s i d e r e d in r e l ative i s o l a t i o n

from o t h e r

While these pa rts ma y be i n t e r r e l a t e d in v a r i o u s

ways so that the learning
has c o n s e q u e n c e

(or level of learning)

for the l e a rning of others,

sible to c o n sider the parts s e p a r a t e l y

of o n e part

it is stil l p o s ­

(4).

In this study each of the nine m a t h e m a t i c a l
will be c o v e r e d in one
devot ed

learnin g unit.

to one m a t h e m a t i c a l topic,

Each

contains

co mpe tencies p r e s c r i b e d for th a t topic.
list of the m a t h e m a t i c a l c o m p e t e n c i e s

Th e

to pic s

learning unit,
the m a t h e m a t i c a l
fol lowing

is a

i n t r o d u c e d unde r each

of the n i n e le arning units.
1.

Measurement
1)

To learn the concep ts of vo lum e and area of v a r i o u s
g e o m e t r i c solids.

2)

To learn ho w to apply the c o n c e p t of s i m i l a r i t y
measurement.

3)

T o learn the m e t r i c s y s t e m of m e a s u r e m e n t and to be
able to c o n v e r t E n g l i s h to m e t r i c and vice versa.

4)

To learn the c o n c e p t s of r e l a t i v e e r r o r and th e
great e s t p o s s i b l e e r r o r in m e asurement .

to

109

2.

3.

5)

To learn that m e a s u r e m e n t of areas and volu m e is
approximate.

6)

To learn about angles,
ent geometric figures.

7)

To learn some b a s i c in for mati on a b o u t c o o rdinate
g e o m e t r y and map reading.

8)

To learn about linear m e a s u r e m e n t an d scaling.

9)

To be able to u tiliz e the acqu ired
p r o b l e m solving situations.

and sum of angles of d i f f e r ­

kno wle dge

in

Syste ms of Num e r a t i o n s
1)

To learn some m o t i v a t i o n and hi s t o r y behind
study of n u m e r a t i o n systems.

the

2)

To learn to interp ret a n u m e r a t i o n system u s in g
d i f f e r e n t symbols.

3)

To learn about the prop e r t i e s of p o s itional
of numeration.

systems

4)

To learn to w r i t e a numeral

in e x p a n d e d notation.

5)

To learn about posi t i o n a l
base o t her than ten.

6)

To learn to add, subtract, m u l t i p l y
nume ral s in base o t h e r than ten.

7)

To learn to c onvert nu merals in b a s e ten into
n u m erals in other bases and vice versa.

systems of num e r a t i o n

two or

wi th

more

8)

To learn to add an d su btract in b a s e twelve.

9)

To be abl e to app ly the a c q u i r e d kn ow ledge
p r o b l e m solving situations.

in

Sets and Set Re lations
1)

To learn to ide ntify e l e m e n t s of a set.

2)

To learn to ide ntify subs ets (proper and otherwise)
of a g iven set as we ll as w h e t h e r the subsets are
d i s j o i n t or intersecting.

110

3)

To

4)

To
learn to identify and d e s c r i b e the i nters e c t i o n
and/or union of s e t s .

5)

To

learn to identi fy the comp l e m e n t of a set.

6)

To

learn to identif y

7)

To
learn to u tilize Venn d i a g r a m s to de pi c t the
relat i o n s h i p be t w e e n two or mo re Bets.

finite and infin ite sets.

8)

To learn to describe, using set notation,
and/or i n ter section of two or more sets.

9)

To learn about the conc ept of greater than and
less tha n as they relate to set relation.

10)

4.

learn about equi v a l e n t and equal sets.

the union

To learn to u t i l i z e c orrect ly the symb o l i z a t i o n
common ly used to d e s c r i b e sets and set relations.

Th e Whole Nu mber S y s t e m
1)

To learn the d e f i n itions an d some pro p erties of the
whole number system.

2)

To learn and ap ply a formal d e f i n i t i o n of ad dition
of w h o l e numbers and the b a s i c pro perties of a d d i ­
tion such as closure, commutative, and a s s o c iative
properties.

3)

To learn about o r d e r relatio n

4)

To learn the b a s i c pr operties of the o p e r a t i o n of
m u l t i p l i c a t i o n such as the closure, commut ati ve,
a s s o c iative and d i s t r i b u t i v e properties.

5)

To learn about a d d i t i v e and m u l t i p l i c a t i v e identity.

6)

To learn about the o pe r a t i o n s of sub traction and
di v ision in r e l a t i o n to a d d i t i o n and mu lti plica tion.

7)

To learn and apply their k n o w l e d g e of the addition,
subtraction, m u l t i p l i c a t i o n and d i v i s i o n algorithms
for w h o l e numbers.

8)

To learn the d e f i n itions of prim e and co mpo site
numbers and d i v isors and to learn the F u n d a m e n t a l
T h e o r e m of Arithmetic, and the prime f a c t o r i z a t i o n
theorem.

for the w h o l e numbers.

9)

To learn a bou t nu mbe r pattern s by d e v e l o p i n g
formula s for sums of number sequences.

The Rational Numb e r S y s t e m s —
Fractions
1)

To learn some m o t i v a t i o n and h i s t o r y beh ind the
c o n s t r u c t i o n of fractions.

2)

To learn a formal d e f i n i t i o n of frac tions (as
or d e r e d pairs of w h o l e numbers b e l o n g i n g to the
same eq uiv a lence set).

3)

To learn the formal d e f i n i t i o n of a d d i t i o n and
m u l t i p l i c a t i o n of fractions.

4)

To learn

5)

To learn ho w to apply the ide nt i t y and inverse
pr o p e r t i e s of a d d i t i o n and m u l t i p l i c a t i o n of
fractions.

6)

T o learn some of the basic p r o p e r t i e s of the
r a t ional num bers such as the order, fractional
re pres entation , commuta tive , associat ive , and
d i s t r i b u t i v e properties.

7)

T o learn about the least comm o n d e n o m i n a t o r and the
g r e a t e s t c om m o n fac tor s and a p p l y i n g this kno wl e d g e
in ari t h m e t i c operations.

8)

To be able to apply the a c q u i r e d know l e d g e of
fr actions in p r o b l e m solving situations.

about s u b t r action and d i v i s i o n of fractions.

The Ra tional Num b e r S y s t e m -Decimals
1)

To learn some m o t i v a t i o n and h i s t o r y behind
c o n s t r u c t i o n of decimals.

2)

To learn a formal d e f i n i t i o n of d e c i m a l s in terms
of p l a c e -value structure, and apply this k n o w l e d g e
in decim al expansion.

3)

To learn a bou t the four b asi c a r i t h m e t i c o p e r a t i o n s
w i t h decimals.

112

7.

4)

To learn about s c i e n t i f i c n o t a t i o n and
application.

5)

To learn h o w to co n v e r t d e c i m a l s into fractions
and vice versa.

6)

Est imate sums, differen ces , p ro duct an d quoti e n t s
of two or more d e c i m a l n umbers to the nearest
specifi ed place.

7)

To learn a bout ra tes and percents and be able to
co nver t d e c i m a l s into p e r c e n t s and vice versa.

8)

To learn to co n v e r t d e c i m a l s w r i t t e n in base ot her
than 10 to their e q u i v a l e n t in base 10.

9)

To be able to appl y the a c q u i r e d k n o w l e d g e abou t
decimals in p r o b l e m solving situations.

Rela tions and F unct i o n s
1)

2)

8.

its

To learn the d e f i n i t i o n of the reflexive, symme t r i c
and tra n s i t i v e p r o p e r t i e s of a r e l a t i o n and to be
able to d e t e r m i n e w h e t h e r a r e l a t i o n p o s s e s s e s any
of them.
To learn the d e f i n i t i o n of e q u i v a l e n c e rel ati on on
a set and to be able to d e t e r m i n e if a given r e l a ­
tion is an e q u i v a l e n c e relation.

3)

To learn and apply a formal d e f i n i t i o n of a r e l ation
and to ident ify fu nction
as spe cia l relation.

4)

To learn and apply the d e f i n i t i o n s of domain,
and invers e of a r e l a t i o n and function.

5)

To

learn to plot a g rap h of a g iven relation.

6)

To

learn to sketch a n o n - l i n e a r

range

function.

St at is tic s and P r o b a b i l i t y
1)

To learn the d e f i n i t i o n of p r o b a b i l i t y of an event
in a sam ple space.

2)

To learn a b o u t p r o b a b i l i t y of i n d e p endent and
d e p e n d e n t events.

113

9.

3)

To learn to c ompute the p r o b a bilit y of o c c u r r e n c e
of at least on e event.

4)

To learn about some cor rect us age of sampling
procedures.

5)

To learn the d e f i n i t i o n s of and m e t h o d s of
c o m p u t a t i o n of basic d e s c r i p t i v e statis tical
data such as mean, mode, medi a n and range.

6)

To learn about the stat ist ica l m e a s u r e of v a r i a b i l ­
ity of data.

7)

To learn some b as ic i n f o r mat ion abou t stat ist ica l
inference.

8)

To be able to ap ply the acquir ed k n o w l e d g e about
s t a tistics and prob abilit y in p r o b l e m solving
situations.

Mathematical

Systems

1)

To learn the formal d e f i n i t i o n of a m a t h e m a t i c a l
s y s t e m and be able to identify e x a m p l e s of such
system.

2)

To learn the r udiments of clock arithmetic, the
d e f i n i t i o n of c o n g r u e n c e d e r i v e d fro m it, and to
o b s e r v e that c o n g r u e n c e is an e q u i v a l e n t relation.

3)

To learn the d e f i n i t i o n of a m a t h e m a t i c a l field
and some of the basic p r o perties of a field, and
be able to id entify example s of a field.

4)

To learn some of the b a s i c pro p e r t i e s of m u l t i ­
plicative and additive inverse of a field.

5)

To learn to c ompute the add ition and m u l t i p l i c a t i o n
table in d i f f e r e n t m o d u l a systems.

Findings
Item by item c o m p a r i s o n b etween the c r i terionrefe rence list and the topic listed above shows that the
following topics are not includ ed in the m a t h e m a t i c s c u r ­
r icu lum of the e x p e r i m e n t a l program:

114

1.

D i v isibility rules

2.

Exponents

3.

Real Numbe rs

4.

Square Roots

5.

Basic concepts of Geometry.

However,

as n oted earlier,

the real number system,

which includes studies of sq uare roots,

and g e o m e t r y will

be c o v e r e d in the second co urse duri n g the junior year.
The two topics

left

(exponent and d i v i s i b i l i t y rules)

are

usually cov ere d under the study of the w h o l e num b e r system
or m a t h e m a t i c a l systems.
should be i n c orporated

These topics could have b e e n and

into the first year course as they

are h i g h l y r elated and also h eavily emphas ize d in e l e m e n t a r y
ma t h e matics textbooks.

The p r o s p e c t i v e teacher c o u l d

learn

the u t i l i z a t i o n of m a n i p u l a t i v e m a t e r i a l s as a m e t h o d of
instructing these two topics to e l e m e n t a r y school children.
As a whole,

of the t h i r t y - t w o to pics included in the

c r i t e r i a - r e f e r e n c e list that w e r e app l i c a b l e to the subjects
covered

in the first year course,

percent).

It is noted,

however,

thirty were i n c luded

(94

that m a n y topics are

included in the e x p e r i m e n t a l p r o g r a m that are not incl uded
in the c r i t e r i o n - r e f e r e n c e list;

these topics m u s t be

included as they fac ilitate the d e v e l o p m e n t of the requir ed
topics.

For example,

le arnin g that m e a s u r e s of a r e a and

115

v o l u m e are a p p r o x i m a t e w i l l

facilitate u n d e r s t a n d i n g of the

co n cepts of r e l a t i v e e rror and g r e a t e s t p o s s i b l e error.
Based on the above,

it is c o n c l u d e d that the m a t h e ­

m a t i c a l c o m p e t e n c i e s p r e s c r i b e d by the experi m e n t a l p r o g r a m
are suf f i c i e n t in m e e t i n g the need of the fu t ure e l e mentary
schoo l teach er in arithmetic.

C o m p a r i s o n of the E x p e r i m e n t a l G r o u p
and b t h e r F r e s h m a n feroups on
^oqn 1 1 1vie an3 A f fective ~
*
Behaviors To ward
Mathematics
The t h i r t y - e i g h t
majo r s who p a r t i c i p a t e d
selecte d

fresh man e l e m e n t a r y educ a t i o n
in the e x p e r i m e n t a l p r o g r a m w e r e

from those stu den ts w h o v o l u n t e e r e d

It is reasonable,

therefore,

to participate.

to q u e s t i o n w h e t h e r these s t u ­

dents d i f f e r e d in t heir entry c h a r a c t e r i s t i c s

from the other

freshman s t u d e n t s at M i c h i g a n State Un ive rsity.
gator was p a r t i c u l a r l y i n teres ted

in k nowing

The i n v e s t i ­

if these s t u ­

dents who v o l u n t e e r e d and w e r e s e l e c t e d d i f f e r e d

from other

freshman st udents on their c o g n i t i v e and a f f e c t i v e behav i o r s
in mat hematics.
As p a r t of the o verall e v a l u a t i o n of the TTT p rogram
at M i c h i g a n S t a t e Unive rsi ty,

the e v a l u a t i o n t e a m select ed

three f i r s t - t e r m f r e shman gr oup s w i t h d e c l a r e d maj o r s

in:

(1) e l e m e n t a r y e d u c a t i o n — bu t di d not v o l u n t e e r for the
experim ent al program,

(2) m a t h e m a t i c s and s econdary

116

education,

and

(3) mathematics.

F r o m each of these groups,

a simple r a n d o m sample of stude nts was se lected and thoBe
who p a r t i c i p a t e d from each of the selec ted grou ps c o n s t i ­
tuted the three " c o m p arison groups."
In this inves tig ati on,
in det e r m i n i n g whether:

the rese a r c h e r wa s interested

(1) the e x p e r i m e n t a l group diffe red

in their initial b e h a v i o r s toward m a t h e m a t i c s

from other

freshman gr oups w i t h s imilar a c a d e m i c m a j o r - - e l e m e n t a r y
educa tio n majors,

(2) the e x p e r i m e n t a l g r o u p di ffered from

freshman gro ups w i t h d e c l a r e d inter est in m a t h e m a t i c s - m a t h e m a t i c s - s e c o n d a r y e d u c a t i o n m aj o r s and m athematics
majors.

Instrum e n t a t i o n
The

fol low in g i n s t r ument s w e r e a d m i n i s t e r e d du rin g

the fall t e r m of 1971 -19 72 to the e x p e r i m e n t a l grou p and
the three " c o m p arison groups":
1.

MSU b a s i c skill and p l a c e m e n t tests

in ar ithmetic

and math ema tics .
2.

Dut ton A t t i t u d e Scale.

3.

A t t i t u d e scales d e v e l o p e d by The International Study
of A c h i e v e m e n t

in

Mathematics.1

lA copy of this sc ale is to be found in A p p end ix D.

117

The MSU b a s i c skill and p l a c e m e n t tests are used
by the U n i v e r s i t y to assess e n t e r i n g fre shman ab ility in
mathematics.
The Dut t o n Att itude Sca le was d i s c u s s e d earlier.
The A t t i t u d e s Scales devel o p e d by The Inter n a t i o n a l
Study of A c h i e v e m e n t in Mathemati cs,

w e r e c o n s t ructed

to

measur e student at ti tude s toward:
1.

m a t h e m a t i c s as a process,

2.

d i f f i c u l t i e s of l e a rning math ematics,

3.

place of m a t h e m a t i c s

4.

school and school learning.

in society,

The c o e f f i c i e n t s of reprodu c i b i l i t y o b t a i n e d
Scale An alysis

for these scales ranged

from the Gu ttm an

from a low of

.88 to

a high of

.95 w h e n tested on A m e r i c a n p r e u n i v e r s i t y - y e a r

students.

These c o e f f i c i e n t s were c o n s i d e r e d a c c e p t a b l e

by G u ttman

(17:118).

Data An al ysis
Summary of stat istical data of the e x p e r i m e n t a l
group and the "com parison groups"
are shown in T a b l e

4.

on each test i n s t r u m e n t

Table 4
Means and Standard Deviations on Entry Data for the Experimental Group and
Three Freshman Comparison Groups

Group

Experimental
Group

Elementary
Education
Majors

Mathematics
Secondary
Education Majors

Mathematics
Majors

Estimate
of Pooled
Variance

1)" Mean
S.D.

32.41
4.37

30.83
5.85

36.56
2.68

37.00
2.75

16.35

2)

Mean
S.D.

15.68
6.08

13.36
6.56

23.74
4.50

25.18
4.77

30.02

3}

Mean
S.D.

5.57
1.61

5.85
1.92

7.96
0.81

7.43
1.90

2.72

4)

Mean
S.D.

6.92
2.IB

5.97
2.26

8.65
3.48

9.32
3.63

8.99

5)

Mean
S.D.

8.46
2.68

8.31
3.37

9.47

9.97

3.02

9.45
3.52

6)

Mean
S.D.

14.05
2.68

14.11
3.81

14.39
2.66

13.97
3.68

10.39

7)

Mean
S.D.

8.51
2.34

8.86
3.70

9.14
3.67

9.26
3.37

11.25

aThese numbers refer to the following:
1) = MSU Arithmetic Test
2) = MSU Mathematics Test
3) = Dutton Attitude Scale
4)
= Attitudes Toward Mathematics as a Process
5)
= Attitudes Toward Place of Mathematics in Society
6)
= Attitudes Toward School and School Learning
7)
= Attitudes Toward Difficulties of Learning Mathematics.

119

In order to d e t e r m i n e the signif i c a n c e of the
dif fe rence b e t w e e n the e x p e r i m e n t a l g r o u p and each of the
"com par iso n grou ps, "

the Du n n e t t t-test was u s e d . 1

To d e t e r m i n e w h e t h e r the o b s e r v e d t -ratio was
signi fic ant at the 0.05 level of confidence,
desig ned by D u n n e t t
Table

t-tables

(35:873) w e r e utilized.

5 shows the t-ratios o b t a i n e d from ap pl ying

Dunnett t-t est to the e n t r y data of the expe ri m e n t a l g roup
and the

"c omp aris on group s."

t-ratios

2The follo win g formula was used in c o m p u t i n g the
for e a c h test.
M
- M.
4- = -g*B
i___
/2 M S e r r / n

Where M eXp i s

the mea n sc ore of the e x p e r i m e n t a l group

is the mean s c ore of "c omp aris on group" i
h is the harm oni c mean, w h i c h is equal to:

4
l/n^ + l/n2 + l/n3 + 1/n^
and MS
e

is an unbi ase d e s t i m a t o r of the poo led va ria nce
(It is in fact the value of the me an squares
of e r r o r o b t a ined from w i t h i n - g r o u p data.
A p p e n d i x P incl ude s a s ummary of the analy sis
of v a r i a n c e for each test result.)

120

Table 5
t-Values for Mean Comparison of Experimental Group and the
Three "Comparison Groups" on Entry Characteristics

Elementary
Education
Majors

Mathematics
Secondary
Education Majors

Mathematics
Majors

l)a

1.71

4.51*

4.99*

2)

1. 50

5.20*

6.13*

3)

0.75

6.35*

4.95*

4)

1. 38

2. 50*

3.48*

5)

0,44

0.81

0.97

6)

0.29

1.94

1.90

7)

0.08

0.45

0.11

aThese numbers refer to the following:
1) - MSU Arithmetic Teat
2) - MSU Mathematics Test
3) ■ Dutton Attitude Scale
4)
= Attitudes Toward Mathematics as a Process
5)
■ Attitudes Toward Difficulties of Learning Mathematics
6)
« Attitudes Toward Place of Mathematics in Society
7)
- Attitudes Toward School and School Learning.
•Significant beyond the

.05 level.

121

Findings
1.

Th e M S U B asic Skill Test in A r i t h m e t i c .--This

test as sess ed the student k n o w l e d g e in a r i t h m e t i c in general.
The m e a n score of the e x p e r i m e n t a l group on this test was
32.41, w h i c h was not s i g n i f i c a n t l y d i f f e r e n t
score of the e l e m e n t a r y e d u c a t i o n majors

from the m e a n

(M = 3 0 . 8 3 ) .

How­

ever, w h e n the e x p e r i m e n t a l g r o u p was c o m p a r e d w i t h the
m a t h e m a t i c s - s e c o n d a r y e d u c a t i o n maj o r s
m athe mat ics m a j o r s

( M * 37.00),

(M = 36.56)

and w i t h

the t-rat ios w e r e highly

si gnificant in favor of the sec ondary and m a t h e m a t i c s majors
(p < .001).
2.

Th e MSU B asic Skill Test in M a t h e m a t i c s .— The

content of this test is d e s i g n e d to ass ess g eneral m a t h e ­
matical k n o w l e d g e w i t h e m p hasis on a lgebra and geometry.
The m e a n sc ore of the e x p e r i m e n t a l g r o u p w a s

15.68, w h i c h

was 2.32 hig her than the score of the e l e m e n t a r y educa t i o n
majors,

a d i f f e r e n c e not si g n i f i c a n t at the 0.05 level.

However,

the e x p e r i m e n t a l group scored s i g n i f i c a n t l y

than the m a t h e m a t i c s - s e c o n d a r y e d u c a t i o n m a j o r s
and the m a t h e m a t i c s m a j o r s

lower

( p < .001)

( p < .001).

On the e v i d e n c e p r o v ided by the tw o m e a s u r e s cited
above,

it is c o n c l u d e d that there are no si g n i f i c a n t d i f f e r ­

ences b e t w e e n the e x p e r i m e n t a l g r o u p and o t h e r
ele mentary e d u c a t i o n m a j o r s

freshman

on their c o g n i t i v e b e h a v i o r s

toward a r i t h m e t i c and ma thema tics.

122

The a r i t h m e t i c and m a t h e m a t i c a l k n o w l e d g e of the
students in the e x p e r i m e n t a l gr oup w h e n en tering college
was s i g n i ficantly lower than the k nowledge of students wit h
specified intere st in the subject

(the m a t h e m a t i c s - s e c o n d a r y

edu cat ion m a j o r s and the m a t h e m a t i c s m a j o r s ) .
3.
tude

A r i t h m e t i c A t t i t u d e S c a l e .--The D u t t o n A t t i ­

Scale

w as utili zed to assess

toward mat hem ati cs.

the students'

feelings

P o s s i b l e scores on this scale ra nge

from 1.0 to 10.5.
W h e n the mean score of the e x p e r i m e n t a l group
(M = 5.57) wa s com pared w i t h the m e a n score of the e l e mentary
edu cation m a j o r s group
(t = 0.75)

(M = 5.85),

was not s i g n i ficant at the 0.05 level.

exper ime nta l g r o u p scores,
lower

( p < .001)

however,

(M = 7.43).

w e r e s i g n i ficantly

(M = 7.96)

and th ose of the m a t h e m a t i c s

The r e l atively h i g h scores of these two

groups were e xpe ct ed since they ha v e an e x h i b i t e d
in the subject.

The

than those of the m a t h e m a t i c s - s e c o n d a r y

education m a j o r s
majors

the t-ratio obtai ned

On the o t her hand,

intere st

the e x pe rimental g roup

and the elem e n t a r y e d u c a t i o n ma jor s have r e l atively

low

scores w h e n c o m p a r e d w i t h the scores of third and fourth
year ele m e n t a r y education majo rs

in other studi es

(48,

49).

This ma y be du e to the fact that the two fr eshman groups
base their atti tudes

solely on the e x p erience of their

pre-c oll ege education,
in other studi es w e r e

w h i l e the at titudes of

those st udents

inf lue nce d by their c o l l e g e training.

123

4.

A t t i t u d e s Towa rd M a t h e m a t i c s as a Process

(8 i t e m s ) .— This

scale inquired about de gre e to w hich

math ematics is v i e w e d as a fixed and given,
times

(a low score),

for all

or as so me thin g that is developing,

and con stantly c h a nging
on this

once

(a hi gh s c o r e ) .

Pos s ible scores

scale rang e from 0 to 16.
A n a lysis of data per t a i n i n g to this scale reveal ed

no sign ifi can t d i f f e r e n c e b e t w e e n the m e a n score of the
ex peri men tal g r o u p

(M*=6.92)

elementary educa tion grou p

and the m e a n score of the

(M = 5.97).

experimental g r o u p had a s i g n i f i c a n t l y

However,

the

lower attitud e towa rd

m athem ati cs as a pro ce ss than di d the seco nda ry ed uca tio n
majors

(M**8.65)
5.

M athe mat ics

and the m a t h e m a t i c s maj o r s

(M=9.32).

A t t i t u d e s Towa r d D i f f i c u l t i e s of Learn ing
(7 i t e m s ) .--This scale ref erred to the p e r c e i v e d

care of learning mathema tics .

A low score indicates

that

the st udent views m a t h e m a t i c s as a d i f f i c u l t subject to
comprehend, w h i l e a high scor e indi cat es that students v i e w
mathem ati cs as a subject that can be learned by most.
sible scores r ange

Pos­

from 0 to 14.

A n a l y s i s of dat a r e v ealed no s i g n i ficant d i f f e re nces
between the mean score
mean s core

of

of the

any of the three

e x p e r i m e n t a l g r o u p and the
"comparison groups."

The

mean ran ged from 8.51 for the e x p e r i m e n t a l g r o u p to 9.26
for the m a t h e m a t i c s majors.

124

6.

Att itudes Toward P l a c e of M a t h e m a t i c s

in Socie ty

(8 i t e m s ).--This sca le repres ent s an e x p r e s s i o n of the
belief tha t m a t h e m a t i c s has an impo rtant role
A low scor e

in our society.

indicates a judgment that m a t h e m a t i c s

is of

little v a l u e and a h i g h score repr e s e n t s an e x p r e s s i o n of
the bel ief

that m a t h e m a t i c s has a vital role.

scores rang e

Possible

from 0 to 16.

A n a l y s i s of d a t a relevant to this sc ale reveale d no
si gnificant d i f f e rences betwe en the scores of
tal g r oup and the sco r e s of any of the three
groups."

Th e mean scor e s ra nge d

the e x p e r i m e n ­
"co mpa ris on

from a low of

ele ment ary ed uca tion g roup to a hig h of 9.26

8.31 for the

for the m a t h e ­

matics majors.
7.

A ttitudes Towa rd Scho o l and School Lear nin g

(11 i t e m s ).--This s c a l e inquires
students toward sch ool

into the fe elings of the

in general.

A low scor e

indicates

dislike of school a n d general d i s s a t i s f a c t i o n w i t h school
learning,

while a h i g h score i ndicates e n j o y m e n t of school

and feelings of c h a l l e n g e in learning.
sible sco res

for this

The r a n g e of p o s ­

scale is fro m 0 to 22.

T h e mean s core of the e x p e r i m e n t a l g r o u p on this
scale w a s

14.05, w h i c h was not si gnificantly d i f f e r e n t

the mean score
groups."

of

from

an y of the thre e other " com parison

Th e rel a t i v e l y high scores on this scal e i n d i ­

cate a h i g h posit ive att itu de by all groups t o w a r d the
im portance of school and the e x p e r i e n c e it p r o v i d e s .

125

Con clusions
On the e v i den ce p r o v i d e d by the an alysis of data of
the seven a t t i t u d e scales d e s c r i b e d above,

it can be c o n ­

cluded that the freshman st udent s w h o p a r t i c i p a t e d
experime nta l p r o g r a m di d not di ffe r s i g n i f i c a n t l y

in the
from othe r

freshman e l e m e n t a r y e d u c a t i o n maj o r s w h o did not volun t e e r
for the e x p e r i m e n t a l p r o g r a m on their attit u d e s
arithmetic,

(2) m a t h e m a t i c s as a process,

of learning m a thematics,
and

(5) scho ol and school

toward:

(1)

(3) d i f f i c u l t i e s

(4) plac e of m a t h e m a t i c s

in society,

learning.

Th e e x p e r i m e n t a l group,

on the other hand,

have s i g n i f i c a n t l y less p o s i t i v e attit u d e s

tended to

toward a r i t h m e t i c

than eit h e r the m a t h e m a t i c s - s e c o n d a r y educatio n ma jor s or
the m a t h e m a t i c s majors.

T h es e two grou p s al so tended

to

view m a t h e m a t i c s as d e v e l o p i n g and con s t a n t l y c h a n g i n g w h i l e
the e x p e r i m e n t a l g roup ten ded

to view it as a rigid subject

with rules to foll ow in sol ving problems.
On the tests of a tt itudes
learning m a t h e m a t i c s ,
school and school

toward d i f f i c u l t i e s of

place of m a t h e m a t i c s in societ y and

learning,

th ere w e r e no si gn ificant d i f ­

ferences b e t w e e n the scores of the students
mental g r o u p and eith e r the stude nts

in the e x p e r i ­

in the m a t h e m a t i c s -

secondary e d u c a t i o n maj o r s or the m a t h e m a t i c s majors.
To summarize,

the e n t ering c o g n i t i v e and a f f e c t i v e

behaviors towa rd m a t h e m a t i c s of the st udents w h o vo l u n t e e r e d

126

to participate in the experime ntal p r o g r a m were similar to
those of other fresh me n w i t h similar
elem ent ary school
however,

teachers.

intere st of be coming

These e n t e r i n g behaviors,

we re significant ly diffe r e n t

from those of other

freshman groups w i t h sp ecified in terest in ma thematics.

E v a l u a t i o n of the E x p e r i m e n t a l G r oup
Pe r f o r m a n c e on tne C r i t e r i o n R e f erenced M e a s u r e s

In thiB part of the study,

re s u l t s of p r e - and p o s t ­

test sco res on the c r i t e r i o n - r e f e r e n c e d measures were
analyzed to e v a l u a t e the extent of a c c o m p l i s h m e n t of the
experi men tal g r o u p on the p r e scribed m a t h e m a t i c a l
tencies.
1.

compe­

The eva l u a t i o n was carried out in two parts:
To d e t e r m i n e the si gnificanc e of gain in
ac h i e v e m e n t on the pre 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 betwee n pre- and p ost-tes t

2.

scores.

To d e t e r m i n e w hether a speci f i e d degree of m a s t e r y
over these c o mpetencies has been achieved.

Hypothe ses T e s t e d
The follow ing m u ltivariat e hyp o t h e s e s and ass o c i a t e d
un ivar iat e h y p o t h e s e s wer e tested:
Al.

There
the

will

be

poet-test

experimental
measures.

a significant
means
group

and
on

difference

the

the

pre-test

between
means

of

criterion-referenced

the

127

Symbolically:

where

is the p o s t - t e s t mea n on m e a s u r e i, and
is the p r e - t e s t mean on m e a s u r e i.

The a ss o c i a t e d univ a r i a t e hypotheses a l s o tes ted were:
the post- t e s t mean of the e x p e r i m e n t a l g r o u p w i l l s i g n i f ­
icantly dif f e r

from th eir p r e - t e s t mean on the c r i terion-

referenced m e a s u r e in:
1.
2.
3.
4.
5.
6.
7.
8.
9.

AS.

Measurement
N u m e r a t i o n Systems
Sets and Set Rel ations
W h o l e Nu m b e r s
F ractions
Deci mals
Rela tions and Func tions
P r o b a b i l i t y and St atistics
M a t h e m a t i c a l Systems.

There
the

will

be

poat-teet

no

meang

(80 p e r c e n t ) on
measure a .

significant

the

and

the

difference
mastery

between

le v e l

criterion-referenaed

128

Symbolically:

The ass o c i a t e d uni v a r i a t e h y p o t h e s e s also tested were:
the p o s t - t e s t mean of the ex pe rimental g r o u p wi ll be at
least equal to the m a s t e r y level

(80 percent)

on the

c r i t e r i o n - r e f e r e n c e d m easure in:
1.
2.
3.
4.
5.
6.
7.
8.
9.

Measurement
N u m e r a t i o n Systems
Sets and Set Re lati ons
Wholfc Nu m b e r s
Fractions
Decimals
R e l a t i o n s and Funct i o n s
P r o b a b i l i t y and S t a tistics
M a t h e m a t i c a l Systems.

Data A n a l y s i s
Data c o l l e c t e d through the a d m i n i s t r a t i o n of preand post-test
deve loped by
were u t i l i z e d

forms of the c r i t e r i o n - r e f e r e n c e d measures
this investi gat or
to

tes t

for use in the present study

H y p otheses A1 and A 2 .

Th e e x p e r i m e n t a l group scor es on these mea su res are
presented

in A p p e n d i x G.

Data inc lud ed

this sec tio n w e r e d r a w n from A p p e n d i x G.

in the tables in

129

Pre- and post - t e s t means,
mean di f f e r e n c e s

stand ard deviations,

and

for the c r i t e r i o n - r e f e r e n c e d me asures are

shown in Ta ble 6.
Univ a r i a t e and m u l t i v a r i a t e ana lysis of variance
techniques w e r e uti liz ed in the a n a l y s i s of data related
to Hy potheses A1 and A2.
In the m u l t i v a r i a t e analy sis
of the instru cti ona l
observed

of variance,

the effect

tr eatment on all c riter ion measure s was

simu ltaneously,

between these measures.

tak i n g

into account

the c orrelation

The m u l t i v a r i a t e test co nside red

student's resp ons e to all m e a s u r e s as a single response,
thus prov i d i n g in fo rmati on a b out the total effect of the
treatment.

The uni v ariate hypotheses,

on the o t her hand,

examined the stud ent re spons e to each m eas ure separately.

Findings
Hyp ot hesis A l .--The data

in T a b l e 6 show gains ma de

by the e x p e r i m e n t a l group on all c r i t e r i o n - r e f e r e n c e d m e a s ­
ures.

The increase

ranged

from 2.35 to 32.01 points.

When

the co lum n vect o r of mean d i f f e r e n c e s was tested against
zero col umn 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 was

26.83 w h i c h was hi ghl y s i g n i ficant
this result,
that

there

poet-test
group

on

Based on

the m u l t i v a r i a t e H y p o t h e s e s A1 w h i c h stated

will
means

the

(p < 0.0001).

be
and

a

significant
the p r e - t e n t

criterion-referenaed

difference
means

of

measures

between

the

the

experimental

was accepted.

Table 6
Means and Standard Deviations of Pre- and Post-Test Scores on the Nine
Criterion Measures for the Experimental Group

Variable

Pre-Test
Mean
S.D.

Post-Test
Mean
S.D.

1)

Measurement

63.97

21.33

66.32

23.41

2.35

2)

Numeration Systems

64.79

24.59

75.45

22.54

10.66

3)

Sets and Set Relations

59.63

23.65

83.03

12.53

23.40

4)

Whole Numbers

55.37

17.18

79.00

16.54

23.63

5)

Fractions

62.00

18.92

80.97

16.22

18.97

6)

Decimals

68.45

18.82

81.53

15.12

13.08

7)

Relations and Functions

50.92

16.52

82.55

15.44

31.63

8)

Probability and Statistics

56.26

18.15

76.76

16.70

20.50

9)

Mathematical Systems

33.52

25.44

65.53

20.74

32.01

Mean
Differences

131

The obtained probability on the multivariate test
prompted consideration of the univariate hypotheses.
Table 7 summarizes the findings for each univariate
hypothesis that was also tested.

Results of the analysis

indicated the following:
1.

On the univariate test of measurement,

the

differences between pre- and post-test means on this
criterion-measure were not significant at the 0.01 level
of confidence.

The univariate hypothesis associated with this

test which stated that the post-test m e a n of the experimental
group will be significantly different from their pre-test
mean on the criterion-referenced test in measurement was
rejected.
2.

The instructional treatment of the integrated

content-methods course had a positive effect on the students
performance on the other eight criterion-referenced measures.
Significant differences in favor of the post-test means were
noted between pre- and post-test means on the criterionreferenced measures in:
a.

Numeration systems

b.

Sets and Set Relations

c.

Whole Numbers

d.

Fractions

e.

Decimals

f.

(p < 0.005)
(p < 0.0001)

(p < 0.0001)

( p < 0.0001)
(p < 0.0001)

Relations and Functions

( p < 0.0001)

Table 7
Multivariate Analysis of Variance for the Experimental Group on Differences
Between Pre- and Post-Test Scores on the Nine Criterion Measures
Multivariate F = 26.8335

Variable

p < 0.0001

Between
Mean Square

Univariate
F

Significance
Probability

208.4474

0.8454

0.3639

4316.4474

8.8528

0.0052

Sets and Set Relations

25740.0263

40.6567

0.0001

4)

Whole Numbers

21221.1579

99.7837

0.0001

5)

Fractions

13680.0263

54.7736

0.0001

6)

Decimals

6500.2368

35.5532

0.0001

7)

Relations and Functions

38021.1579

114.6630

0.0001

8)

Probability and Statistics

15969.5000

53.7865

0.0001

9)

Mathematical Systems

38912.0000

71.0263

0.0001

1)

Measurement

2)

Numeration Systems

3>

133

g.

Probability and Statistics

h.

Mathematical Systems

Based on these results,

(p < 0-0001)

( p < 0.0001).

the univariate hypotheses, which

stated that the post-test mean of the experimental group
will be significantly different from their pre-test mean
on the criterion-referenced measure in:
a.

Numeration Systems

b.

Sets and Set Relations

c.

Whole Numbers

d.

Fractions

e.

Decimals

f.

Relations and Functions

g.

Probability and Statistics

were accepted.
Hypothesis A 2 .--The vector column of differences
between post-test means on the criterion-referenced measures
and the mastery

level of 80 percent was tested against a

zero column vector.

The multivariate F value associated

with this test was 12.68 which was highly significant
(p <0.0001).

The multivariate Hypothesis A2 which stated that

the p o s t - t e s t

means

significantly

different

on

o f the

experimental

from

the m a s t e r y

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

group will
level

not

be

(80 p e r c e n t )

was rejected.

The obtained probability on the multivariate test
prompted consideration of the univariate hypotheses.
Table 8 summarizes the findings for each univariate

Table 8
Multivariate Analysis of Variance for the Experimental Group on Differences
Between Post-Test Scores and Mastery Level (00 percent) on the
Nine Criterion Measures
Multivariate F =12.6829

Variable

p < 0.0001

Between
Mean Square

Univariate
F

Significance
Probability

7115.7079

12.9798

0.0010

Numeration Systems

787.6053

1.5500

0.2210

3)

Sets and Set Relations

348,0263

2.2167

0.1450

4)

Whole Numbers

38.0000

0.1388

0.7117

5)

Fractions

36.0263

0.1369

0.7135

6)

Decimals

88.5263

0.3874

0.5378

7)

Relations and Functions

247.6053

1.0390

0.3147

8)

Probability and Statistics

398.1316

1.4284

0.2397

9)

Mathematical Systems

7960.5263

18.5065

0.0002

1)

Measurement

2)

135

hypothesis that was also tested.

Results of these analyses

indicated the following:
1.

The poBt-test mean in Measurement

was significantly

( M = 66.32)

(p < 0.0001) below the mastery level.

post-test mean in Mathematical Systems
significantly below the mastery level.

The

(M “ 65.53) was also
Based on these

results the univariate hypotheses associated with testing
the significance of difference between the post-test means
and the mastery level on the criterion-referenced tests in
(a) Measurement,
2.

and

(b) Mathematical Systems were rejected.

At the 0.05 level of confidence,

there were no

significant difference between the post-teBt mean and the
mastery level of 80 percent on each of the following
criterion-referenced measures:
a.

Numeration Systems

(p < 0.2210)

b.

Sets and Set Relations

c.

Whole Numbers

d.

Fractions

e.

Decimals

f.

Relations and Functions

g.

Probability and Statistics

( p < 0.1450)

( p < 0.7117)

(p < 0.7135)
( p < 0.5378)
( p < 0.3147)
( p < 0.2397).

Based on these results the univariate hypotheses associated
with testing the significance of difference between the
post-test means and the mastery level on the criterionreferenced measure in the above seven topics were accepted.

136

Analysis of Teat Results
Table 9 shows the percent of students in the
experimental group who scored 80 or more

(mastery level)

on each of the criterion-referenced measures.

Pre-test

results show that students performed better on traditional
topics with which they have had previous experience than on
topics which were introduced for the first time such as Sets
and Set Relations,
Statistics,

Relations and Functions,

and Mathematical Systems.

Probability and

It was also noted

that on the related topics, Whole Numbers, Fractions,

and

Decimals, many students became progressively more able as
they learned the essentials on one unit to improve their
performance on the next unit.

Only 13 percent of the

experimental group scored 80 or higher on the pre-test
of Whole Numbers, while 21 percent scored 80 or higher
on the pre-test of Fractions,

and 37 percent attained the

80 percent or higher level on the pre-test of Decimals.
The student performance on the pre-tests are obviously
influenced by their performance on the post-tests of
previous units.

The experimental group showed significant

improvement on achievement of the mathematical competencies
prescribed by the program.

However,

it is not sufficient

that these students improve their knowledge of mathematics,
it is more important that they attain a certain level of
achievement that would indicate mastery over that topic.

137

Table 9
Percentage of Students in the Experimental Group (N - 38)
Attaining the Pre-Established Mastery Level

Pre-Test
(A)

Measure

Post-Test
<%>

1)

Measurement

24

39

2)

Numeration Systems

37

50

3)

Sets and Set Relations

21

68

4)

Whole Numbers

13

61

5)

Fractions

21

79

6)

Decimals

37

74

7)

Relations and Functions

8

74

8)

Probability and Statistics

16

58

9)

Mathematical Systems

5

39

Bloom

(4) suggested an accuracy level of 80 percent on

each formative test as an indication of mastery.

On the

mathematical topics specified in the experimental program,
students achieved mastery over all but two topics, Mea s u r e ­
ment and Mathematical Systems.

Measurement was the first

topic introduced and most students spent much time famil­
iarizing themselves with the new surrounding and becoming
acquainted with the manipulative materials in the m athe­
matics

laboratory.

This,

of course, minimized the amount

of time spent on the mathematical activities associated
with this topic.

Mathematical Systems, on the other hand,

was the last topic to be taught.

Only three days were

allocated for its instruction and the contents of this topic

138

were completely new to most students
below 50 on the pre-test).

(82 percent scored

Many students did not finish,

in the limited time, all the activities in the unit file.
Mathematical Systems was the only topic in which the
instructional designs developed by the student were not
implemented with the children at the elementary school
mathematics laboratory due to the end of the academic year
of the elementary school.
Overall,

analysis of pre-test results indicated

lack of understanding of basic mathematical concepts.

While

most students did comparatively well on computational prob­
lems, most had difficulties with problems dealing with the
mathematical principles underlying the operations of addi­
tion,

subtraction, multiplication,

and division, as well as

the basic properties of these operations.
Post-test results reflected the emphasis placed in
this course upon insuring that the prospective teachers
understand the basic mathematical concepts they are expected
to teach children.
Effect of the Experimental Program on the
Basic Mathematical Understandings and
Attitudes Toward Mathematics
In this part of the study,

the effect of the

mathematics component of the experimental program upon the
basic mathematical understandings and attitudes toward
mathematics of the experimental group were analyzed.

139

These results are reported under two major headings:
(1) growth in basic mathematical understandings, and
changes in attitudes toward mathematics.

(2)

Within each

heading related data were analyzed.

Growth in Basic Mathematical
Understandings
The two forms of the test,

"A Test of Mathematical

Understanding," were utilized in an attempt to measure
the basic mathematical understandings possessed by the
experimental group prior to and after completing the
integrated content-methods course in mathematics education.
Hypothesis B1
The hypothesis related to this aspect of the study
was stated as:

Th e r e

wil l

test o f b a s i c m a t h e m a t i c a l
test

scores

o f the

be a e i g n i f i a a n t
understanding

experimental

group and

difference

between

the

on a

post-

th e i r p r e - t e s t

scores.

Data Analysis
Raw scores of the experimental gr oup on both formB
of the test of mathematical understandings are included in
Appendix H.
Pre- and post-test means,
changes

standard deviations and

(difference between means)

computed and are shown in Table 10.

on these tests were

140

In order to determine the significance of the
difference between the pre- and post-test means

(correlated),

a t-test was u s e d . 3
The resulting t-ratio was compared with the "t" in
a table designed for use in determining the significance of
"t."

These data are included in Table 10.

sThe following formulas were used to compute the
significance of the difference between correlated means
obtained from tests administered to the same group.

Where X^ - X 2 = difference between pre- and post-test means,
SE^ = standard error of the difference between
correlated means
SE— was computed by the following formula:

Where

SE^ = standard error of pre-test,
S E 2 = standard error of post-test, and
r^ 2

11 correlation between pre- and post-tests.

141

Table 10
Pre- and Post-Test Results of the Experimental Group
on the Test of Mathematical Understandings

Pre-Test

Post-Test

38

38

38.1316

44.0263

Standard deviation

4.7883

4.3027

Standard error

0.7872

0.7074

Number of students
Mean

Correlation between pre- and post­
test scores

0.7541

Standard error of difference
between means

0.5294
11.1347*

Observed t-value

•Significant at the 0.05 level

(t.05 <37) "

1.69).

Findings
The post-test mean was 5.90 points higher than the
pre-test mean.

When this mean difference was tested against

the hypothetical zero gain,

the resulting t-ratio was 11.13,

which was highly significant

(p < .001).

Based on the results of this criterion measure,

it

was concluded that the post-test scores were significantly
higher than the pre-test scores.
stated that t h e r e w i l l
of b a e i a m a t h e m a t i c a l
acorea

o f the

was accepted.

be

Hypothesis Bl, which

significant

understanding

experimental

group

a nd

difference

between

on

a teet

the p o a t - t e a t

their pre-teat

acorea ,

142

Analysis of Pre- and Post-Test
Results
Pre-test resul ts.--The concepts which caused
students most difficulties on the pre-test were:

principles

underlying number operations such as properties of addition
and multiplication

(items 17,

22, 29, 47), meaning of a

partial product in multiplication and remainder in division
(items 28, 38), converting decimals into fractions and vice
versa

(item 41),

than 10

fundamental operations with bases other

(items 7, 11), set vocabulary and set operations

(item 55), and measurement of related geometric figures
33, 48).

(items

The incorrect responses selected by the students

on the pre-test indicate previous teaching procedures which
have emphasized computational aspects and drill procedures
rather than understanding of basic arithmetic concepts.
For example, when asked to choose the sentence that best
tells why the answer in the division problem

3
4

2
3

(5 i ^- =6— ), is

larger than 5, 82 percent of the students said because "in­
verting the division turned j upside down" which indicates
a memorization of rule rather than understanding of the
3

concept that the divisor — is less than 1.
On the fifty-five-item pre-test,

the scores ranged

from a low of 32 to a high of 49 with a mean score of 38.13.
Post-test re sults.— The majority of the students
were able to improve their understanding of basic m a t h e ­
matical concepts during the integrated content-methods course.

143

The empha sis , during this course, on teaching mathematical
concepts rather than on drill work and computational skills,
increased the students capacity to analyze problems and to
follow reasoning.

Improvement was noted on problems related

to operations with Whole Numbers, Fractions,

and Decimals.

About 30 percent of the students still had difficulties with
problems related to Measurement of areas and volumes of
geometric figures, and on Set Operations
On the fifty-five-item post-test,

(items 50, 53, 55).
the scores ranged

from a low of 33 to a high of 51 with a mean score of 44.02.
Changes in Attitude Toward
Mathematics
The 1962 Revised Dutton Arithmetic Attitude Inven­
tory

(Form C) was utilized in an attempt to evaluate changes,

if any,

in the attitudes of the prospective elementary

school teachers in the experimental group toward mathematics
which occurred during the academic year 1971-1972 during
which the experimental program was implemented.
Hypothesis B2
The hypothesis related to this part of the study was
stated as:
arithmetic
o f the

T h e r e will
attitude

experimental

he s i g n i f i c a n t

differences

inventory

between

group

their p r e - t e e t

and

on tin

the p o e t - t e s t
ecoree.

ecoree

144

Treatment of Data
Responses to the fifteen statements on the first
part of the attitude inventory were tabulated according
to item.

Each item on the scale was assigned a scale value

(from 1.0 which represents an extremely negative attitude
toward arithmetic to 10.5 which represents an extremely
positive a t t i t u d e ) .

The individual score was obtained by

finding the average or median scale of the statements which
the student selected.
A composite report of the results from the admin­
istration of this arithmetic attitude inventory has been
included in Appendix H.

Data included in the tables in

this section were drawn from Appendix H.
Findings
Pre- and post-test means,

and standard deviations

for scores on the attitude scale were computed.

The mean

difference for the experimental group was tested against
hypothetical

zero gain through the use of t-test for sig­

nificance of difference between correlated means.

These

data are presented in Table 11.
The obtained t-value of 6.59 on the attitude scale
was highly significant.
group
mean

The mean score of the experimental

(M = 7.07) was significantly higher than the pre-test
(M = 5.57).

145

Table 11
Pre- and Post-Test Results of the Experimental Group
on the Dutton Attitude Scale

Pre-TeBt
Number of students

Post-Test

38

38

Mean

5.5684

7.0737

Standard deviation

1.8154

1.5206

Standard error

0.2985

0.2500

Correlation between pre- and post­
test scores

0.6665

Standard error of difference
between means

0.2283

Observed t-value

6.5935*

•Significant beyond the 0.0S level

(t Q5 (37) ■= 1.69) .

Based on the result of this criterion measure,
Hypothesis B2 which stated that there will be s i g n i f i c a n t
difference
post-teat
test

on

an a r i t h m e t i c

scores

of

the

attitude

inventory

between

experimental

group and

the ir

the

pre­

s c o r e s was accepted.

Related Questions
In addition to the statements
Inventory,

in the Attitude

other questions were included regarding attitudes

toward arithmetic.

Space was provided for the students to:

(1) estimate their general feeling toward arithmetic,

(2)

indicate the grade level in which attitude toward arithmetic

146

was influenced most,

and

(3) list two things liked and two

things disliked about the subject.
Findings
1.

General feeling toward ari t h m e t i c .--Each student

was asked to circle a number between 1 and 11 to show his or
her overall feeling toward arithmetic

(1 representing

extreme dislike and 11 representing extreme l i k e ) .

A

summary of student judgment of individual attitude toward
arithmetic on the pre- and post-tests

is shown in Table 12.

Table 12
Students

(N-38)

FeelingB About Arithmetic in General

Pre-Test

Post-Test

1

1

0

2

0

1

3

2

0

4

0

3

5

3

0

6

7

5

7

4

3

B

4

11

9

11

5

10

3

4

11

3

6

Feeling About Arithmetic in General
Extreme Dislike

Extreme Like

147

The mean score of the experimental group on the pre-test
was 7.52 while it was 8.00 on the post-test.
that the students'

It was noted

judgment was considerably higher than

the scores obtained from the Attitude Inventory, where the
mean pre-test score was 5.57 and the mean post-test score,
7.07.

Dutton

(48:420)

attributes such a result to the

averaging of both favorable and unfavorable items checked
on the scale by each individual to secure the overall value
of the inventory,
2.

Grade where attitudes were d e v e l o p e d .— Feeling

toward arithmetic is developed in all grades.

However,

the most crucial years for the students in the experimental
group seemed to be when the students were in the third
through sixth grade,
post-test data

as reported in both the pre- and

(see Table 13 b e l o w ) .

consistent with Dutton's

findings

These results are

(48).

148

Table 13
Grade Level Where Students'

(N ■* 38) Attitudes Were Developed

Grade Level

Pre-Test

Post-Test

1

2

2

2

3

4

3

5

5

4

6

6

5

3

4

6

5

4

7

2

3

8

3

3

9

2

2

10

3

3

11

2

2

12

2

1

3.

Aspects of arithmetic liked or disliked m o s t .--

Students in the experimental program were asked to list two
aspects of arithmetic liked most and two aspects liked least.
This technique was used to give equal treatment to favorable
and unfavorable feelings.
In tabulating the data collected at the beginning
and at the end of the academic year 1971-1972,

it was noted

that the challenge presented by arithmetic was the most
frequent positive response given by the students both at the
beginning and at the end of the school year.

Story problems

were the aspect of arithmetic disliked by most students
Table 14).

(see

149

Table 14
Aspects of Arithmetic Students

(N ** 38) Liked and Disliked Most

Pre-Test

Post-Test

Aspects of Arithmetic Liked Most:

1.

The challenge presented by arithmetic.

21

18

2.

Has practical application.

14

15

3.

Stimulating, enjoy working with numbers.

6

7

4.

Necessary for everyday life.

8

10

5.

Satisfaction in working out problems.

4

7

6.

Solving word problems.

3

5

7.

Algebra.

3

0

a.

Games about arithmetics.

1

4

17

12

Aspects of Arithmetic Disliked Most:

1.

Story problems,

2.

Teachers.

9

4

3.

Boredom and frustration.

6

3

4.

Memorizing rules.

6

5

5.

Drill and busy work.

5

2

6.

Proofs.

5

1

7.

Set theory

0

4

8.

Long division.

4

1

9.

Metric system.

0

3

.0.

Measurement.

0

2

150

It was noted by the investigator that on the
pre-test,

the students tended to choose one of the st ate­

ments on the Attitude Inventory

(first 15 statements)

as

the aspect of arithmetic liked or disliked most, while in
the post-test, many students selected new aspects that they
had confronted for the first time during the experimental
course such as games about arithmetic,
metric system.

In general,

set theory,

and

the results obtained in this

study in relation to this particular question are similar
to those obtained by Dutton in a study of attitudes of pro s­
pective elementary school teachers toward-arithmetic

(48).

Comparison of the Experimental Group with
Elementary Teacher Education
"Group on Mathematical Understandings
and Attitudes toward Arithmetic

The two equivalent forms of the test "A Test of
Mathematical Understandings" and the Dutton Arithmetic
Attitude Scale were administered to a group of prospective
elementary teachers in the regular teacher education program
who are enrolled in the methods course

(Education 3 2 5 E ) .

This particular class was chosen for three reasons:

(1)

it

was taught during the same term as the experimental inte­
grated content-methods course,

(2) it waB taught by a member

of the TTT program also involved in the experimental course,
and

(3) all the students in that course had had the m a t h e ­

matics content course

(Mathematics 201) within that school

151

year.

The pre-tests were administered on the first day of

the term and the post-tests during the last day of the term.
Data collected through the administration of the two
equivalent forms of the test of mathematical understandings
and the attitude scale were utilized in an attempt to co m­
pare the basic mathematical understandings and attitudes
toward arithmetic between the experimental group after
completing the first-year trial of the mathematics curric­
ulum of the experimental program and a group of prospective
elementary teachers in the regular teacher education program
after completing their required mathematics education train­
ing.

Summary of the data collected is presented in Table 15.

Table 15
Pre- and Post-Teat Results of the Experimental Group and the
Regular Methods Course (Education 325E) Students on the
Test of Mathematical Understandings (MU) and
Dutton Arithmetic Attitude Scale (AA)

Group

Pre-Test
Mean
S.D.

Post-Test
Mean
S.D.

Variable

Number

Experimental

MU
AA

3B
3B

3B.13
5.57

4.79
1.82

44.03
7.07

4. 30
1.52

Regular Methods
Course (Education 325E)

MU
AA

21
21

40.90
6.10

5.51
2.14

41.29
6.25

4.78
2.03

152

Hypothesis Cl
The first hypothesis related to this part of the
study stated:
experimental

"The a d j u s t e d m e a n p o s t - t e s t
group

mean p o s t - t e s t
school

teachers

p r o g r a m on

a

will

ecoree

he at

of a group

enrolled

test

least

in

the

equal

scores

to

of basic mathematical

the

the a d j u s t e d

of prospective
regular

of

teacher

elementary
education

understanding. "

Findings
The analysis of covariance was utilized for the
analysis of data.

This statistical technique, known to be

particularly applicable to any experiment,
present one,

such as the

in which groups could not be randomized or

equated before treatment, made it possible for the inves­
tigator to adjust the outcomes of the experiment
mathematical understandings)
ation

(the p r e - t e s t ) .

elementary teachers,

(gains in

in terms of a source of va ri­

The scores of fifty-nine prospective

thirty-eight in the experimental group

and twenty-one uppe^-classmen in the regular methods course
(Education 325E) on a test of basic mathematical und ers tan d­
ings were used for the analysis.

Data are presented in

Table 16.
Wh e n the F-ratio was applied to the adjusted "among
groups" and "within groups" variance, F wa s highly signifi­
cant

( p <0.00 1)

in favor of the experimental group.

concluded therefore,

It was

that the two final means, whe n initial

153

difference was allowed for, did differ significantly in
favor of the experimental group.
the a d j u s t e d m e a n
will

be

at

of a g r o u p
teacher

least

poet-teet
equal

to

of pr o sp ec t iv e

education

program

Thus, Hypothesis Cl,

ecoree

the

of

adjusted

elementary
on

i he e x p e r i m e n t a l

a

test

mea n p o e t - t e s t

teachers
o f basic

in

the

that
group
scores

regular

mathematical

u n d e r s t a n d i n g was accepted.

Table 16
Summary of the Analysis of Covariance for the Scores of the Experimental
Group and the Regular (Education 325E) Students on
the Test of Mathematical Understandings

Source of
Variation

D.F.

SSy

SSXy

SSX

SS£

MS

1

101.5801

-102.7957

104.0179

269.1336

269.1336

Within groups

56

1141.2594

985.4398

1456.1516

474.3703

8.4709

Total

57

1242.8475

882.6441

1560.1695

743.5039

Among groups

269.1336
8.4709

31.7716

Critical value of F at .05 level = 4.02
at .01 level = 7.10

154

Hypothesis 02
The second hypothesis related to this part of the
study stated:
arithmetic

There

attitude

scores

of

scores

of a group

enrolled

the

in

wi l l

be a s i g n i f i c a n t

inventory

experimental

the

between

group and

of p r o s p ec t iv e

regular

teacher

difference

in

the a d j u s t e d p o s t - t e s t
the a d j u s t e d p o s t - t e s t

elementary

teachers

education p r o g r a m .

Findings
Analysis of covariance was utilized for the analysis
of data.

The outcomes of the experiment

(changes in a t t i ­

tudes toward arithmetic) were adjusted in terms of the
initial source of variation

(the pr e-t est).

The scores of fifty-nine prospective elementary
teachers,

thirty-eight in the experimental group and twenty-

one in the regular methods course
used for this analysis.

(Education 3 2 5 E ) , were

Data are presented in Table 17.

When the F-ratio was applied to the adjusted "among"
and "within" variances,

it was noted that the observed F

(FQba *= 13 .65) was highly significant in favor of the e x per­
imental group

(p < 0.001).

Based on this evidence,

it was

concluded that the two final means, when initial difference
was allowed for, did differ significantly in favor of the
experimental group.
be a s i g n i f i c a n t
inventory

between

Thus,

difference

Hypothesis C2, that there w i l l
on

the a d j u s t e d

an

arithmetic

post-test

attitude

scores

o f the

155

experimental
methode

group

oouree

and

thoee

(E d u c a t i o n

o f the

3Z5E)

etudente

in the

regular

was accepted.

Table 17
Summary of Analysis of Covariance for the Scores of the Experimental
Group and the Students in the Regular Methods Course
(Education 325E) on Dutton Arithmetic
Attitude Inventory

Source of
Variation

D.F.

SSy

SSXy

SSX

SS£

MS

1

9.1234

-5.9050

3.8220

18.2466

18.2466

Within groups

56

167.7461

140.8284

213.5621

74.8801

1. 3371

Total

57

176.0695

134.9234

217.3841

93.1267*

Among groups

„

1 8.2466

,„

F ■ 1~ 371 - U -6464
Critical value of F at .05 level ■ 4.02
at .01 level = 7.10

156

Correlation Analyala
Thus far, the data collected from the experimental
program have been utilized to determine the relative effect
of the mathematics component on the mathematical achievement,
understanding of basic mathematical concepts, and attitude
toward arithmetic.

In addition,

attention has been given to

the question of whether these effectB are related or whether
they are influenced by o t her factors such as the level of
high school mathematics preparation and grade point averages.
For this phase of the study,

scores from all pre-

and post-tests as well as other background data of the
experimental group were used to calculate an intercorre­
lation matrix.

The resulting 36 by 36 matrix is included

in Appendix J.
To determine w hether the correlation coefficient
between two variables is significantly different from zero
at the 0.05 level,

the following t - ratio was used

= rxy

»T^r!
xy
where r

is the correlation coefficient between variables

x and y.

With thirty-eight subjects

(df**36), a coefficient

wh ich is more than 0.321 or less than -0.321 is considered
to be sufficient for significance at the 0.05 level of
confidence.

157

Analysis of the intercorrelation matrix revealed
the f ollo wing:
1.

On the nine criterion-referenced t e s t s , the

following results wer e noted:
a.

There were significant correlations between preand post-test scores on each measure.

b.

There were significant correlations between all
post-test scores except the tests on Sets and Set
Relations/and on Relations and Functions.

c.

There were significant correlations between the
post-test scores on the test of mathematical
understandings and each criterion-referenced pos t­
test except the test on Relations and Functions
(r - 0.271).

d.

There were significant correlations between the
post-test scores on the Dutton Attitude Scale and
each criterion-referenced post-test.
2.

There was significant correlation between p r e ­

test scores on the test of mathematical understandings and
the arithmetic attitude scale
nificant correlation
two measures

( r = 0 . 454),

There were sig­

between the post-test scores on these

( r * 0.668).

The correlations between pre- and

post-test scores on the test of mathematical understandings
(r *=0.754),

and the arithmetic attitude scale

were also significant.

As expected/

( r =0.667)

the correlation

158

coefficient between scores on the arithmetic attitude scale
and the student rating of his general feeling toward arith­
metic were high
r » 0.786).

(for p r e - t e s t , r - 0.615;

for post-test,

The significance of these correlations are

consistent with results obtained in similar studies by
Dutton

(49) and Litwiller
3.

(82).

On the attitude scales toward mathematics,

negatively significant correlations were noted between
student performance on the test of mathematical under­
standings and
(r “ -0.561),

(a) attitudes toward mathematics as a process
and

(b) attitudes toward difficulties of

learning mathematics

( r »-0. 323).

Positive correlations

were noted between performance on test of mathematical
understandings and
learning

(a) attitudes toward school and school

( r »0.439),

and

mathematics in society
4.

(b) attitudes toward place of

(r *= 0.669).

Significant correlations were obtained between

the number of mathematics courses taken in high school
and

(a) pre- and post-test scores on the nine criterion-

referenced measures,

(b) pre- and post-test Bcores on the

test of mathematical understandings,

(c) pre- and post-test

scores on the arithmetic attitude scale, and

(d) high school

grade point average.
5.

The final grade the students received on the

integrated content-methods course was significantly correlated

159

to post-teat scores on the criterion-referenced tests,
test of mathematical understandings,

and arithmetic attitude

scale.
6.

Scores on the MSU basic skill test in m a t h e ­

matics were significantly correlated to scores on the pr e­
test

(r«0.4B)

understandings.

and post-test

(r ° 0,521) of mathematical

Scores on the MSU arithmetic test were not

significantly correlated to any of the criterion-referenced
measures or the test of basic mathematical understandings.
In all, the correlation data indicated significantly
positive relation among the criterion-referenced tests,

the

standardized test of basic mathematical understandings,

and

the Dutton Arithmetic Attitude Scale.
High school mathematics preparation and overall
grade point average were significantly correlated to p er­
formance on pre-tests

of

mathematical understandings,

attitudes toward mathematics.
content-methods course,

and

Final grade on the integrated

as assigned by the faculty members

responsible for the course, was highly correlated with the
post-test scores on the test of mathematical understandings
( r * 0.572)

and the Dutton Arithmetic Attitude Scale

(r = 0.657) .

160

Evaluation of Student Reaction to the
Mathematics Component of the
Experimental Program
Prospective elementary teachers'

reactions to

different aspects of the mathematics component of the
experimental program are an important facet of the formative
evaluation process.

If the student reacts positively to new

methods and procedures of instruction, he or she may be
motivated to learn mor e under these methods,

and may utilize

similar procedures in future teaching strategies.
In this part of the study,

the experimental group's

responses to a questionnaire w e r e analyzed to determine
their reaction to specific aspects of the program.

The

el even—item questionnaire was administered during the last
day of classes after completing the integrated contentmethods course.
A copy of the questionnaire is included in A p pen­
dix E.

The first ten questionnaire items referred to

particular aspects of the program with scaled responses
(strongly agree, agree in general, undecided, disagree in
general,

and strongly disagree).

The experimental group's

response distribution is shown in Table 18.
Analysis of student responses to the questionnaire
indicated the following:
1.

When asked if more activities using manipulative

materials should be used in the integrated content-methods

Table 18
Frequency Distribution of Experimental Group Response to Student Evaluation Questionnaire

Item

Agree in General

Undecided

Disagree in General

Strongly Disagree

1

0

16

12

10

0

2

9

21

4

4

0

3

8

23

5

2

0

4

4

11

12

10

1

5

3

8

7

17

3

6

4

18

6

9

1

7

8

16

12

2

0

8

0

5

2

26

5

9

1

1

4

25

7

10

13

16

3

3

1

161

Strongly Agree

162

course,

42 percent agreed with the statement,

were undecided,

and 26 percent disagreed.

32 percent

This seems to

indicate the group feels the amount of activities using
manipulative materials was sufficient.
2.

About 80 percent of the students suggested more

time be spent on methods of teaching elementary school mathe­
matics.

This is probably due to the group realizing the

need for such instruction during the clinical experience.
3.

Thirty-one students

(82 percent)

thought more

time should be spent on planning teaching strategies to be
used at the elementary school

(clinical e x p e r i e n c e ) .

This

response is consistent with their reaction that more time
be spent on methods of teaching mathematics.
4.

The group as a whole seemed undecided on whether

more time should be spent in the mathematics laboratory at
the elementary school.

It was noted that the four students

who "strongly agreed" with that statement had also scored
very high on the mathematics tests and the attitude scale.
A majority of the students seemed to feel that one-half hour
per wee k of teaching experience is sufficient at this stage
of their education.
5.

When asked if there should be more lectures

about mathematical content,
statement,
undecided.

29 percent agreed wi th the

53 percent disagreed,

and 18 percent were

163

6.

While the students did not wa nt more lecture

time on the mathematical c o n t e n t , the majority

(58 percent)

wanted more lectures on methods of teaching mathematics.
Problems, encountered by the students at the clinical
experience, may have prompted this reaction.
7.

Students seemed to have enjoyed films related

to the teaching of elementary mathematics.

Only 5 percent

of the students felt that the films were of no value.
8.

A majority of students

(82 percent)

felt the

number of weekly hours assigned to the integrated contentmethods course should not be increased.
9.

Only 5 percent of the students agreed there

should be more time spent on paper and pencil problem
solving activities,

84 percent disagreed with the statement,

and 11 percent were undecided.
10.

Eighty-two percent of the students liked the

idea of working with elementary school teachers on planning
of strategies for teaching mathematics at the elementary
school.

The students suggested that more time be spent

on such activities.
Overall,

the student reaction to the items on the

questionnaire were quite consistent and seemed to indicate
their satisfaction with the methods and procedures followed
in the integrated content-methods course.

They felt, however,

164

that not enough emphasis is placed on planning strategies
to be used with the elementary school pupils.
At the questionnaire's end, the students were asked
to make suggestions that may help improve the mathematics
component of the experimental program.
cent)

Thirty-four

(88 per­

studentB offered some comments about the program.

The

responses varied.
Many students wanted more time spent on preparing
their lesson plan with the pupils in elementary school.
Others stressed the need for more feedback from the faculty
members on their work in the university and at the elemen­
tary school.

Most students felt that the textbook assigned

to the integrated content-methods course did not relate to the
procedures followed in the mathematics laboratory,
not clear.

and was

Following are some of the students' comments:

PerhapB a question-and-answer sum-up of the
week would be good on one of the last days.
Just
to be sure people know what's going on.
Would
also be nice for every person to get one of the
conferences [with faculty members] around midterm
if possible.
Otherwise, I thought the math lab
turned out pretty well.

I really liked it when the Allen School
teachers came and helped us wit h our strategies.
The activities are usually pretty good,
although some seem trivial at times.
The
method of presenting the materials, using folders,
is superb1 Whoever thought of that deserves a gold
star.
The textbook was useless, as far as I was

165

concerned.
I never used it except for reference
and when 1 used it for that, I usually found it
inadequate.
I am strongly against having more paper and
pencil work.
All my years in Math, I have done
written work.
The math lab this term surprised
me in showing me with manipulative materials
how little I really understood math concepts.
My greatest problem came in thinking of
strategies to use in teaching my math lesson;
at times it was very difficult to think of activ­
ities that would hold the interest of the
students [pupils at the elementary sc h o o l ] .
More emphasis should be on the groups and
their plans.
More feedback from professors as
to additional ideas for teaching from their
experiences.
The stress on individual work
and creativity was very good.
Perhaps the course needs to be a little
more structured at times.
More lectures or p r o f e s s o r s 1 explanations
on certain subjects or materials.
More time at Allen Street School.
Before post-tests are given, a question
and answer session woul d be helpful to those
wh o did not understand some materials even
after doing all activities.
Nuffield guides should be part of the
Math Lab.

166

Summary
Analysis of the data collected during this
investigation revealed the following results:
1.

The mathematical content of the experimental

program adequately meet the needs of the future elementary
school teacher.
2.

The mean post-test

group was significantly higher

score of the experimental
than the mean pre-test score

on the criterion-referenced measures in:
Systems,

(b) Sets and Set Relations,

(d) Fractions,

(e) Decimals,

(g)Probability

and Statistics,

3.

(a) Numeration

(c) Whole Numbers,

(f) Relations and Functions,

The mean post-test

and

(h) Mathematical Systems.

score of the experimental

group was not significantly different than the mean pre-test
score on the criterion-referenced measure in Measurement.
4.

The mean post-test score of the experimental

group was not significantly different from the mastery level
(score of 80 or higher)
in:

on the criterion-referenced measures

(a) Numeration Systems,

(c) Whole Numbers,

(d) Fractions,

tions and Functions,
5.

(b) Sets and Set Relations,

and

(e) Decimals,

(f) Rela­

(g) Probability and Statistics.

The mean post-test score of the experimental

group was significantly below the mastery level on the
criterion-referenced measures in:
(b) Mathematical Systems.

(a) Measurement,

and

167

6.

There was significant difference on a test of

mathematical understandings between the post-test scores of
the experimental group and their pre-test scores.
7.

There was a significant improvement on an

arithmetic attitude inventory between the post-test scores
of the experimental group and their pre-test scores.
8.

The adjusted mean post-test score of the

experimental group was significantly higher than the
adjusted mean post-test score of a group of prospective
teachers in the regular teacher education program on a test
of mathematical understandings and on an arithmetic attitude
inv ent ory .
9.

There wer e significant correlations between:

(a) pre- and post-test scores on the criterion-referenced
m e a s u r e s , (b) post-test scores on the test of mathematical
understandings and the arithmetic attitude scale,

{c ) number

of high school courses in mathematics and pre- and post-test
scores on the test of mathematical understandings,

(d) pre-

and post-test scores on the mathematical understandings test
and high school grade-point average.
10.

The experimental group expressed desire for

more participation in clinical experience concurrent with
the laboratory oriented integrated content-methods courses.

CHAPTER V
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

The preceding chapters were devoted to a discussion
on the current significance of the problem, a delineation of
its purpose, and a description of the procedures followed in
evaluating the effect of the mathematics curriculum of the
experimental program upon cognitive and affective behaviors
of a group of prospective elementary school teachers.
Chapter V, the final chapter of this report,
(1) a general summary of the study,
and

is devoted to:

(2) major conclusions,

(3) recommendations for future action and research.
Summary
This thesis reports the results of a formative

evaluation of the mathematics component of an evolving
elementary teacher education program at Michigan State
University.

This section contains a summary of this

evaluation.

168

169

The Mathematics Component of the
Experimental Program
An experimental elementary teacher education program
was initiated at Michigan State University in the fall of
1971.

The program is funded and staffed by the "Trainers

of Teacher Trainers

(TTT)" project,

and is based on several

aspects of the Behavioral Science Teacher Education Program
(BSTEP) Model developed at Michigan State University in
1966.

Although the program has many inovative facets in

human and academic curricular areas,

this evaluation is

devoted to the mathematics component of the experimental
program.

The objectives of the experimental program in

the field of mathematics education are to provide the
prospective elementary school teacher with:

(1) an adequate

knowledge of the mathematics he or she would be required to
teach,

(2) an adequate knowledge and technique in teaching

the mathematics to elementary pupils,

(3) the opportunity

to experiment with these teaching skills,

and

(4) un der­

standing of human development and the nature of learning
mathematics adequately well, as to adopt appropriate pro­
cedures to facilitate learning of mathematics.
Purpose
The major purpose of this investigation was the
formative evaluation of the mathematics component of an
experimental elementary teacher education program at

170

Michigan State University.
sought to

Specifically,

this investigation

(1) analyze and evaluate the adequacy of the

mathematical content of the experimental program in meeting
the future needs of the elementary school teacher in ma the­
matics,

(2) evaluate the effect of the instruction,

aB

prescribed by the mathematics component of the experimental
program, on the prospective elementary teachers who partic­
ipated in the program,
tencies,

in relation to the specified compe­

(3) assess whether the students have achieved a

degree of mastery over these competencies,

(4) evaluate the

basic mathematical understandings of the students who
participated in the program prior to and after instruction,
in order to assess the effectiveness of the prescribed
mathematics treatment on their general mathematical knowl­
edge,

(5) assess the experimental program's effect on the

attitudes toward mathematics of the students who partic­
ipated in the program,

(6) compare the students in the

experimental group with students

in the regular teacher

education program in relation to their mathematical under­
standings and attitudes,

and

(7) determine the relationship

between selected variables and achievement in mathematics.
Review of Literature
The increased concern of mathematics educators from
all over the world with the necessity of improving m a the­
matics education has been very influential in the development

171

of new t e a c h e r educa t i o n p r o g r a m s and in the impro vem ent
of m ethods of evaluation.
S i n c e teach er s and their e d u c a t i o n are the pri nc i p a l
substance beh ind any
ment of e d u c a t i o n a l

'fort m a d e
s

for the u l t i m a t e i m p r o v e ­

terns, e d u c a t o r s have d evo ted a

great deal of time to the im p r o v e m e n t of teache r education
programs,

d e v e l o p i n g cr iteria

for the t r a ining of p r o s p e c ­

tive t e a che rs of m a t h e m a t i c s at both the pri m a r y and the
secondary

level.

Chang es

in the c ontent of cu rricula h a v e

been a c c o m p a n i e d by e x p e r i m e n t s
teaching methods,

in the d e v e l o p m e n t of ne w

and by new r e s earch on the m a t h e m a t i c a l

co mp etencies needed by e l e m e n t a r y school
elem ent ary

teac her s and

school children.

C u r r i c u l u m res earch and e v a l u a t i o n has con tinued to
progress.

Whil e summa t i v e e v a l u a t i o n

is still regarded as

an a d e quate and neces s a r y m e t h o d to m a k e d e c i s i o n s about
cur r i c u l u m adoptio n and e f f e c t i v e use,
techniques

are c o n s i d e r e d more and m o r e

curr i c u l u m s p e c i alists duri n g

fo rm ative e v a l u a t i o n
impo rtant by m o s t

the d e v e l o p m e n t of a te a c h e r

education pr o g r a m and also for ins t r u c t i o n and student
learning.
C o n c e r n e d by the i m p o r t a n c e of
surement of achievement,

many

i mprovi ng the m e a ­

r e s e a r c h e r s and educators

point ou t the need for c r i t e r i o n - r e f e r e n c e d t esti ng as a
part of c u r r i c u l u m eval uat ion .

They i n d icate

that

172

c o n ventional

testing instruments,

altho ugh effective

in

d i f f e r e n t i a t i n g among i n d ividual student's per formance,
are not always efficient, nor even valid,
student's p e r f o r m a n c e on s p e c i f i e d
Re sear chers have pro posed
refer enc ed m e a sures

for assessin g

learning objectives.

a m o r e extensive use of criterion-

in the a s s e s s m e n t of the degr e e of

comp ete nce attai ned by a p a r t i c u l a r student.

This

type of

m e a s u r e m e n t is re latively new in educa t i o n bu t the d e v e l o p ­
me nt of i n s t r u c t i o n a l tec hno log y

and the rece nt em phasis on

cur r i c u l u m res earch and c u r r i c u l u m eval u a t i o n have str ess ed
the need for the kin d of i n f o r m a t i o n made avail a b l e by

the

use of c r i t e r i o n - r e f e r e n c e d measures.
The attitude s

tow ard m a t h e matics

that the p r o ­

spe ct ive teache rs h o l d are almo s t as im portant as cogni t i v e
learning in mathemati cs,
instruction,
If, w h i l e
like

since math ematics,

is inten ded to form a base

learning mathema tics ,

for the subject,

further

like any school

for future

learning.

the stude nt acquire s

learning

is unlikely,

a dis­
and part

of the p urpose of in s t r u c t i o n is lost.
At t i t u d e s
ext ens ively

toward m a t h e m a t i c s have been

in vestigated

in r e l ation to p e r s o n ality ch aracteristics,

teacher's a tti tudes and effe cti vene ss,
and the new m a t h e m a t i c s curricula.

students'

achievem ent

173

Hypotheses
The ef fec t of the m a t h e m a t i c s compo n e n t of the
experim ent al p r o g r a m on the cognit ive and a ffective b e h a v i o r
of the e x p e rimental group were assess ed by the following
hypotheses:
A2.

There
the

will

be

poot-teat

experimental

a

significant

meana
group

and
on

the

the

difference
pre-teat

between

meane

of

the

criterion-referenced

meaureo.

The un ivariate hypo t h e s e s ass ociated w i t h this
m u ltivariate hy pothesis were:
The me an p o s t - t e s t score of the experi m e n t a l g r o u p
will be s i g n i ficantly hig h e r than the m e a n p r e -test score
on the c r i t e r i o n - r e f e r e n c e d mea sures
a.
b.
c.
d.
e.
f.
g.
h.
i.
>12.

in:

Measurement
Numeration
Sets and Set Rel ati ons
Wh ole Numbers
Fra ctions
Decimals
Re lat ions and Functions
P r o b a bility and Sta tistics
M a t h e m a t i c a l Systems.
There

will

poet-test
on

the

be

no

means

oignificant
and

the

difference

mastery

criterion-referenced

level

between

(80 p e r c e n t )

measures.

The uni v a r i a t e hyp o t h e s e s a s s o c i a t e d w i t h this
m u ltivariate h y p othesis were:

the

174

The m e a n p o s t - t e a t score of the e x p e r i m e n t a l g r o u p
will be at least equal to the ma s t e r y
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
a.
b.
c.
d.
e.
f.
g.
h.
i.

Bl.

(80 percent)

on

in:

M e a s u remen t
Nume r a t i o n
Sets an d Set Relati ons
Whole Nu m b e r s
Frac tio ns
Decimals
Rel ati ons and Funct i o n s
P r o b a bi lity and Stati sti cs
M a t h e m a t i c a l Systems.

There

will

basic

mathematical

test

be

scores

pre-test
B2.

level

There

be

arithmetic
of

of

significant

difference

understanding

the

experimental

on

between
group

a teet
the

and

of

poet-

their

scores.

will

scores

a

a

significant

attitude
the

difference

inventory

experimental

between

group

and

on

an

the

post-teet

their

pre-test

scores.

The foll owing two h y p o t h e s e s were tes t e d to compa re
the e x p e r i m e n t a l grou p and studen ts e n r ol led

in the regular

teac her education p r o g r a m on b asic m a t h e m a t i c a l u n d e rstanding
and at tit udes
Cl.

The

to ward arithmetic.

adjusted

mean

experimental
adjusted

group

mean

prospective

post-test
will

post-test

elementary

be

scores
at

of

the

l ea st

equal

to

of a

group

of

scores
teachers

enrolled

in

th e

th e

175

regular

C2.

teacher

education

basic

mathematical

There

will

arithmetic
poet-test
adjueted

a

of

poet-teet
teachers

on

a

test

of

understanding.

eignificant

attitude
ecoree

elementary
education

be

program

difference

inventory
the

between

experimental

ecorea

of a group

enrolled

in

the

in
the

an
adjusted

group

and

the

of prospective
regular

teacher

program.

The Int e grated C o n t e n t - M e t h o d s
Co urs e
A te am of mathema t i c i a n s ,
and e l e m e n t a r y sc hool teache rs
mathematical experience
men t a l program.

Nine

for the

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

formu lat ed the int egr ate d
first yea r of the e x p e r i ­

learning un its w e r e d e s i g n e d in

acco r d a n c e w i t h guid e l i n e s p r o p o s e d by the BSTEP Model.
Ea ch

lea rn ing uni t w a s devote d to a m a t h e m a t i c a l topic

deem e d n e c e s s a r y

for ele m e n t a r y school t eacher education.

The topics were:
1.
2.
3.
4.
5.
6.
7.
8.
9.

Measurement
N u m e r a t i o n Systems
Sets and Set Re la tion s
Whole Numbers
Fract i o n s
Decimals
R e l a t i o n s and Fun ctions
P r o b a b i l i t y and St atistics
M a t h e m a t i c a l Systems

176

The learning units had the f ollowing common
1.

G o a l s and obje ctiv es

features:

(mathematical competencies)

for that topic.
2.

E x p e r i e n c e s and s t rategies utilizing manipulative,
a u d i o - v i s u a l and o t her i nstr u c t i o n a l materials

to

achieve these objectives.
3.

C r i t e r i o n - r e f e r e n c e d tests
to assess the students'

(two equ i v a l e n t

forms)

pre- and p o s t - t r e a t m e n t

b e h a v i o r s on the spe cif ied m a t h e m a t i c a l
competencies.
Th e c r i t e r i o n - r e f e r e n c e d tests, d e v e l o p e d by this
investigator,

y ielde d m e a s u r e m e n t s that w e r e dire ctl y

i nte rpretable in terms of the spe cified mathe m a t i c a l
compete nci es.
structed

The m eth o d in w h i c h these tests were c o n ­

insured their c ontent validity.

co e f f i c i e n t s

The reliab ili ty

for these tests var i e d from 0.77 to 0.93, w hich

is a c c e p t a b l e for c r i t e r i o n - r e f e r e n c e d tests.
The

integrated c o n t e n t - m e t h o d s cour s e met eigh t

hours a w e e k for nine weeks.
mat ics

It was c o n d u c t e d in a m a t h e ­

labora tor y e qui pp ed w i t h m a n i p u l a t i v e and o t h e r

instructional materials.

The student s w o r k e d in grou ps of

four on the activ iti es pres c r i b e d in the learning unit.
Th e n at th e end of each w e e k they planned, w i t h the
a s s i s t a n c e and s u p e r visi on of instructors and elemen tar y
school teachers,

instructional des ign s to be used w i t h

177

elementary school c h i l d r e n the fo llo wing week.

Eac h m e m b e r

of the g r o u p was r e s p o n s i b l e for pa rt of the te aching of
four or five ele m e n t a r y school children.
clinical experience,

D u r i n g the

students spent on e full m o r n i n g per

week in an elementary school.

T h r e e hours w e r e

spent

w orking w i t h or o b s e r v i n g c l a s s r o o m teachers.

Th e remainin g

hour was

five pupils

spent teachi ng m a t h e matics

to four or

in a m a t h e m a t i c s lab oratory setting,

and r e c e i v i n g feedback

from teach er e d u c a t o r s wh o o b s e r v e d the t e a c h i n g experience.

Students in the Study
T h e stude nts in the e x p e r i m e n t a l p r o g r a m wer e

fresh­

men e l e m e n t a r y e d u c a t i o n maj ors wh o v o l u n t e e r e d and w e r e
selected to p a r t i c i p a t e in the program.
two e n t e r i n g
selected.
dropped

freshmen volunteered,

Initially,

forty of w h o m were

At the b e g i n n i n g of the school-year,

from the program.

fifty-

two student s

The r e m a i n i n g th irty - e i g h t p r o ­

spective elem e n t a r y school teachers, w h o p a r t i c i p a t e d

in

the first trial impl e m e n t a t i o n of the m a t h e m a t i c s component,
compr i s e d the e x p e r i m e n t a l group for this study.

Other

groups of students w e r e utilized for com p a r i s o n purposes
and for tes ting the r e l i a b i l i t y of the m e a s u r i n g instrum ent s
developed in this study.

178

In s t r u m e n t a t i o n
The

following i n s t r ument s w e r e d e v e l o p e d or selec ted

for the coll e c t i o n of data:
ac h i e vement me asures

(1) N i n e 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.

Dossett's 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
allel

f o r m s ) , (3) Dutt on A r i t h m e t i c A t t i t u d e

(two p a r ­
Inventory,

(4) A t t i t u d e Scales T o w a r d D i f f e r e n t Asp ect s of Ma them ati cs,
devel o p e d by The International S t u d y of A c h i e v e m e n t in
Mathematics,

and

(5) M S U Basic S k i l l Tests

in A r i t h m e t i c

and Ma the mat ics.
S t a t i stical Analysis
M u l t i v a r i a t e and u n i v a r i a t e analysi s of v a r iance
were used to det ermine

the eff ect of the instructional

tr eatment u p o n the e x p e r i m e n t a l g r o u p p e r f o r m a n c e on the
c r i t e r i o n - r e f e r e n c e d tests.
was

Th e

t-test for corr e l a t e d means

used in c omparing chan ges in the e x p e r i m e n t a l g r o u p and

a grou p of stude nts

in the re g u l a r teacher e d u c a t i o n p r o g r a m

on their m a t h e m a t i c a l
mathemat ics .

u n d e r s t a n d i n g and att itu de s

toward

The Du n n e t t t-test w a s used to assess s i g n i f ­

icant d i f f e r e n c e s b e t w e e n the e x p e r i m e n t a l group and other
freshman g r o u p s on their en tering c o g n i t i v e and af fective
b eha v i o r s

tow a r d mathema tic s.

Th e Pea r s o n p r o d u c t m o m e n t

c o r r e l a t i o n co e f f i c i e n t was u t i l i z e d in the relati o n s h i p
analysi s r e p o r t e d in this study.

179

The 5 p ercent

level of significa nce was chos en for

acc ept ing or re je cting the resea rch hy potheses

in this study.

Limit a tions of the Study
The pres e nt study contains several
must be kept in mind when

limitations whi ch

int erp reti ng the results of this

investigation.
1.

While

there are several goals

that p ertain to the

formative eval u a t i o n of an educa tio nal program,
study e v a l u a t e d only
the e x p e r i m e n t a l
2.

this

the m a t h e mati cs componen t of

teacher e ducation program.

E v a l u a t i o n of the p r o g r a m was confin ed so lely to
those p r o s p e c t i v e e l e mentar y

teachers w h o v o l u n ­

teered and we re se lect ed to partic ipate in the first
year trial implemen t a t i o n of the program.
3.

The study did not at tem pt to evalu ate the eff ect
the int e grated content - m e t h o d s

course on the t each­

ing b e h a v i o r of its recipients

in ele mentary school

setting.
4.

The study di d not attemp t to ev aluate the effect
of the experi m e n t a l p rogram on the m a t h e m a t i c a l
c o mpetency of the school ch ildren wh o w e r e

taught

by the e x p e r i m e n t a l p r o g r a m participants.
5.

of

The ext e n t to w h i c h
a d equately m e a s u r e d

the ev aluativ e instruments
the effects of the integ rat ed

c o n t e n t - m e t h o d cou rse and the clinic al exp e r i e n c e

180

was also a limitation.

The inst ruments used in

this study had the inherent
a nd- p e n c i l

limi tations of pa per-

tests.

Findi ngs

of the Study

Analy sis of the M a t h e m a t i c a l
C ont ent of the E x p e r i m e n t a l
P rogram
A s corecar d of the m a t h e m a t i c a l

topics sugge s t e d

for the p r e p a r a t i o n of e l e m e n t a r y sc hoo l teach ers was
co nstructed b a s e d on the r e c o m m e n d a t i o n s of m a t h e mati cs
educators,
studies,

n a t i o n a l l y - r e c o g n i z e d advi sor y groups,

and el ementary

teacher's guides.

school m a t h e matics

r e s earch

textbooks

and

The s c ore card ser ved as a criteria-

referen ced list for assessing

the ade quacy of

the

math ematics c o n t e n t of the e x p e r i m e n t a l program.

The

investi gat or noted that the m a t h e m a t i c s comp o n e n t of the
e x perimental p r o g r a m included 94 per c e n t of the topics on
the d e v e l o p e d c r i t e r i a - r e f e r e n c e d

list,

topics not s u g g e s t e d by special ist s,

and a l s o other

but i n cor porated

facilitate the d e v e l o p m e n t of othe r requ ire d
was concl u d e d that the m a t h e m a t i c a l

topics.

to
It

competencies prescribed

by the e x p e r i m e n t a l p r o g r a m were suf f i c i e n t
needs of the future ele m e n t a r y sc hoo l

in meet ing

teacher

the

in mathematics.

181

Comp arisons of the E x p e r i m e n t a l
Group w i t h O t h e r frresnman Groups
on E n t e r i n g C o g n i t i v e a n d
Af fective Be haviors To wa rd
M athe mat ics
S tuden ts
with three
in:

in the e x p e r i m e n t a l group wer e com pared

fi rst-term fr es hm an gro ups w i t h d e c l a r e d majors

(1) elem e n t a r y e d u c a t i o n — b u t did not v o l u n t e e r

the e x p e r i m e n t a l program,
education,

and

(3) mathema tic s.

were a d m i n i s t e r e d to all
of 1971-1972:

(2) m a t h e m a t i c s

and

and se co ndary

following

four grou p s during

(1) MSU b a s i c ski ll

ar ith metic and ma the mati cs,
Scale,

The

(2)

for

in struments

the Fall

and p l a c e m e n t

term

tests

Du tton A r i t h m e t i c A t t i t u d e

(3) A t t i t u d e Scales Towa r d D i f f e r e n t A s p e c t s of

M a t h e matics

(developed by

Achievement

in M a t h e m a t i c s ) .

the I n t e r n a t i o n a l Study of
R esults o b t ained sho wed

the e n t e r i n g co gn itive and a f f e c t i v e b ehav iors
matics of the students w h o v o l u n t e e r e d

those of o t her

however,

freshmen w i t h s i m i l a r

school teachers.

to p a r t i c i p a t e

in the
from

interes t of becomi ng

T h e s e e n t e r i n g behaviors,

w e r e s i g n i ficantly d i f f e r e n t

freshmen w i t h s pecifi ed i n t e r e s t

from those of other

in m a t h e m a t i c s

m a t h e m a t i c s and s econdar y e d u c a t i o n majors,
m a j o r s ).

that

toward m a t h e ­

e x p e r i m e n t a l p r o g r a m w e r e not s i g n i f i c a n t l y d i f f e r e n t

elementary

in

(the

and ma t h e m a t i c s

182

E v a luation of the E x p e r i m e n t a l
Group P e r f o r m a n c e on the
Cr i t e r i o n - R e f e r e n c e d Mea s ures
The eff ect of the instruction al treat m e n t w a s
evaluated by analy z i n g

the e x p e r i m e n t a l g r o u p p erformance

on the c r i t e r i o n - r e f e r e n c e d measures.
Hypothesis A 1 was

tested to d e t e r m i n e

the s i g n i f ­

icance of gain in a c h i e v e m e n t on the p r e s c r i b e d mat hem a t i c a l
competen cie s be t w e e n pre- and post - 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 measures.
F i n d i n g s .- - The m u l t i v a r i a t e

test in dic ated

that the

overall dif ference b e t w e e n pre-

and p o s t - t e s t me ans was

highly s i g n i f i c a n t

A n a lysis

hy potheses

(p < .0001).

of the univ ari ate

ass o c i a t e d w i t h this test yi e l d e d p r o b a bilities

that ind icated s i g n i f i c a n t gain b etween pre- and post- 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 m e a s u r e s
1.

Numeration

2.

Sets and S e t Re lat ions

3.

Whole Numbers

4.

Frac tions

5.

Deci mal s

6.

Re lat ions

7.

P r o b a b i l i t y an d St atistics

8.

Mathematical

(p < .005),
(p < .0001),

( p < .0001),

( p < .0001),
( p < .0001),
and Func tions

There was,
of M e a s u r e m e n t

in:

(p <

Syst ems

however,
.3639).

( p < .0001),
(p < .0001),

and

( p < .0001).

no s i g n i f i c a n t gain on the test

183

Hy pot hesis A2 w as

tested to det erm ine w h e t h e r a

sp ecified de gr ee of m a s t e r y
the post-test)

(a scor e of 80 or high er on

was achieved.

F i n d i n g s .--The m u l t i v a r i a t e

test i ndicated that the

overall dif ference b e t w e e n p o s t - t e s t means an d the m a s t e r y
level was hi ghly s i g n i f i c a n t
univariate hypo the ses

(p < .0001).

Analy s i s of the

a s s ociated with this

test indi cat ed

that the e x p e r i m e n t a l group's p o s t - t e s t sco res were not
signi ficantly d i f f e r e n t

from the maste ry level on the

c r i t e r i o n - r e f e r e n c e d measu res in:
1.

N u m e r a t i o n Sys t e m s

(p < .2210),

2.

Set and Set Relati ons

3.

W h o l e Numbe rs

4.

Fr ac tion s

5.

Decima ls

6.

Rel ations and Fun ctions

7.

P r o b a bil ity and Statis tic s

(p < .1450),

( p < .7117),

( p < .7135),
( p < .5378),
( p < .3147),

and

(p < .2397) .

The post- t e s t scores of the e x p e r i m e n t a l group were,
however,

s i g n i ficantly below the mas te ry

c r i t e r i o n - r e f e r e n c e d me asu res
and

(2) M a t h e m a t i c a l Systems

in:

level on the

(1) M e a s u r e m e n t

( p < .001),

(p < .0002).

B a s e d on these results,

it was c o n c l u d e d that the

i n s t r uctional t r e a t m e n t pr oduced positi ve results,

that the

ex p erimental group a c h i e v e d m a s t e r y over the m a t h e m a t i c a l
compete nci es p r e s c r i b e d in seve n of the n i n e m a t h e m a t i c a l
topics of the i n t e g r a t e d content - m e t h o d s course.

184

The E f f e c t of the E x p e r i m e n t a l
Pro gram on the b a s i c M a t h e m a t i c a l
Un derstanding s and A t t i t u d e s
Toward M a t h e matics
T w o h y potheses w e r e

tested to e v a l u a t e

the e f f e c t

of the i n t e g r a t e d c o n t e n t - m e t h o d s course a n d the clinical
experi enc e on 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
attitudes

toward mathemat ics .

The first#

an d

H y p oth esis B1

(related 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 ) # w a s
stated as:
teat

of

test

scores

There

baeic
of

will

he

a significant

mathematical
the

difference

understanding

experimental

group

between

and

their

on

the

a
post-

pre-test

scores.

F i n d i n g s .- -Testing this hypothesis#

the inve sti gat or

found that the mean p o s t - t e s t scores w e r e n u m e r i c a l l y

as

well as s i g n i f i c a n t l y h i g h e r than the m e a n p r e - t e s t scores.
Based on this data the h y p o t h e s i s was acce pted.

It was

co ncluded that the i n t e g r a t e d c o n t e n t - m e t h o d s co urs e had
a s i g n i f i c a n t p o s itive e f f e c t on the m a t h e m a t i c a l u n d e r ­
standing of
results

the e x p e r i m e n t a l group.

A n a l y s i s of p r e - t e s t

indicated lack of u n d e r s t a n d i n g of b a s i c m a t h e m a t ­

ical p r i n c i p l e s u n d e r l y i n g
and r e f l e c t e d the students'

the

four a r i t h m e t i c operations#

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

emph a s i z e d c o m p u t a t i o n a l skill.

P o s t - t e s t results

indicated

a d e f i n i t e i m p r o v e m e n t in under s t a n d i n g n u m b e r properties,
m u l t i p l i c a t i o n and d i v i s i o n algorithms,
decimals

and fractions.

a n d o p e r a t i o n s with

The empha sis p l a c e d on the structure

185

of the n u m e r a t i o n syst ems d u r i n g the in t e g r a t e d
c ont ent - m e t h o d s

cour s e was cle arly r e flected on the

student p e r f o r m a n c e on related
Hyp o t h e s i s B2
mathematics)
difference
the

an

poot-tcat

pre-teat

arithmetic

ecorea

in the post-test.

(related to the at titud e toward

was s t a t e d as:

on

items

of

the

There

will

attitude

be

a

eignifleant

inventory

experimental

group

between
and

their

ocoree.

F i n d i n g s .- - T h e Dutton A r i t h m e t i c A t t i t u d e Scal e was
used to assess

the change of atti tude of the e x p erimen tal

group d u r i n g the first year of their teacher education.
P ossi ble

scores on this scale

dislike)

to 10.5

range from 1.0

(extreme like).

(extreme

On the p r e - t e s t the

e x p e r i m e n t a l g rou p's mean s c o r e was

5.57 w h i c h

reflects

the n e g ative attit u d e s mos t of these studen ts had toward
arithmetic.

On the post - t e s t

the m e a n was 7.07.

diff e r e n c e wa s h i g h l y s i g n i ficant
this result Hyp o t h e s i s B2 wa s

(p < .001).

1.

the

foll owing

Based on

accepted.

R elated q u e s t i o n s .— In con n e c t i o n w i t h
study,

The

the attitu de

results were obtained:

Fe elings

towa rd a rit h m e t i c are

However,

the most c rucial y e a r s

seem e d to b e when they were
the si xth grade.

formed in all grades.
for the students

in the third through

186

2.

The cha llenge pres e n t e d by a r i t h m e t i c and its
pra ctical

ap plic a tion w e r e the aspe cts most liked

about arithmetic.
3.

Story problems,

teachers,

and m e m o r i z i n g rules

wer e the aspects m o s t disliked a bout arithmetic.

In summary,

the mat h e m a t i c s

compone nt of the

expe r imental p r o g r a m had a po sitive ef fec t on im proving
the e x per imental group 's atti tudes

toward mathematics.

C o m parison of the Ex perimental
Group w i t h a R e g u l a r El ementary
Edu cat ion Gro up on M a t h e m a t i c a l
U n derstandings and A t t i t u d e s
To ward Mathematics
The e x p e r i m e n t a l grou p was c o m p a r e d with a grou p of
students

in the regul ar elem ent ary e d u c a t i o n p r o g r a m w h o had

co mple ted the ma t h e m a t i c s
in the meth ods

course c o n currently w i t h

c o n t ent-methods course.
first,

to

scores
the

prospective
teacher

of

the

adjus ted
e lemen tary

education

the inte gra ted

Two h y p otheses w e r e tested.

Hypothe sis Cl, wa s

post-test
equal

content cour se and were e n r o l l e d

stated as:

experimental

mean

post-test

teachci's

program

on

a

The
grou}>

uill

scores

enrolled

test

adjusted

of

in

basic

of
the

be
a

The

mean
at

least

group

of

regular

mathematical

understanding.

F i n d i n g s .- - The anal ysi s of cov a r i a n c e was used
the an alysis of data, w i t h

the p r e -te st scores

as the

for

187

covariate variable.
signi ficant

Th e o b t a i n e d F r a tio w a s hig h l y

(p < .001)

in favor o f the e x p e r i m e n t a l group.

It was t herefore concluded that the two final m e a n s , when
initial d i f f e r e n c e s w e r e a l l o w e d for, di d dif fer s i g n i f i ­
cantly in favor of the e x p e r i m e n t a l group.

Thus,

Hypothesis Cl was accepted.
H y p o t h e s i s C2 w h i c h sta t e d that:
aignif icant

difference

between

adjuoted

group

the

and

the

proopeative
teacher

an

arithmetic

poet-teat

adjueted

elementary

education

in

ecoreo

poot-teet
tcachere

of

eaoree
enrolled

There

will

attitude
the
of
in

a

experimental
group
the

of

regular

program.

for s i g nificanc e of d i f f e r e n c e s b etween

final means.

a

inventory

F i n d i n g s .— The analysis of c o v a r i a n c e was
to assess

he

The F rat io o b t a i n e d

related to this hypoth esis was

initial di f f e r e n c e s

a llowed for,

adjusted

from the a n a l y s i s of d a t a

highly s i g n i f i c a n t

It was there f o r e concl u d e d that

utilized

( p < .001).

the e x p e r i m e n t a l group, w i t h
had s i g n i f i c a n t l y more

posi tive a t t i t u d e tow a r d a r i t h m e t i c than did a g r o u p of
students

in the regu l ar e l e m e n t a r y

Thus H y p o t h e s i s C2 wa s

t eacher e d u c a t i o n program.

accepted.

C o r r e l a t i o n Ana lysis
Data c o l l e c t e d on m a t h e m a t i c s a c h i e v e m e n t ,
m a t h e m a t i c a l un der standings,
ma tic s w e r e

and a t t i t u d e s

stud ied to d e t e r m i n e w h e t h e r

basic

toward m a t h e ­

these effec ts w e r e

188

related or inf l u e n c e d by o t h e r

factors,

such as the level

of h i g h - school m a t h e m a t i c s p r e p a r a t i o n and grade point
averages.
tests,

For this p urpo se scores

fr om all pre- and p o s t ­

as w e l l as other b a c k g r o u n d d a t a of the e x p e r i m e n t a l

grou p were used to c alcul ate an i n t e r c o r r e l a t i o n matrix.
all,

In

the c o r r e l a t i o n data i n d i c a t e d significantly positive

relations b etween

the c r i t e r i o n - r e f e r e n c e d me as ures,

n o r m - r e f e r e n c e d tests of m a t h e m a t i c a l
the atti tude scale.

the

unde rstandings,

and

H i g h - school m a t h e m a t i c s p r e p a r a t i o n

and over all grade p o i n t a verage w e r e signi f i c a n t l y c orre­
lated to m a t h e m a t i c a l perfor man ce,

underst andi ngs ,

and

atti t u d e s .

Student R eactions to the
M a t h e m a t i c s C o m p o n e n t of
the E x p e r i m e n t a l P r o g r a m
The
wer e as ked

th ir ty-e igh t students
to re act

in the e x p e r i m e n t a l g r oup

to the m ethods and p r o ce dures used in

the impl e m e n t a t i o n of the int e g r a t e d content - m e t h o d s
the m a t h e matics
Overall,

course,

lab or ator y and the clinica l experience.

reactio ns

to the e l e v e n - i t e m q u e s t i o n n a i r e see med

to indic ate a g e n e r a l s a t i s f a c t i o n w i t h
aspects of the program.

However,

the d i f f e r e n t

the students

felt that

not enough e m p h a s i s was p l a c e d on plann i n g instr uctional
st r ategies

to be used wit h e l e m e n t a r y

school pupils.

The

ma j o r i t y of the students o f f e r e d s o m e comments abou t the
i m p r o v e m e n t of the program.

189

C onc lusions

The analy si s of the da ta gat hered in this study
and p r e s e n t e d in the p r e c e d i n g chapters appears
a number of conclusions.
evidence ob tained

to w arrant

T h e s e conclusions are base d on

from the findings of the p resent study

and the inve s t i g a t o r ' s o b s ervation s and in ter pre tat ions of
these results.
1.

A n a l y s i s of the m a t h e m a t i c a l

con ten t of the

experime nta l p r o g r a m ind icated a strong agreemen t w i t h
present e l e menta ry

school m a t h e matics

the

cont ent and with re c­

ommendat ion s of p r o f e s s i o n a l organizations of mat hematics
educators

and research groups.

It was

concluded,

therefore,

that the m a t h e m a t i c a l c o mpetencies pre s c r i b e d in the e x p e r i ­
mental p r o g r a m w e r e s u f f i c i e n t in m eetin g
future ele m e n t a r y
2.

Since

the needs of

teachers of arithmetic.
the gr oup of

freshman p r o s p ective e l e m e n ­

tary teachers w h o co mprised the e x p erimental group v olun­
teered and were s e l ected to the program,
to d etermine if they differ significantly
chara cte ris tics

it wa s

important

in their entering

f r o m other fresh man groups on co gnitive and

affective b e h a v i o r s

toward mathema tic s.

Th ree groups of

freshmen w e r e used for com p a r i s o n purposes:

(1) a group

of e l e m e n t a r y e d u c a t i o n majors ,

(2) a group of m a thematics-

secondary e d u c a t i o n majors,

(3) a grou p of mathematics

majors.

and

Results of the analy s i s

ind icated that the

190

e x p e r i m e n t a l group d i d not d i f f e r

in their c ognitive and

affec tiv e beh avior tow a r d m a t h e m a t i c s
the same p r o f e s s i o n a l interests
majors).

from a grou p w i t h

(e.g.,

e l e m e n t a r y educa t i o n

The e x p e r i m e n t a l g r o u p ' s co gnit ive b e h a v i o r toward

m a t h e m a t i c s was s i g n ificantly

lower than tha t of

students w i t h m a n i f e s t e d i n t e r e s t in m a t h e m a t i c s
m a t h e m a t i c s - s e c o n d a r y educ a t i o n majors,

freshman
(e.g.,

and m a t h e m a t i c s

majors).

These two gr oup s also h a d signi f i c a n t l y hi g h e r

attitude s

toward a r i t h m e t i c than di d the e x p e r i m e n t a l group.

3.

On the c r i t e r i o n - r e f e r e n c e d me asures,

e x p e r i m e n t a l group s h o w e d s i g n i f i c a n t gains
(p < .005)

and Functions,
Systems.

in achievement

on the m e a s u r e s of N u m e r a t i o n Systems,

Set Relations, W h o l e Numbers,

Fr actions,

the

Sets

Decimals,

P r o b a b i l i t y and Statistics,

and

Relation

and Mat hem a t i c a l

The e x p e r i m e n t a l g r o u p disp l a y e d a positive,

not significant,
4.

but

g a i n on tests of Mea surement.

When p o s t - t e s t scor e s were c o m p a r e d with

the

mastery level, w h i c h was a s c o r e of 80 or hig h e r on the
post-test,

results sho w e d that the e x p e r i m e n t a l g r oup

attai ned the p r e s c r i b e d mastery
ation Systems,
Fra ctions,

level on m e a s u r e s of N u m e r ­

Sets and Set Rel a tions, W h o l e Numbers,

Decimals,

ity and Statistics.

Rel ations

and Functions,

and P r o b a b i l ­

The e x p e r i m e n t a l groups di d not reach

the level of m a s t e r y on the m e a s u r e s of M e a s u r e m e n t and
M a t h e m a t i c a l Systems.

191

5.

The e x p e r i m e n t a l group sho w e d s i g n i f i c a n t gains

on a test of ba sic m a t h e m a t i c a l und ers tand ing s.
6.

The e x p e r i m e n t a l g r oup sh owe d s i g n i f i c a n t

positive g ains
7.

in attitudes

toward mathemat ics .

Wi th ini tia l d i f f e ren ces a l l o w e d for,

the

exp eri men tal group s c o r e d signi f i c a n t l y hig h e r on a test of
basic m a t h e m a t i c a l unde r s t a n d i n g s
dents in the regular el ement ary
8.

than d id a g r o u p of s t u ­

teacher e d u c a t i o n program.

Wi th i nitia l d i f f e rences a l l o w e d for,

the

experim ent al group e x h i b i t e d s i g n i f i c a n t l y more p o s i t i v e
attitudes

to war d m a t h e m a t i c s

in the r e g u l a r elem e n t a r y
9.

than did a g r o u p of st ud en ts

tea che r e d u c a t i o n program.

The e x p e r i m e n t a l group e x h i b i t e d desi r e

for

more p a r t i c i p a t i o n in clinical e x p e r i e n c e c o n c u r r e n t l y wit h
the la boratory o r i e n t e d i n t egrated c o n t e n t - m e t h o d s

On

the basis of changes

course.

in m a t h e m a t i c a l a c h i e v e m e n t

on the c r i t e r i o n - r e f e r e n c e d m e a sure s and the tests of b a s i c
m a thematica l understa n d i n g s and attitudes,

results of data

analysis p r o v i d e d e n c o u r a g i n g signs

(1) c o m b i n a t i o n

of emphasi s

on m a t h e m a t i c a l c ontent and c o m m i t m e n t

making m a t h e m a t i c s
be achieved,
mathemat ics

that:

toward

u n d e r s t o o d by p r o s p e c t i v e tea cher s can

(2) u n i fyin g theory of te aching and
concurrently with

learning

labora tor y and c l i n i c a l

ex perience p rovide p o s itive m e t h o d s of im pr ovin g

the

cognitive and affec t i v e be hav iors of the p r o s p e c t i v e

192

el eme ntary teacher s toward mathemati cs,

(3) the use of

m ani pu l a t i v e m a t e r i a l s is eff ect ive in teaching prospe cti ve
element ary teachers
ex pec ted to teach,

the b a s i c mathem a t i c a l concep ts
and

(4)

they are

the clin ical ex perience provides

a framework from which the p r o s p e c t i v e teachers

could apply

the theor eti cal conte nt of their courses.

D i s cussion
On the c r i t e r i o n - r e f e r e n c e d tests in Measurement,
the e x p e r i m e n t a l group's p o s t - t e s t m e a n

(M = 66. 32) was not

sign ifi can tly d i f f e r e n t from the p r e - t e s t mean
The p o s t - t e s t m e a n remain ed s i g n i ficantly

( M =63.97).

(p < .0001)

below

the m a s t e r y level

(a score of 80 or h i g h e r ) .

may be a t t r i b u t e d

to two reasons:

first topic to be

taught in the i n t e g r a t e d content - m e t h o d s

course in a s etting

These results

(1) M e a s u r e m e n t was the

(mathematics laboratory)

unf a m i l i a r to

most if not all

the students.

the f irst week,

as the in structors

a c q u a i n t e d with

the new s u r r o unding a n d familiarizing t h e m ­

selves w i t h the

functions of m a n i p u l a t i v e and other i n s t r u c ­

tional ma terials.

The stude nts s p e n t much of
intended,

beco min g

This m i n i m i z e d the amount of time spent

on m a t h e m a t i c a l ac ti vities

as sociat ed with Me asu rem ent ,

and

(2) A mo r e p l a u s i b l e reason may hav e been that the po st-test
in M e a s u r e m e n t w a s more d i f f i c u l t than the pre-test.
Analy s i s of pre-

and post - t e s t items

in dic ated that more

em pha sis on c o m p rehe nsion of m a t h e m a t i c a l conce pt s related

193

to p rec ision in m e a s u r e m e n t w a s pla ced on the p o s t - t e s t
while the p r e - t e s t e m p h a s i z e d c o m put ations of probl ems
related to the above concepts.

Analysis of r e l i a bi lity

estimates of the c r i t e r i o n - r e f e r e n c e d tests re vealed that
the c o r r e l a t i o n b etw een pre- an d p o s t - t e s t results
Measurement

(r = 0.64)

was the

lowest for the nine m e a s u r e s

and that the re li ability c o e f f i c i e n t for p o s t - t e s t
was also the

in

(r = 0.77)

lowest for all measures.

On the c r i t e r i o n - r e f e r e n c e d test in M a t h e m a t i c a l
Systems,

the e x p e r i m e n t a l g r o u p e xhib ited a highly s i g n i f ­

icant i m p r o v e m e n t
the p r e - t e s t

<p < .0001)

(M *= 33. 52)

be t w e e n their p e r f o r m a n c e

and the post- t e s t

ever the p o s t - t e s t m e a n was signi f i c a n t l y
the m astery
factor:

level.

only

(M = 65.53),
( p < .0002)

on
how­

belo w

This was attributed to the fo ll owing

five class hours were a l l o c a t e d to this

topic,

mos t s t u dents did not finish the p r e s c r i b e d a cti v i t i e s
this l imited time,

and the c o n t e n t of this

the m a j o r i t y of the students.

topic w as

the

not able to teach

it to the ele mentary school pup i l s due to the end of
school year.

new to

Mathem a t i c a l Syste ms wa s

last topic and the e x p e r i m e n t a l group wa s

eleme nta ry

in

the

The studen ts w e r e aware t h a t they

wou ld not h a v e to teach this

topic and it could b e

tha t they

did no t pu t the same level of ef for t into learning it as
they did w i t h othe r topics.

194

Obs ervations
1.

A c h i e v e m e n t tests,

a t t i t u d e scales,

and students

q u e s t i onna ires a r e im po rtant aspects of the fo rma tive e v a l u ­
ation p r ocess of an e v o l v i n g e d u c a t i o n a l program.
measures,

however ,

paper-and-pencil
the soc ial

suffer

tests.

fr om the i nhe re nt

Thes e

limit atio ns

of

T h e y do not e xplain the na tur e of

i n t e r action that takes p l a c e in the c l a s s r o o m

which m u s t be k n o w n if one is to p l a c e the e f f e c t i v e n e s s of
the p r o g r a m in p r o p e r pr ospective.
tions w e r e made by

teachers

course and du ring

in the i n t egrated

the clinic al exp e r i e n c e

at the e l e m e n t a r y school's m a t h e m a t i c s
A.

fo llo wing o b s e r v a ­

this i n v e s t i g a t o r as a r e s u l t of o b s e r v ­

ing the p r o s p e c t i v e ele m e n t a r y
content - m e t h o d s

The

laboratory.

D u r i n g the i n t e g r a t e d c o n t e n t - m e t h o d s course,

it was noted t h a t as the c o u r s e p r o g r e s s e d att e n d a n c e
improved.

Fr e q u e n t l y st uden ts

through the m a t h e m a t i c s
wo r k i n g with
class.

could be found b r o ws ing

l a b oratory

look ing at books or

the m a n i p u l a t i v e m a t e r i a l s b e f o r e and a fter

O f ten studen ts w o u l d stay a fter c l a s s -hours

c o m p l e t e some i n s t r u c t i o n a l a c t i v i t i e s or
their w o r k w i t h
some of

instructors.

to

to di scusss

Man y students w o u l d borr o w

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

from the m a t h e m a t i c s

l a b oratory to u s e in p r e p a r i n g their i n s t r u c t i o n a l design.
St udents sha r e d their e x p e r i e n c e at the e l e m e n t a r y sc hool
with o t h e r m e m b e r s of their g r o u p and it was

also n o ted

19 5

that each small grou p tended to form a "leader" who w o u l d
take the i n i tiati ve in aski n g q u e s t i o n s and pre paring
mat erials and act as a g r o u p leader in p l a n n i n g instruc ­
tional designs.
attitudes

Stude nts wh o w e r e shy or ha d negative

toward m a t h e m a t i c s tended to become

"followers"

and took p a s s i v e roles in the group activities,

although

such si tuations w e r e m i n i m i z e d toward the end of the course.
S inc e the students

in the e x p e r i m e n t a l group w o r k e d

in subg rou ps of four dur i n g the i n t egrated co n t e n t - m e t h o d s
course,

it is e x p e c t e d that the students'

p e r f o rmance o n

the c riterion m e a s ures w i l l no t be in fluen ced only by
instru cti onal

the

tre atment b u t also by the interac tio n b e t w e e n

member s of each subgroup.

In this study such interaction

eff ec t due to the s u b g r o u p

forma tion was not eval u a t e d

two reasons:

(1) the stu den ts

tended to cha n g e

for

from one

subgro up to a nother as the course pro g r e s s e d and they
tended to know each o t h e r better,

an d

(2) d u r i n g the c l i n ­

ical e x p e r i e n c e the int e r a c t i o n b e t w e e n membe rs of the same
sub gr oup ceas e d

to exi st since on any p a r t i c u l a r day onl y

one m e m b e r of each g r o u p was pr e s e n t at the el ementary
school.

D u r i n g the cl i nical expe r i e n c e an interaction

b e t w e e n studen ts prevailed.
In future studies,
constant,

if the subgroups

remained

then i n t e r a c t i o n w i t h i n su bgroups m u s t be

e x p l a i n e d in o r d e r to assess the o v e r a l l e f f e c t of the
inst r u c t i o n a l

treatment.

196

B.

At the elementary

exp e r i e n c e took place,
teacher and pupi ls
improve d during

interaction between

at the m a t h e matics

the nine-week

p ros pective t e a c h e r was very
from her

lesson plan.

term.

the clinica l

the p r o s p ective

lab oratory clearly
At

the beginning,

the

nervous a n d te nded to di v e r g e

This usua lly o c c u r r e d when she tried

to control the c h ildren's

n o n - m a t h e m a t i c a l behaviors.

A p p a r e n t l y she f e l t she c ould

instruct t h ese child ren only

if they were q u i e t l y listening.

She ten d e d to show the

pupils, with m a n i p u l a t i v e mate rials,
cepts

school, w h e r e

the m a t h e m a t i c a l co n ­

the pupils we r e to learn but did no t allow the m to

w o r k with the materials.

This,

however,

chan ged as the

lessons went by an d the p r o s p e c t i v e t e a c h e r let the pupils
h a n d l e and play w i t h the m a n i p u l a t i v e m a t e r i a l s w h i l e acting
more as a guide ready to a s s i s t the p u p i l s
m ath ema ti cal concepts.

In the short

teache rs spent teaching,

in d i s c o vering

time the prospec tive

they showed a w o r t h y change

their

teaching abilities.

built

through

The

the o p p o r tunities

confid enc e

in

they a t t a i n e d was

to try d i f f e r e n t techniques

wi t h children.
2.

Th e Ha wth orne e f f e c t v irtually

cr e a s e d learning on the pa rt of the st udents.
enthusiasm,

insures

The interest,

an d ca pabilities of the p r o j e c t staff no doubt

i n f l u e n c e the students'

p e r f o r m a n c e as well

a tti tudes toward mathematics.
six ma them a tics

educators

in­

as their

The p r e s e n c e of five to

to assist the t h i r t y - e i g h t

197

students

in the experi m e n t a l grou p w a s

in the success of

the progr am,

results of this study.
been doubt ful h a d
cators

This

in p l a n n i n g

their instructional s t r a t e g i e s to be

use d with

The p r e s e n c e of t e a c h e r
to provide an

to each and every p r o s p e c t i v e

students'

teachers

to the c ha n g e in a f f e c t i v e

at the eleme ntar y school

m a t h e matics

The p r e s e n c e of the

teacher to as sis t the p r o s p e c t i v e

behavior toward mathem ati cs.

feedback

de gr ee of success w o u l d have

laboratory.

childr en c o n t r i b u t e d h eavily

ed ucators

as e v i d e n c e d by the posit iv e

there b e e n only one or two t e a c h e r edu­

in the m a t h e matics

e le m e n t a r y school

an i mportant factor

imm ediate

teacher a f t e r the

lab orat ory t e a c h i n g se s s i o n also i n f l u e n c e d the

fu ture action.

i mpo rtant qu esti on:
impl eme nta ble ?

T h e s e obs ervations

involve d in such

to an

To w h a t ex ten t is such a p r o g r a m

In a large u n i v e r s i t y

State Unive rsi ty,

lead

such as M i c h i g a n

if the n u m b e r of p r o s p e c t i v e
a p rogram swell ed

teachers

to five or s i x hundred,

it w o u l d dema nd at least thirty m a t h e m a t i c s e d u c a t o r s

to

p rovide these p r o s p ectiv e

teach ers w i t h similar e x p e r iences

as w e r e

study.

described

in this

p r o s p e c t i v e teachers,

no less

w o u l d be ne ede d to pro vide
the nece s s a r y

For such a l a r g e number of

than twe l v e e l e m e n t a r y schools

th ese p r o s p e c t i v e te a chers w i t h

cl ini cal experience.

198

Hecommendations

The fo llowing
inve stig ato r's

r e c o m m e n d a t i o n s are base d on the

interpr e t a t i o n s of the analysis of data

collected in this study and his personal o b s ervations
carrying out the p resent evaluation.

The r e c o mmendations

have been o r g a n i z e d in two categories:
for future act i o n and

Re comm endation s
Acti on

in

(1) r e c o m m e n d a t i o n s

(2) r e c o m m e n d a t i o n s

for future research.

for Fu ture

It is r e c o m mended that the m a t h e m a t i c s

pr e p a r a t i o n

of p r o s p ective e l ementary school

teachers sh ould

the content an d met hod s courses,

thus teaching

tive teacher the m a t h e m a t i c a l co ntent
w o u l d be e x p e c t e d to teach
It is r e c o m mended

the p r o s p e c ­

in the same form he

it.
that such i n t egrated content-

methods cour se be taught in a m a thematical
w h e r e ma nipu lative,

integrate

audio-visual,

laboratory

setting

and in str uctiona l mate ria ls

are av ail able and utilized.
It is r e c o m mended

that the p r o s p e c t i v e e l e men tary

school teacher study the theories of

teaching and

learning

mathem ati cs w i t h laboratory and cl inical exp e r i e n c e so as to
be able to rel a t e theory to practice.

This

type of clinical

e x perience s h o u l d b e g i n as early as p ractical
edu cat ion of

the p r o s p e c t i v e teacher.

in the

199

It is reconcnended that the p r o s p e c t i v e

teacher's

clinical e xp e r i e n c e pr o v i d e hi m the op p o r t u n i t y to teach
inner-city pupils

as w e l l as rural pupils,

the hi ghly

m otivated pupil as well as the u n m o t ivated pupil,
hostile pupil as well as the co o p e r a t i v e pupil,

the

the less

able as well as the mor e able.
It is r e c o m mended that time be alloc a t e d

for short

lectures and/or d e m o n s t r a t i o n s by a teacher educa t o r for
topics of mathem a t i c a l con cepts

that the p r o s p e c t i v e t e a c h ­

ers m i g h t not d i s c o v e r for th em selves
time.

These topics may include:

than ten,

in the limited class

div isi on in ba ses other

the con cep t of c o n g r u e n c y in m a t h e m a t i c a l systems,

p robab ili ty of depen d e n t events,
and symbolization.

Such

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

language

lectures or d e m o n s t r a t i o n s may

facilitate faster and b e t t e r u n d e r s t a n d i n g by the p r o ­
specti ve teachers of re l a t e d concepts.
It is rec o m m e n d e d that for each
(topic)

learning unit

a short film or v i d e o - t a p e re cor ding of an exampl e

of instructional a c t ivities on this topic be shown.

Such

films or vid eo- tapes w o u l d serve three m ajor objectives:
(a) enable the p r o s p e c t i v e teacher to observ e e x p e r i e n c e d
teachers w o r k i n g w i t h c h i l d r e n on m a t h e m a t i c a l activities,
(b) enable the p r o s p e c t i v e tea cher to o bserve c h i ldr en
w o r k i n g on the learning situation,

and

(c) en able

teacher educator to d e m o n s t r a t e d i f f e r e n t met h o d s
instruction.

the
of

200

It is r e c o m m e n d e d that the p r o s p e c t i v e

teachers

work in small gr ou ps of four or five on d e v e l o p i n g i n s t r u c ­
tional d esigns
groups,

to be used w i t h children.

W o r k i n g in such

the p r o s p e c t i v e t eacher m u s t take an acti ve role

in

planning the i n s t r u c t i o n a l d e s i g n s ince he or she is r e s p o n ­
sible for teachi ng one part.
It is r e c o m m e n d e d that m a t h e m a t i c a l a c t ivities
ut ilizing m a n i p u l a t i v e m a t e r i a l s be used w h e n e v e r p o s s i b l e
by the p r o s p e c t i v e

teacher s

in p l a n n i n g their i n s t ructional

design.
It is r e c o m m e n d e d that instr u c t i o n a l a c t i v i t i e s be
added to the unit on M e a s u r e m e n t to pro vid e the pr o s p e c t i v e
teacher wi th b e t t e r k n o w l e d g e and u n d e r s t a n d i n g of the
metric syst e m of m e a s u r e m e n t and its relatio n

to the English

system.
It is r e c o m m e n d e d that th ose p r o s p e c t i v e teachers
who do n ot at t a i n an acc e p t a b l e de g r e e of m a s t e r y o v e r

the

p re 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 be g iven a d e f e r r e d
grade until they show e v i d e n c e of a t t a i n m e n t of the de s i r e d
degree of competency.
It is re c o m m e n d e d

that the m a t h e m a t i c s e du cators

care fully iden tif y and d e f i n e the c o m p e t e n c i e s
es sential
that,

for elem e n t a r y scho ol

teach ing of m a t h e m a t i c s

for the p r o s p e c t i v e e l e m e n t a r y

ma t hematical education,

that are
and

teacher to c o m p l e t e his

he m u s t ev idence a t t a i n m e n t of no

less than 80 p e r c e n t of these competencies.

201

The r e lativ ely
eleme nta ry sc hool
participate

low n u m b e r of freshme n p r o s p e c t i v e

teache rs

{fifty-two)

w h o v o l u n teered

in the e x p e r i m e n t a l p r o g r a m suggest s

to

the need

for m o d i f y i n g the p r o c e d u r e by w h i c h the student s are
app ro ached a b o u t p a r t i c i p a t i n g in the program.
pro g r a m to be s u c c e s s f u l in r e a c h i n g pro s p e c t i v e

For a new
teachers,

an agg r e s s i v e r e c r u i t m e n t p r o c e d u r e should be es tablished.
Instead of w r i t i n g a letter to all inc om ing freshmen w i t h
dec la red in tere st in e l e m e n t a r y e d u c a t i o n sol i c i t i n g their
p a r t i c i p a t i o n in the program,

it is r ec o m m e n d e d that mem b e r s

of the p r o j e c t staff c o n t a c t these stud ent s i n d ividually or
in groups d u r i n g the students'
week,

v i s i t to campus,

or the b e g i n n i n g of the sch o o l year.

orientation

This w o u l d allow

the s t u d e n t an o p p o r t u n i t y to ask q u e s t i o n s and rec eiv e an
immediate response.

It alBO w o u l d allow the p r o j e c t staff

to make a s pecial e f f o r t at r e c ru iting m a l e and m i n o r i t y
stude nts

to p a r t i c i p a t e

in the program,

thus o b t a i n i n g a

more r e p r e s e n t a t i v e samp le of the p o p u l a t i o n of ele m e n t a r y
educ a t i o n majors.
Recommendations

for Fut u r e R e s earch

B e c a u s e of the posit ive
study,

results o b t a i n e d

in this

it is r e c o m m e n d e d that the study be r e p l i c a t e d w i t h

another sample of p r o s p e c t i v e e l e m e n t a r y school

teacher s wh o

v o l u n t e e r to p a r t i c i p a t e in the e x p e r i m e n t a l program.

202

It is r e c o m m e n d e d that a for mat ive eval u a t i o n be
conducted on the se con d integrated content-methods cour se
(algebra and geometry)

of the e x p e r i m e n t a l program;

and that

c r i t e r i o n - r e f e r e n c e d measures be d eveloped to assess the
effec t i v e n e s s of that course on the students'
on the p r e s c r i b e d m a t h e m a t i c a l

achievement

competencies.

It is r e c o m m e n d e d that an e v a l u a t i o n of the ef fe ct
of the m a t h e m a t i c a l

in s t r u c t i o n by the p r o s p e c t i v e

teachers

on the c o g n i t i v e be hav i o r s of the e l e m e n t a r y school childre n
wh o w e r e

taught by t hese pr o s p e c t i v e teachers be conducted.
It is r e c o m mended that a l o ngitudinal study of

the

t h irty-eigh t p r o s p e c t i v e tea cher s who p a r t i c i p a t e d in the
e xpe rim en tal p r o g r a m be

ini tiated to p r o v i d e answers

to the

following questions:
1.

To w h a t ext e n t do these p r o s p e c t i v e
m a t h e m a t i c a l co mpetencies,
matics,

and/ or improve

teachers retain

a t t i t u d e s toward m a t h e ­

the teachin g pro c e d u r e s

d e v e l o p e d as a re sul t of this e x p e r i m e n t a l p r o g r a m
du ring

their s tudent teach ing and during

their

t e a ching pr of ession?
2.

Are

there any s i g n i f i c a n t d i f f e rences b e t w e e n

the

g r o u p of pro s p e c t i v e e l e m e n t a r y s c h o o l teach ers who
p a r t i c i p a t e d in the e x p e r i m e n t a l p r o g r a m an d a group
of p r o s p e c t i v e e l e m e n t a r y school

teache rs in the

r e g u l a r t each er e d u c a t i o n p r o g r a m on their co gni tive

203

and a f f e c t i v e b e h a v i o r s

toward m a t he matics,

i m m e d i a t e l y after both groups c o m plete

their

undergraduate education?
3.

Will thi s group of p r o s p e c t i v e teache rs be m o r e
s u c c e s s f u l in:

(a) t e a c h i n g mathem a t i c s ,

st and ing of e l e m e n t a r y school children,
ting in pr o f e s s i o n a l l y ,

(b)

and

under­

(c)

fit­

than r e g u l a r l y e d u c a t e d

teachers?
4.

Will this group of p r o s p e c t i v e
in clined

toward a d o p t i n g

teach ers be m o r e

s t u d e n t - c e n t e r e d activities

in their el emen tary school m a t h e m a t i c s c l a s s e s ?
Finally,

it is r e c o m m e n d e d that a research study be

c o n d u c t e d to a s c e r t a i n

the e f f e c t of the cl inic al exp erie nce

on the cogn i t i v e and a f f e c t i v e b ehavio rs
of

toward mathematics

the e l e m e n t a r y e duca t i o n m a j o r s w h o p a r t i c i p a t e

e x p e r i m e n t a l program.

The P o s t - t e s t O n l y Control

D e s i g n sugge s t e d by C a m pbell
priate.

R a n d o m assignment,

ti cipate in the e x p e r i m e n t a l

in the
Group

and S t a n l e y 1 w o u l d be a p p r o ­
of

those wh o v o l u n t e e r

program,

(thirty-five to forty s t u d e n t s

to pa r ­

in to two groups

in each g r o u p w o u l d suffice)

w o u l d provide an e x p e r i m e n t a l and a c o n t r o l group.
g rou p s should r e c e i v e the same

treatme nt exc e p t tha t

Both
the

D o n a l d T. C a m p b e l l and C. Stanley, E x p e r i m e n t a l and
Q u a s i - E x p e r i m e n t a l Designs for Resea r c h (Chicago:
Ran d
M c N a l l y , 1466).

204

ex p erimental g ro up w o u l d be i n v o l v e d in clinica l p r a ctice
of teach ing m a t h e m a t i c s once a w e e k
school children.
methods courses,

Concurrent with

cant di f f e r e n c e s b e t w e e n
or affective b eh aviors

to both groups.

the two groups on

that influence

of the ex periment.

and attitude

Any s i g n i f i ­

their cogn itive
can be a t t r i b u t e d

the cl inic al experience.

e x p e r i m e n t a l design w o u l d cont rol

the effect of

integrated content-

toward m a t h e m a t i c s

to the ef fec t of

variables

their

c r i t e r i o n - r e f e r e n c e d measures,

scales w o u l d be a d m i n i s t e r e d

mo stly

to a g rou p of e l e m e n t a r y

Such

for m o s t c onfounding

the sour c e of internal validity

The e x c l u s i o n of p r e - t esting eli m i n a t e s

interaction between

In conclusion,

it seems

testi ngs

and treatment.

that the present study has

p oin ted o u t the need for o b j e c t i v e and critical e v a l u a t i o n
of ele m e n t a r y
crucial

school

teach er e d u c a t i o n programs.

Thi s

is

if these p r o g r a m s are to train teachers w h o u nder­

stand hum an d e v e l o p m e n t and the nat ure of ma themati cal
learning s u f f i c i e n t l y well

so that they

niz e and foster co nd i t i o n s

that

mathematics.

are able

fac i l i t a t e

to recog­

learning of

b i b l i o g r a p h y

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Withnell, M. C.
"A C o m p a r i s o n of the M a t h e m a t i c a l
U nd er 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 T e a c h e r s
in C o l l e g e Having D i f f e r e n t M a t h e m a t i c s R e q u i r e ­
ments."
Doctoral d is sertation, C o l o r a d o State
College, 1967.
Abstract.
Dissertation A b s t r a c t s ,
28:4941, 1967.

89.

White, M. J. A.
"A Study of the C ha n g e of A c h i e v e m e n t
and A t t i t u d e T o w a r d A r i t h m e t i c by P r o s p e c t i v e
E l e m e n t a r y School T e a c h e r s U n d e r the C o n d i t i o n s
of T e l e vi si o n. "
D o c t o r a l dissert at io n, W a yn e State
University, 1963.
A n n Arbor, Mich.:
University
M icrofilms, 1964, No. 64-5114.

90.

Woodall, P. G.
"A Study of Pupils* A c h i e v e m e n t s and
A t t i t u d e s in the SMSG and the T r a d i t i o n a l M a t h e ­
matics Programs of the L e w i s t o n School District,
1960-1965."
D o c t o r a l di ssertation, U n i v e r s i t y of
Idaho, 1966.
A n n Arbor, Mich.:
University M i c r o ­
films, 1967, No. 67-5389.

215

91.

Yasui, R. Y.
"An A n a l y si s of A l g e b r a i c A c h i e v e m e n t
and M a t h e m a t i c a l A t t i t u d e B e tw e en the Modern and
T r a d i t io na l M a t h e m a t i c s P r o g r a m s in the Senior
High School:
A L o n g i t u d i n a l Study."
Do ctoral
dissertation, U n i v e r s i t y of Oregon, 1967.
Ann
Arbor, M i c h . : U n i v e r s i t y Microfilms, 1968,
No. 68-4017.
Unpublished Materials

92.

"A Brief Description, Progress, and F ut u r e of T T T . "
P rogress Report.
M i c h i g a n State University,
u n p u b l i s h e d paper, July 30, 19 71.

93.

Begle, E. G.
"The Rol e of R e s e a r c h in the I mprovement
of M a t h e m a t i c s Education."
Paper p r e s e n t e d at the
F i r s t I nt er n a t i o n a l C o n g r e s s on M a t h e m a t i c a l E d u c a ­
tion.
Lyon, France, August, 1969.

94.

Glennon, Vincent, P. Peck, and S. Willoughby.
"Proposed G u i d e l i n e s for the P r e p a r a t i o n of
T e a ch e rs of M at he m a t i c s . "
A d d r e s s to the NCTM.
Chicago, April, 1972.

95.

Golding, E. W.
"Report on T e a c h e r Education, Wi th
Special R ef er en ce to M a t h e m a t i c s Learning."
I nt er n a t i o n a l S tudy G r o up for M a t h e m a t i c s Learning,
U ni v e r s i t y of Sherbrooke, Canada, 1968.

96.

Guba, E g o n G.
"An O v e r v i e w of the E v a l u a t i o n Problem."
P a p e r p r e s e n t e d to the A m e r i c a n E du c a t i o n a l
R e s e a r c h A s s o c i a t i o n , February, 1969.
Bloomington:
School of E du cation, U n i v e r s i t y of Indiana.
(Mimeographed.)

97.

Henderson, Judith, and Perry Lanier.
"C o mpetency/
P e r f o r m a n c e B ased Te a c h e r Education:
A P a p er for
Discu ss io n. "
M i c h i g a n St ate University, January,
1972.

98.

M c S w e e n y , Mary Ellen.
" E v a l ua ti on of the TT T E l e m e n ­
tary C o m p o n e n t — A s p e c t s of the Internal Evaluation."
U n p u b l i s h e d paper, M i c h i g a n State University, 1972.

99.

Wasylik, E.
"A L a b o r a t o r y A p p r o a c h to M a t h e m a t i c s for
Low A c hievers:
An E x p e r i m e n t a l Study."
A working
paper, Un i v e r s i t y of Alberta, 1970.

216

100.

Wilkinson, Jack.
"The L a b o r a t o r y M e t h o d as a T e a c h i n g
S t r a t e g y in G r a d e s 5-8."
P a per 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, O c t o b e r 17,
1970.
Miscellaneous

101 .

Alpert, R . , G. Stellwagon, and D. Becker.
"Psycho­
logical Fa c t o r s in M a t h e m a t i c s Ed uc at io n. "
Report
s um ma ry in N e w s l e t t e r , No. 15, SMSG, Sta n fo rd
U n i v e r s i t y , 1963.

102 .

"An A n a l y s i s of N e w M a t h e m a t i c s P r og ra ms ."
National
C o u n c i l of T e a c h e r s of Ma th ematics, W ashington,
D . C . , 1964.

103.

Bell, Max, K. Fuson, and B. Hammond.
"Teach in g
A c t i v i t y - O r i e n t e d C o u r s e for P r e s e r v i c e and Inservi ce E l e m e n t a r y Teachers."
U n i v e r s i t y of
Chicago, 50th annual m e e t i n g of NCTM, 1972.

104.

Cox,

105.

" Ca li fo r ni a M a t h e m a t i c s C ou nc il ( T h e) ."
R e p r i n t of the
S t r a n d ' s Report, 1967-1968, P a r t II, p. 34.

106.

Davis, Edward.
"A Study of the A b i l i t y of S e l e c t e d
S ch o o l Pupils to P er ce i v e the Plane S e c t i o n s of
S e l e c t e d S olid Figures."
U n i v e r s i t y of Georgia,
Athens.
S M A C , An ERIC Center, 1972.

107.

Ebel, R o b e r t L.
"Some L i m i t a t i o n s of C r i t e r i o n Referenced Measurement."
American Educationul
R e s e a r c h Asso ci at i on .
ERIC M i cr of il ms , 1970,
ED 038670.

108.

Herriot, Sarah T.
"The S l o w L e a r n e r Project:
The
S e c o n d a r y S c ho o l 'Slow Learner' in M a t h e m a t i c s . "
SM SG Report No. 5, Stanford, 1967.

109.

M or risett, L. N., and J. V i n s o n h a l e r , eds.
M a t h e m a t i c a l L e a r n i n g . M o n o g r a p h s of the Socie ty
for R e s ea rc h In c h i l d D ev el opment, Vol. 30, No. 1,
1965.

R i c h a r d C . , and Julie S. Vargas.
"A C o m p a r i s o n
of Ite m S e l e c t i o n T ec h n i q u e s for N o r m - R e f e r e n c e d
and C r i t e r i o n - R e f e r e n c e d T e sts."
ERIC M icrofilms,
ED 010517, 1966.

217

110 .

R ajaratnam, N . , L. J. Cronbach, and G. C. Gleser.
" G e n e r a l i z a b i l i t y of S t r a t i f i e d - P a r a l l e l Teats."
P s y c h o m e t r i k a . Vol. 3 0 r March, 1965, pp. 39-56.

111 .

Suppes, Patrick.
"The A b i l i t y of E l e m e n t a r y - S c h o o l
C h i l d r e n to L e a r n the N e w M a t h e m at ic s. "
In
M a t h e m a t i c s In E l e m e n t a r y E d u c a t i o n . Edited by
N i c h o l a s J. Vigilante.
Toronto:
The M ac M i l l a n
Co., 1969, pp. 351-358.

112 .

Torrance, E. P., et al.
" C h a r a c t e r i s t i c s of
M a t h e m a t i c s T e a c h e r s that Af f e c t Students*
L ea rning."
R e p o r t No. CRP-10 20 , 1966, U n i v er si ty
of Minnesota, Co nt ra ct No. O E C - S A E - 8 9 9 3 , U.S.
O ff i c e of Education.

APPENDIX A

SET O F N I N E PRE - A N D P O S T - T E S T F O R M S OF
THE CRITERION-REFERENCED MEASURES

219

MEASUREMENT

1.

2.

PRE-TEST

In one hour And a half, the minute hand of a clock rotates through
an angle of:
a)

60°

b) 90°

d)

540°

e) 720°

c) 360°

The distance between two towns on a m a p is 9 cm. (centimeters). If
3
the scale of that m a p is:
— cm. - 10 km. (kilometers), the actual
distance is:

3.

a)

50 km.

b) 60 km.

d)

120 km.

e) none of these

c) 75 km.

The measurement of a line segment was stated to be

(2--

± ^y)

inches,

This implies that the Begment is:

4.

5.

6.

a)

as

long as

3 inches or as

short as

2 inches

b)

as

17
longas

2—

inches or

15
as short

as

2 -j j inches

c)

as

longaa

2 -7—

inches or

as short

as

2 ^ 7 inches

d)

exactly 2 1 inches long

e)

none of the above

16

16

A carpenter needs six wooden boards each 2 feet 8 inches long.
If
wood is sold by the foot, what is the least number of feet that
must be purchased?
a)

10

b)

12

d)

16

e)

18

c) 14

A photograph measures 3 by 6 incheB.
It is enlarged so that the
shorter measure will be 16 inches.
The length of the enlarged
longer measure will be:

Of

a)

B inches

b)

19 inches

d)

48 inches

e)

80 inches

c) 32 inches

the following w hich is the shortest?
a) 30 incheB

b)

20 centimeters

d) one yard

e) one meter

c) one decimeter

220

7.

8.

9.

If a box is 10 units high, 6 units wide, and 4 units deep, how many
cubes will fill this box if each cube is 2 units on each side?
a) 240

b)

120

c) 60

d) 30

3) none of these

The surface of a cube is 150 square yards.
this cube in cubic yards?
a) 50

b)

100

d) 200

e)

250

c)

125

There is a geometric figure of the shape and dimensions of the
adjoining drawing.
What is its area in square inches?
a) 2 5

5"

7"

e) none of these
10.

What is the volume of

What is the area in square centimeters of the shaded portion of the
adjacent figure?
(The circle is inscribed inside the square.)
a) 400 - 100 (3.14)
b) 400 (3.14) - 4.00
c) ~
4
d)

(400)

100 (3.14) - 100

e) none of these
b

20 cm.—

—

MEASUREMENT

POST-TEST

The rectangle below consists of a square of area 16, and a rectangle
of area 12.
What is the distance PQ?
a)

b) 4
c)

P

3
Area * 16

5

d) 6
e)

7

'

Area
v\ = 12
\
\
V
\
K

C
What is the height of a rectanble block that is 3 feet wide and
8 feet long, if its volume is equivalent to that of a cube with
an edge of 6 feet?
a) 12 feet

b) 9 feet

d)

e) none of these

3 feet

c)

6 feet

In the figure below, X is the center of the circle and XY is
perpendicular to XZ.
If the area of the circle is 36tt , what
is the area of the triangle XYZ?
a) 12
b)

14

c) 16
d)

18

e) 20
The sum of the measure oi the four angles in a quadrilateral
a)

90°

d)

360°

b) 180"

is:

c) 270°

e) depends on the size of the quadrilateral

The circumference of a circle is 0rr meters.
circle in square meters?
a)

8tt

b) 16n

d)

64TT

el 32

What is the area of the
c) 32ti

Which of the following is the nearest approximation of one yard?
a)

one kilometer

b)

one centimeter

d)

one meter

e)

39 centimeters

c) one millimeter

222

7.

8.

9.

How many cubic feet of water can fill a cylinder if the radius
of its base is 2 feet and its height is 10 feet?
c) 31.4

a)

125.6

b) 62.8

d)

15.7

e) cannot be determined from the above
jnformation

If a scale of a ma p is 1 centimeter * 60 kilometers, what on a map
would represent an actual distance of 720 kilometers?

If

a)

12 centimeters

b) 12kilometers

d)

18 kilometers

e) none

of

c)18 centimeters

these

j | | | is equal to ONE U N I T , what is the area of the adjacent

figure?
a) 6 units

10.

b)

7 units

c)

8 units

d)

20 units

e)

21 units

The "greatest possible error"

in measurement is defined as:

a) The smallest unit of measure used in the measurement
b) One-half the smallest unit of measure used in the measurement
c) One-tenth the smallest unit of measure used in the measurement
d) Any fraction of a whole unit of measure used in the measurement
e) None of the above.

PRE-TEST

SYSTEMS OF NUMERATION

Suppose that in place of the number system, a symbol system was
developed in which the following digits were used:

A,

l

,

r,

y,

Z,

A,

D , 7, v,

6 x

representing respectively 0, 1, 2 , 3, 4, 5, 6 , 7, 8 and 9.
The digit A in the symbol system is used in the same fashion as 0
in the decimal system, e.g., EA - 40.
A.

Which of the following is equal to 1 0 2?
a) LAA
d)

B.

□ □ □

b)
X

lAAA

c) LAy

e ) XXXy

Which of the symbolic representation is equivalent to
two-t h i r d s .

e) none of these
C.

What is the value of L T V PLUS V p ?
a) □ E

b)

r

d) TAA

e) XX r

c) rrr

ae

The decimal expansion of the numeral 14 3.25 is?
a)

1(100) + 4(10) + 3(1) - 2(10) - 5(1)

b) 1 (100) + 4(10) +■ 3(1) - 2(10) - 5(100)
c) 1 (1000) + 4 (100) + 3(10)
d)
e)

+ 2 (1) + 5(0)

1 (102) + 4(10*) + 3(10°) + 2(10_l) + 5(10“ 2)
none of the above is correct

224

3.

4.

5.

In the numberal 7,698,500,000
symbols is in the 10 * place?

(base 10), which of the following

a)

7

b) 6

d)

5

e) 0

Which of the following
a)

100,000 {baBe2 )

d)

100 (base 5)

c) 9

is the largest?
b)

10,000 (base 3)

1,000 (base 4)

c)

e) 10 (base 10)

W h i c h of the following numerals is not equal to the others?
a)

100,000

d)

51

(base 2)

(base 7)

b) 210

(base 4)

e) 121

(base 5)

c) 36

(base 10)

6 . In what base is 213 + 552 • 1205?

7.

a)

ten

b) nine

d)

seven

e) six

in the following equation, what is the value of X?
43

fl.

(base5) - 24

(base 5) « X

a)

11

b) 12

d)

14

e) none of these

(base 5)
c) 13

When working with base twelve, we need 12 symbols, so we will use
0, 1, 2, 3, 4, 5, 6 , 7, 8 , 9, T, E, where T stands for ten and E
stands for eleven.
A.

B.

9.

c) eight

What is:

BT7

(base 12)

a)

1226

(base 12)

d)

T E 6 (base 12)

In the numeral ET62
base 10 is:

+ 319

b) 1004

(base 12) equal to?
(base 12)

c) EE4

(base 12)

e) none of these
(base 12),

the actual v alue of T in

a)

10 (12 2 )

b) 10 (12 s)

d)

10 (10 s)

e) none of these

c) 10

(10 2 )

If 12 (base 5) is an odd number (seven), which of the following
is another example of an odd number?
a)

101

(base 3)

b) 100

(base 5)

d)

121

(base 7)

e) 101

(base 9)

c) XVIII

225

10.

11 .

The numbers 312 and 21 are in base 4.

Their product

a)

20212

b) 13212

d)

1212120

e) none of these

(in base 4)

c) 6552

In w hat base is 15 * 2 ■ 6?
a)

ten

b) nine

d)

six

e) four

c) seven

is:

226

POST-TEST

SYSTEMS OF NUMERATION

1,

What is the bas e of this numeration system?

Mil

i±ii

Mii

Mil

Mii

I
A

+
1

2.

a) 1

to) 4

d) A

e) cannot be determined from the above
information

The expanded notation:
5 x 6 5 + 3x6"

+ 4 x

65 + 5 x

62 + l x 6 I + 4 x 6 °

is equivalent to which of the following numerals?

3.

(base 10)

b)

d) 534514

(base 6 )

e) none of these

a) 2

b) 5

d) 4

e) 9

a) 41

(base 10)

(base 9)

d) 1,101

6.

c)

30 + 18 + 24 + 30 + 24

In the decimal number 8943.752, which of the following symbols is in
the 102 place?

The numeral 37

5.

534514

(times 6 )

a) 534514

(base 3)

c) 3

is a different number than;
b)

52

(base 7)

e) 101,101

c) 211

(base 4)

(base 2 )

Jeff said there are 120 hours in a day.
he working w i t h ?

What numeration system was

a) base 9

b) base 8

d) base 2

e) none of these

c) base 4

in binary notation, what is the number w hich follows 11,011

11,100

a) 11,010

b)

d) 100,000

e) 11,110

c)

11,111

(base 2)?

227

7.

6.

9.

10.

11.

What is the decimal equivalent of ET (base 12), where T stands for
ten and E for eleven in the base 12 system?
a)

120

b) 142

d)

122

e) none of these

If 302

(base S) - 133

c) 132

(base 5) - A

a)

440

b) 104

d)

124

e) 204

(base 5), then A is:
c) 114

In which base does the numeral 35 represent an even number?
a)

twelve

b) ten

d)

seven

e) six

c) eight

The multiplication problem 16 x 4 *■ 60 has been computed in which
base?
a)

thirteen

b) twelve

d)

nine

e) eight

Compute the quotient 41

(base 5)

a)

14

b) 13

d)

11

e) 10

t

c) eleven

3 (base 5) c) 12

228

SETS AND SET OPERATIONS

1.

PRE-TEST

If S (th« universal set) - (l, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A - {2, 4, 8}, B - {l, 2, 3, 4}, C - {7, 8}

A(JB

a) Enumerate

A\JB - {
b) Enumerate

}

AflB

Af|B - {
c) Enumerate
AUB

and if

}

A(JB

(complement of A(JB)

- (

d) Enumerate

}

A f) (B UC)

A 0 (B UC)

- {

}

e) Enumerate the set D which is described in set-builder
notation as (xcs| x is even or XCB}
D - {

2.

3.

Let A ■ {a, b, c}.

b) 6

d)

e) 1

3

c) 4

If a set A contains n distinct elements, which of the following
formulas will always giv e the number of non-empty subsets of set A?

d) n - 1

5.

Exactly how m a n y subsets does A have?

a) 8

a) n 2

4.

}

b) 2 (n)

c) 2n - 1

e) cannot be determined from the
information given

Let A » {3, 5, 9},
B ■ {9, 3}
Then A
B represents:

and C ■ {3, 5, 9, 4 } .

a) A

b) B

d) {0}

e) (5)

c) {3, 5, 9, 4}

if set A - {a, b, c} and B - {l, 2, 3}, in how many ways could one
establish a one-to-one correspondence between these two sets?
a) 1

d)

5

b) 3

e) 6

c) 4

229

Consider the set A •* ' (4, 2) , (a, b) } and the set B - {4, 2} . Which
of the following is true?
(n(A) is the number of distinct elements
in set A . )
a) n (A) » n(B)

b) n(A)

d) n (A) < n(B)

e) n (A) ■ 4

Let 5 * {x, y) .
of S is:
a)

{x},

> n(B)

c) n (B)

Then a complete listing of all possible subsets

{y}

b) {x}, {y}, {x, y}

d) fB, { x } , {y }, (x, y}

e)

{(»},{x}

c) x, y
{y}

(x, y)

If a set A has 10 subsets, how man y proper subsets doeB A have?
a)

10

b) 9

d)

1

e) 0

c) 5

Let A be the set of all positive even numbers.
Let B be the set of all the letters in the English alphabet.
Which of the following statements is (are) true7
a) A and B are matching sets.
b)

BC A

c) Both A and B are equivalent sets.
d) Both A and B are infinite sets.
e) A is an infinite set and B is a finite set.

10 .
ii

h i

BSISUUlttUtfUiUii

VI

For aach of the following sets , circle the Homan number
represents the shaded area of the Venn diagram a b o v e .
a)
b)
c)
d)
e)

A U (B f)C)
A 0 (B UC)
A UC
(AUB)Uc
A H (B 0C)

I
1
I
I
I

II
II
II
II
II

III
III
III
III
III

IV
IV
IV
IV
IV

V
V
V
V
V

VI
VI
VI
VI
VI

230

SET AND SET OPERATIONS

POST-TEST

If S (the universal set) - {a, b, c, d, e, f} and A ■ {a, c, d, e}
B = {b, c, e} and C ■ {d, e r f], list the elements of the following
sets.
a) A U B

- {

)

b) A H B - {

)

(complement of A U C )

c) A(JC
d) A

■ {

}

- {

1

e) The set D w hich is described in the set-builder notation as
{xcs| xCA and xcc)

D - {
2*

3.

}

If A ■ {x, y, z, w} , exactly how many subsets does A have?
a) 4

b)

d)

e) none of these

32

8

c)

16

What does the adjacent diagram illustrate?
a) a one-to-one correspondance
b) matching sets

TOM DICK
HARRY

c) set equality
d) all of the above

DICK
— >> TOM
*• HARRY

e) none of the above
4.

5.

6.

Let A - {9, 7, 4 , 2 } ,

B « {2, 4, 7}

and

C - (4, 9}.

a)

A

b) B

d)

{4}

e) none of those

Then B

C -

c) C

How many one-to-one correspondance are there between two two-number
sets?
a)

1

b) 2

c) 3

d)

4

e) cannot be determined from the above
information

How many subsets are there in a four m e m b e r set?
a)

1

b) 4

d)

16

e) none of these

c) 8

231

7.

Which of the following statements about sfets is

(are) true?

a) If A is a subset of B, then B is a subset of A.
b) If A is a subset of B, then B is not a subset of A.
c)

If A is a proper subset of B, then B is not a proper subset
of A.

d)

If A is a different Betfrom
B, then A is a proper subset of
B or B is a proper subset of A.

e) If set A • { (a, b) } end B - {l, 2},
number of elements.
8.

9.

10.

Let A
Let B
Let C
Which

be
be
be
of

the
the
the
the

then A and B have the same

set of all pupils in an elementary school.
set of first graders in that school.
Bet of teachers in that school.
following statements is (are) true?

a)

C is a proper subset of A.

b)

B is a proper subset of A.

c)

A Q B is the empty set.

d)

B O C is the empty set.

e)

Af) B - A.

Which of the following statements is

(are) correct?

S, then the complement
are not in A.

a)

If A is a subset of the universal set
of A is the set of elements in S that

b)

if B is a proper subset of A, then the complement of A is the
set of elements that are common to A and B.

c)

If A and B are two disjoint subsets of S such
that A U B «■ S,
then A is the complement of B and B is the complement
of A.

d)

Onl y (a) and

(b) are correct.

e)

Onl y (a) and

(c) are correct.

Which of the following sets is represented by the shaded portion of
the Venn diagram?
a)

(ADB)Dc

b)

(AUB)UC

)

<AUB)f)C

c

d)

(a O

e) Af|C

b

)UC
(complement of AfjC)

232

PRE-TEST

THE WHOLE NUMBER SYSTEM

M a t c h each statement with the property
illustrates.
1)

3 + (6 + 2) - ( 3 + 6 ) + 2

2)

15 x (8 + 5) - (15 X 8 ) + (15 x 5)

3)

18 x 1 - 1 x 18

4)

2 + ( 3 x 8 ) • (2

5)

18 + (7 + 5) * 5 +

6)

48 x 86 - 86 x 48

7)

14 + 0 - 1 4

8)

25 t 0 - 0

9)

(28 + 8 ) t 7 - (28

10)
2.

3.

a) commutative property
+ 3) x (2+ B)

b)

(18 + 7)

associative property

c) distributive property
d)

identity property

e) false statement
* 7) +

(8 r 7)

4 -0 - 0 - 4

W h i c h of the following sets is

(are) closed with respect to addition?

a)

(Whole numbers less than 50}

b)

(0 ,

1}

c)

(2,

4,

6,

8 ,1 0 , }

d)

(l,

3,

5,

7,9}

e)

(Whole numbers that are multiple of 3}

In the division algorithm:
a)

10, 400,

520, 0

b)

4, 2, 3

c)

1200 , 120 , 2

d)

400, 20,

3

e) none of these
4.

(ies) or definitions it

.

1.

10,998 ^26,

what multiple of 26 are used?

26)10,998
- 10,400
598
- 520
78
-78

0

In the adjacent algorithm, what does the number marked by the arrow
actually represent?
128
a> 2 * 14
14) 1792
b) 20 x 14
1£
39
c) 200 x 14
28 <
d) 14 x 2
112
e) none of these

112

0

233

5.

6.

7.

8.

Which of the following statements is false?
numbers)

(f and g are whole

a)

-f -g - - (f + g)

b)

(-f)

c)

-f-(-g)

-

d)

(-f) (g) -

e)

if f > g ,

then f - g - - ( f + g )

(g-f)

Which of these numbers is

(-g) - fg
(f)

(-g)

(are) prime?

a)

51

b) 14

d)

43

e) 25

c)

1

What is the highest prime to consider as a divisor in the
factorization of 132?
a)

131

b) 13

d)

9

e) 2

c)

Which of the following statements is

11

(are) true?

a)

All prime numbers are odd numbers.

b)

All composite numbers are even.

c) If a prime number divides the product of two natural numbers,
then it m u s t divide at least one of the two numbers.
d) Every natural number has at least two factors.
e) There are finite number of prime.
9.

10.

The sum of the first 50 odd digits is:
1, 3, 5 ............... 99)

(i.e., the sum of

a) 1250

b) 2500

d) 10,000

e) none of these

Which of these are the
a)

5000

prime factorization of 60?

2) 60
2 ) 3 0 - 2 * 2 x 3 x 5
*

c)

b) 60

r— 30

I2

r— 10

1
3

r— 5

I2

3) 14
5
c)

both

(a) and

e)

60 - 10 x 30

(b)

d) neither

(a) nor

(b)

- 2 x 3 x 2 x 5

234

THE WHOLE NUMBER SYSTEM

1.

Match each statement with the property
illustrates.

2.

3.

(ies) of definitions it
a) commutative property
of addition

1)

5 x 12 - 12 x 5

2)

(35 + 10) 4 5 - (35 4 5) + (10 4 5)

3)

5 + (9 + 11) - 11 + (5 + 9)

4)

36 X 92 - (36 x 90) + (36 + 2)

5)

18 4 0 - 0

6)

0 + 35-35

7)

5 + (9 x 3) - (5 + 9) x (5 + 3)

8)

13-0-0-13

9)

11 X 1 - 11

10 )

POST-TEST

b)

commutative property
of m u l tiplication

c) associative property
of addition
d)

associative property
of m u l t iplication

e) distributive property
f)

identity p r o perty

9)

false statement

5 + (11 + 4) - (5 + 11) + 4

Which of the following sets is N O T closed u n d e r multiplication?
a)

{whole number}

b)

(odd natural numbers)

c)

{even natural numbers}

d)

{prime numbers}

e)

{whole numbers that are multiple of 5}

In the division a l gorithm 134,616
a)

4, 2, 1,

3

b)

400,

1, 3

c)

128,000,

d)

4,000,

e)

none of these

20,

6,400,

200,

10,

320,
3

96

32, what multiples of 32 are used?
32)134,816
- 128,000
6,816
6,400
416
320
96
96

0
In the adjacent m u l t i p l i c a t i o n algorithm,
product mar k e d b y the arrow represent?
a)

6 x 342

b)

3 x 342

c)

60 x 342

d)

30 x 342

e)

none of theBe

wha t does the partial
342
x 63
1026
2052 21546

235

5.

6.

7.

8.

Which of the following is (are) true for all
(W is the set of whole numbers)
a)

3+n<5+n

b) 2n > 3n

d)

(m x n)v n - m

e) (-3m)4- (-m) - - 3

m and n e W?
c) 2n + 3 > 5

Which of the following numbers has the greatest number of different
prime factors?
a)

15

b) 16

b)

27

e) 32

c) 25

If p is a prime number, then 1 3 + P is always:
a) a prime number

b) a composite number

d) an odd number

e) divisible by 13

c)

an even number

Which of the following statements is N O T true7
a) 2 is the smallest prime number
b) If a and b are whole numbers and b
whole number q and r such that
a - bq + r

where

ft 0, there

exist a unique

0 ^ r < b

c) Zero is a factor of every whole number
d) If p (a prime number) divides m x n (m and n are natural numbers),
then p divides either m or n or both
e) The
9.

10.

The sum of
(i.e., the

set of whole numbers iBclosed

under substraction.

the first 100 even digits is:
sum of 2, 4, 6 , .............. . 200)

a)

20,200

b) 2,020

d)

1,010

e) none of these

c) 10,100

What does this diagram illustrate?

9

8t4 -2

a) 4 x 2 * 8

b)

d) both a and b

e) none of these

10
c)

8< 9

236

FRACTIONS

Notei
1.

PRE-TEST

r, a, t, u, w are whole numbers with s, u + 0

Mark the correct statements below.
a)

If W is the set of whole numbers, and R is the set of
fractions, then W
R.

b)

The set of fractions is closed under division.

c)

The additive inverse of y is

d) £

S

<

(j)” 1

^

B.U

e) If the product of two fractions is 1, then the two fractions
are called reciprocals of each other.
2.

The shaded portion of the rectangle below represents what part of
the rectangle?

<0* 52 of* -3

,6
,2
cl? o f 5

d) | of 1

e) none of these
3.

The fraction — is equivalent to:
a)

t +u
s +u

d) n. —
u
4.

b)

c)

t 4S
8 4S

fc*
/u
e)\ —
7—
s/u

What sum is represented by the shaded portions of this illustration?

■>&

b) ± 2 .
' 12

d) ±5.
24

e) none of these

C) U

237

5.

IT

t

The difference — - — ia equal to which of the following?

. r -t
a) ■— - 1
a -u

.. r - t
b)
su
none of theae

ue
6.

t

r

The product — x — equal*:

a) t . r + u . s

d)

7.

ru-st
c)■
su

b)

n*”?
(u. a)

c)

u *r

(t.s) • (u.r)

e) none of theae

3
Workmen have — miles of road to build.

1
If they build y mile per day,

how long will it take them to build the road?

8.

9.

. 3 1
a) 4 X 3

. . 3 . 1
b) 4 T 3

,,, 1 , 3
d) 3 ■ 4

. 3 1
6) 4
3

Which o n

of the following

.
a)

125
126

. .126
120

..
}

215
330

144
0) 153

What value replaces n in the

-

10.

fractions is

V

-

* 1 3
C) 3 X 4

in its lowest

(reduced)

, 129
132

equation:

1

•>

f

w 2

d)

3

e) y

c) i

The least common denominator of 7 , 7 and 7 is:
3 4
6

6

a)

3

b)

d)

13

e) 72

c)

12

terms?

238

11.

if the greatest coonon factor of p and q li 2, what Is the least
common multiple of p and q7

b , «
d) 2 pq

e) 4 pq

c) pq

239

FRACTIONS

Note:

POST-TEST

t, u, v are whole numbers with v, t ft 0.

Which of the following statements is

(are)

false?

a) The set of whole numbers is a proper subset of the fractions,
b) T h e set of fractions is closed under multiplication and
division.
U

V

c) The additive inverse of — - -v
u
d)

if — * w - — , then w - 1.
v
v

e) The fraction

s
U + V

m s_
U

1_
V

The shaded portion of the rectangle represents what part of the
rectangle?

1 *3
.)* jOfj
b) T1 of« 73
^ 2 * 3
c) 3 °f 4

d>

2
3
3 °f 3

e) none of these
3.

If u

v,
a)

d)

4.

the fraction ^

t •u
t •v

is not equivalent to:
b)

u - 0
v - 0

e)

u
t
v_
t

c)

0 -u
0 -v

What sum is represented by the shaded portion of this illustration?

240

5.

The difference 1 . -u

. ,1 - u

* '7

What fraction of an acre is Mr. Farmer's vegetable garden?

t 6

^ 9

.1

b >8

C) 2

12
yg-

a) y y

b) T -

d)

e) none of these

5 hours

125

29
122

1036
2042

(reduced)

terms?

123
333

i144
453
11

12

On the number line, what fraction is half-way between y y and yy?
* 23
a) 12
d)

11y

. . 132
) 169

. 23
C) 26

e) none

of these

The lowest common denominator of

is:
4

11.

miles per

C) 4 hourB

Which of the following fractions is in its lowest

*

10.

ej none of these

Dave took a hike of 10 miles, walking an average of 2 y
hour.
How long did the hike take?

*
a)

9.

.) ^ uv

•> ?
d)

8.

c) “

3
2
Mr. Farmer has — acre of land and uses — of it for a vegetable
garden.

7.

„v - u

b> —

d) ^ u
6.

is equal to which of the following?

a) 4

b) 0

d) 48

e) none

b

o

c) 24
of these

Using the number line below, demonstrate the division algorithm
ll * I
1 5 * 5

241

DECIMALS AND PERCENTS

1.

Which of the following are correct?
a) 2.17 i 0.31 - 0.7
d)

2.

PRE-TEST

b)

40

(30%) X (10%) = 300%

- 0.25%

c)

5.701 - 0.37 = 5.664

e) 23.692 + 0.05 + 5 - 28.742

What fraction is represented by 0.82 8 2 8 2 8 2 .....

?

v 82

0

* 02

a> 100

9

C) 99

Q
d)
3.

4.

5.

6.

7.

8.

—

e) none of these

What is

«

(rounded to two places)

a)

189.00

b) 18.90

c) 1.89

d)

0.19

e) none of

these

What is 1.051 - 0.702 + 0.066 a)

0.283

b) 0.415

c) 1.009

d)

1.819

e) none of

these

1
2
Given the numbers 0.12, — , 0.125, — — , and 0.0999, which is the
smallest?
a)

0.12

b)

t

c) 0.125

d)

-

e)

0.0999

4

In an elementary school there were 220 girls.
This was 55% of the
school population.
How many boys were in the school?
a)

270

b) 220

c) 180

d)

121

e) none of

these

Which of the following IS falHe?
a)

10s

d)

1.1 x 1 0 s

t

102

- 10**

b)

- 11,000

10"** - 0 . 0 0 0 1
e)

Which number, on the number line,
a)

0.14

b)

0.10

d)

0.50

e)

0.90

(4.2

c) 2.1 x 10-2 - 0.021

x 10**)x (2.0 x 10**)

- 8.4 x10**

is half-way between 0.08 and 0.2?
c) 0.40

242

9.

Which of the following is the expanded notation for the decimal 53.24?
a)

5 x 102 + 3 x 1 0 1

+2 x

10°

b)

5 x 1 0 1 + 3 x 10°

c)

5 x 10* + 3 x 1 0 1

-2 x

10*

d)

5 x 1 0 1 + 3 x 10°

-2 x

101

+

4

x10"'

-

4

x10*

-

4

x10*

2x 10-1 + 4 x 10-1

e) none of the above
10.

11.

The decimal 0.00074 written in scientific notation is:
a)

74 x 1 0 s

b) 7.4 x lO” **

b)

74 x 10"**

e) none of theae

The decimal 0.42 is written in base five.
ten is:
a)

0.86

b) 0.70

d)

0.21

e) none of theae

c) 74 x 101*

Its equivalent in base
c) 0.24

243

DECIMALS AMD PERCENTS

1.

Which of the following are correct?
a)
c)
e)

2.

— ££-0.41
O. 4

b) 53.005 -0.28 - 25.005

7- - 20%

d) (50%) x(20%) -10%

D

54.823 +0. 7 + 0.02 - 55.723

What fraction is represented by 1.55555..... ?
. 15
"> T
d)

3.

POST-TEST

.. 14
b> T

15
—

e) none of

,15
c) To
these

^ , 0.014 x 0.84
,
j
.
What ia ------ — ---- - (rounded to two places)?
• w a

4.

a)

0.59

b) 0.58

d)

0.5888

e) none of

d)

6.

these

1
2
Given the five numbers! -r^zr , 7%, 0.1, 0.0199, and t t , which is the
largest ?
a)

5.

c) 5.888

b> 7%
0.0199

What is:

c) 0.1

e)

0.407 - 0.32 + 0.076 - ?

a)

0.847

b) 0.651

d)

0.163

e) none of

c) 0.451
these

In a mathematics test, 85% of the students in a class of 60 passed.
How many students did not pasB?
a)

15

b) 50

d)

9

e) none of

c) 51
these

244

7.

8.

9.

Which of the following statements is NOT true?
a)

1 0 # t 102 -10*

b) 10“ 3 - 0 .001

d)

1.1 x 10** - 11,000

e)

c) 3 . 2 x 10“ 2 - - 3 2 0

(2 x 102) x <3.1 x 1 0 3) - 6.2 x 10 s

Which number, on the number line,

is half-way between 0.02 and 0.2?

a)

0.22

b) 0.11

c) 0.40

d)

0.18

e) none of theae

The decimal 24.06 written in expanded notation is:
a)

2x

102+ 4x 1 0 1+ 6

b)

2x

1 0 2+ 4

c)

2x

1 0 1+ 4 x 10°+ 6

x 10“ 2

d)

2x

102+ 4 x 1 0 1+ 6

x 10"1

x 10°

x 1 0 1+ 6

x 102

e) none of these
10.

11.

The decimal 0.0031 written in scientific notation is:
a)

31 x 1 0 ~ 3

b) 3 . 1 x 1 0 s

d)

3.1 x l O 3

e) none of these

The number 2.3 is written in base four.
ten is:

c) 3 . 1 x l 0 ~ S

Its equivalent in base

a)

0.75

b) 2.30

d)

2.75

e) none of these

c) 0.75

245

RELATIONS AND FUNCTIONS

1.

Let (R) be a relation
definitions.

PRE-TEST

(Bet of ordered p a i r s ) .

Consider the following

a) The set of all first members of the ordered pairs making up
the relation (R).
b) The set of all second members of the ordered pairs making up
the relation (R).
c) A relation with the reflexive,
properties.

symmetric and transitive

d) A set of points in the plane corresponding to the ordered
pairs of the relation (R).
e) A relation in which no two ordered pairs h ave the same first
element.
Match the following terms with their definitions from above by
circling the appropriate alphabetical representation.
1) An equivalence relation
2)

The domain of relation

(R)

3) A function

2.

a

b

c

d

e

a

b

c

d

o

a

b

c

d

e

4) A graph of a relation

(R)

a

b

c

d

e

5) The range of relation

(R)

a

b

c

d

e

The relation "is greater
numbers.
Decide whether
and/or transitive.

than" is defined on the set of all natural
this relation is reflexive, symmetric,

a) reflexive
b)

symmetric

c) transitive

3.

d)

reflexive and symmetric

e)

reflexive,

symmetric and transitive

On the set {1, 2, 3, 4}, we will define the relation (R) consisting
of the following elements:
{(1,1),

(2,2),

(3,3),

(4,4),

(1,3),

(3,1),

(2,4),

(4,2)}

Which of the following statements are T R U E ?
a)

(R)

is reflexive

c)

(R) is transitive

b) (R)

is symmetric

d) (R)

is

e) all of the above are correct

an equivalent relation

246

4.

In the adjacent grid, graph the relation

(R) described in Problem 3,

6
5
4
3

2

1
0
The accompanying figure describes a function

(f).

a) The domain of f is:

{

}

b) The range of f is:

{

}

c)

f (b } =

Suppose tho adjacent function table was given,
following pairs would also be on the table?
a)

(0,-2)

b)

(12,144)

c)

(4,10)

d)

(4,8)

Which of the

25
64

e) none of these

7.

Consider the accompanying diagram.
true?

Which of the following is

a) The diagram illustrates a function from
A to B.
b) The diagram illustrates a function from
B to A.
c) There exists a one-to-one correspondence
between A and B.
d) All of the above.
e) None of the above.

A

(are)

B

247

8.

The function g, from A to B is illustrated by the adjacent table.
Which of the following defines the function g(A)?
a)

g(A) - A + A

b)

g(A) = 2A + 2

0

1

c) g (A) - A 2 + 2

9.

d)

g(A) *= 3A + 1

e)

none of these

2

3
4

Which of the following statements is correct7
a)

The inverse of a function

is never a relation.

b)

The inverse of a relation

is never a function.

c)

The inverse of a function

is always a relation.

d)

The inverse of a function

is always a function.

e) none of these
10.

Utilize the graph paper attached to graph the function f defined
on set
A - {l, 2,

...

10}

by

f(A)

= 2A - 5.

B
1
4
7

10
13

248

RELATIONS AND FUNCTIONS

1.

Which of

definitions is

(are) false?

a)

A function is a relation in which no two ordered pairs have
the same first element.

b)

An equivalence
relation
is
a relation wit h
symmetric, and transitive properties.

c)

The range of afunction is the size of
a set of ordered pairs.

d)

A relation is a set of ordered pairs.

e)

2.

the following

POST-TEST

the reflexive,

the object set of

A relation R is symmetric if for all x and y given,
yRx.

The relation "is the son of" is defined on
Decide whether this relation is reflexive,
transitive.

xRy

then

the set of all men.
symmetric, and/or

a) reflexive
b)

symmetric

c) transitive
d) reflexive,

symmetric and transitive

e) none of these
3.

On the set {a, b, c}, we will define the relation
the following elements:
(a,a), (b,b), (c,c), (a,b),

(b,a), (a,c),

Which of the following statements is

4.

(c,a),

consisting of
(b,c), (c,b)

(are) T R U E ?

a)

R is reflexive but

not symmetric.

b)

R is symmetric but

not transitive.

c)

R is transitive but not reflexive.

d)

R is reflexive and

e) R is reflexive,

(R)

symmetric but not transitive.

symmetric,

and transitive.

In the grid below, plot the relation

(R) described in Problem 3.

c ---------------b ---------------a ----------------

a

b

c

249

5.

The accompanying diagram describes a function
a) The domain of
b) Thr range of

(f) is:
(f)

iB:

(f) from A to B.

{
(

c) f(c) «

Suppose the adjacent function table was given.
Which of the
following ordered pairs would also be on the table?
a)

(-3,-20)

b)

(-4,20)

c)

(10,50)

d)

(B,45)

A
-2
0
5
7
•
*

e) none of these

B
-10
0
25
35
•

Which of the following, does the adjacent diagram illustrate?
a) The diagram illustrates a function
from X to Y.

K

b) The diagram illustrates a function
from Y to X.
c) There exists a one-to-one correspondence
between X and Y ,
d) All of the above,
e) None of the above.
The function G, from
to $ is illustrated by the table below.
Which of the following defines the function G ( ^ ) ?

a) G ( * )
b)

G{l¥)

«
-

r¥

+ l
9
+ 1

c) G ( # )

- # S + 1

d) G ( # )

= 2*+

2

e) none of the above

f¥
-1
1
-2
2
-3
3

5
2
2
5
5
10
10

250

PROBABILITY AND STATISTICS

A coin is tossed into the air.
land "heads up"?

PRE-TEST

What is the probability that it will

a)

50

b> f

d)

1.0

e) none of these

One ball is to be drawn at random from a box containing 3 red, 3 blue,
and 4 green balls.
What is the probability that the drawn ball is red?
c) 10

a) 0.1

b) 0. 3

d)

e) none of these

30

If THREE coins are tossed simultaneously into the air,
ways could they land with:

in how many

A. Three heads up

a)

1

b)

2

c)

3

d)

4

e) none of these

B. Two heads up

a)

1

b)

2

c)

3

d)

4

e) none of these

C. One head up

a)

1

b)

2

c)

3

d)

4

e) none of these

D. 0 heads up

a)

1

b)

2

c)

3

d) 4

e) none of these

E. At leaBt two
heads up

a)

1

b)

2

c)

3

d)

e) none of these

4

What is the
A box contains five pieces of paper as shown below.
probability of drawing the two pieces with the numberal 1 on them
in two successive draws without replacement?

© © ®
a) 0.50

b) 0.45

d) 0.05

e) none of these

c) 0.40

A political committee of 10 is to be selected from a population of
60 Democrats and 40 Republicans.
Which of the following provides
the best representative sample?
a) Select 10 names at random from the nameB of the 100 persons
invol v e d .
b) Select at random 5 me n and 5 women.
c) Classify the population involved into 5 age groups* then
select at random one Republican and one Democrat from each
age group.
d) From the Democrats select 6 at random and from the Republicans
select four at random.
e) Mak e a list of the last names of the 100 persons involved in
alphabetical order and select every 10th name.

251

6.

A journalist interested in knowing the attitude of his community
toward a proposed increase in school millage sampled his population
by questioning the first 20 customers of the local barber shop.
Frcxn
their responses he concluded that his community is against the m i l l ­
age increase.
Which of the following best describes the sampling
method used and the conclusion drawn?
a) The conclusion is valid since the sample is representative
of the community.
b) The sample is an unbiased random sample of the male population
and therefore the conclusion is valid for that population only.
c) The sample isunbiased but the conclusion is biased.
d) No valid conclusion can be drawn since the sample is biased.
e) The sample is biased but the conclusion is not since most
people do not like millage increases.

7.

The IQ scores of any large group tends to be normally distributed
about their mean.
From a population of 10,000 college freshmen a
sample of 100 is randomly drawn.
They are tested and their IQ
scores are found to have a mean of 108.
What can be concluded
from this experiment.
a)
The average IQ of the adult U.S. population is 108.
b)
The average IQ of the college students is 10B.
c) The college freshmen's IQ is 8 points higher than the
average persons of the same age.
d) Since the set of college freshmen is a subset of the total
college students, then the average IQ of the college students
is at least
108.
e)
The average IQ of the college freshman is about 108.

Questions 8, 9, and 10
STATEMENT.

ARE TO BE

ANSWERED WITH REFERENCE TO THE FOLLOWING

Suppose that the numbers below represent the scores of 15 students
on a mathematical examination.
90
90
as

85
85
80

75
70
70

65
65
60

55
50
40

8.

The mean score of this test is: ___________

9.

a) The me d i a n of the scores of this teBt is:
b) The mode of the scores of this test is:

10.

The range of the scores of this test is:

POST-TEST

PROBABILITY AND STATISTICS

When tosBing a coin, if the probability that it will land "heads u p ”
is Pr (H), and the probability that it will land "tails up" is
Pr (T), then:
a)

Pr (H) + Pr

(T) - 100

b) Pr

(H) + Pr

(T) - 1

c)

Pr (H) x Pr

(T) - 1

d) Pr

(H) x Pr

(T) - 0

e)

Pr (H) - Pr

(T) - 1

A box contains 50 light bulbs, 10 of which are 50-watt, 15 are
75-watt and the remaining are 100-watt.
What is the probability
on one drawing a 75-watt bulb will come up?
c) 0.25

a) 0.10

b) 0.15

d) 0.50

e) none of these

If THREE coins are tossed simultaneously into the air, what is the
probability that they will land with:
4
3
1
2
e) none of these
a)
d)
A. Three heads up
b)
c)
8
8
8
8
4
3
2
1
d)
a)
c)
e) none of these
B. Two heads up
b)
8
8
8
B
A
1
2
C. One head up
a)
b)
d)
o) none of these
c) 2
8
8
8
B
1
3
4
2
D. 0 heads up
a)
c)
d)
e) none of these
b)
8
8
8
8
E. At least one
head up

a)

1
8

b)

2

8

c)

2
8

d)

£
8

e) none of these

A fair six-sided die (cube) is rolled three times.
What is the
probability of obtaining "five spots up" on each of the three rolls?
b)

^
1
1
d> 6 * 6 X 6

6 + 6+6

c)

3±

e) none of these

Fro m a group of 4 boys and 2 girls, how man y ways are there of
selecting a committee of 2 boys and 1 girl?
a)

12

d) 2

b) 8
e) none of these

c) 4

253

6.

Suppose there are 1300 fifth-grade students in a school system.
You are given the task of estimating their arithmetic achievement
by testing a sample of the population.
Which of the following would
provide you with the best sample for this purpose?
a) From the population, select one boy and one girl and test
them.
b) Select at random one e lementary school in that system and
test all the fifth graders in that part i c u l a r school.
c) In each school, ask the fifth-grade teacher to provide you
with the names of 10 pupils in her class and test them.
d) From each of the e l ementary schools select at random 10%
of the fifth-grade students and test them.
e) No representative sample can be obtained and therefore all
1300 students m u s t be tested.

7.

The m e a s u r e of variability most used

in testing

is:

a)

the range

b) the m o d e

d)

the med i a n

e) the standard de v i a t i o n

QUESTIONS 8, 9, and 10 ARE
FOLLOWING STATEMENT

c) the m ean

TO BE A N S W E R E D WITH REFERENCE TO

THE

The numbers b elow represent the ages of a sample of 20 pupils
in an elementary school.
12
11
11
10

10
10
10
9

9
8
8
8

8
6
7
7

6
6
6
6

8.

The m e a n age of the pupils in this sample is:

9.

a) The median of the ages is:
b) The m o d e of the ages is: ____________

10.

The range of the ages is:

254

MATHEMATICAL SYSTEMS

1.

Which of the following statements is

(are)

PRE-TEST

false?

a) A mathematical system is a set of elements together with
one or mor e binary operations defined on the set.
b) The set of rational numbers with the operations of addition
and multiplication is a field.
c) In clock arithmetic

(mod 12):

(9 + 5) mo d 12 * 14
d) In clock arithmetic

(mod 12):

(A x B) mo d 12 ■* (B x A) mod 12
e)

2.

3.

A numeral is divisible by q if the sum of the
decimal representation is zero m o d 9.

In clock arithmetic,

37

(mod 8) »5.

The

a)

sum

b) product

d)

remainder

e) none of these

What is
a)

result

(8 + 7) mod 4 equal to?
1

c) 8 (mod 4) + 7

b) 3
(mod 4)

d) a and c are correct
e) b and c are correct
4.

Complete the multiplication table
X

0

1 2

0

0

0

1

0

2

0

3

0

(mod 4).
3

0

0

digits in

5 is actually a:
c)

quotient

its

255

5.

Consider the following two tables.
Table 1

A.

B.

Table 2

B

A

B

C

•

A

B

C

A

A

B

C

A

A

A

A

B

B

C

A

B

A

B

C

C

C

A

B

C

A

C

B

Use the above tables to compute the following.
a)

(A0B)

C- #

b)

(ABB)

A-#

c)

(cBc)

c-#

Does Table 1 describe a mathematical system?
yes _____

C.

Does Table 2 describe a mathematical system?
yes

D.

no ______

____

no ______

Docs the set (a, B, c} and the two o p e r a t i o n s B
a field?
yes _ _ _ _ _

no _____

and B

form

256

MATHEMATICAL SYSTEMS

1.

Which of the following statements is

(are)

POST-TEST

true?

a) Clock arithmetic is closed under addition and multiplication.
b) In clock arithmetic

(mod 1 2 ) i

A (B ♦ C) « AB + C
c) The set of whole numbers and the operation of addition define
a mathematical system.
d) There is only one way of establishing a one-to-one
correspondence between set A - {l, 2, 3} and Bet B - {a, b, c) }
e)
2.

3.

(12 x 6) mod 8 - 0

In clock arithmetic
actually at

(mod 9), 53

(mod 9) - 8.

a) remainder

b) partial quotient

d) product

e) none of these

In clock arithmetic

(mod 8), what is:

The result 8 is

( 6 x 3 ) mod

a) 6 (mod 8) x 3 (mod 8)
b) 2
c)

18

d) a and b are correct
e) a and c are correct
4.

c)

Complete the clock addition table (mod 4) below.

partial sum

8-

257

5.

Consider the table below.

a) Does the table above describe a mathematical system?
yes

no

b) Does the operation Q
yes

have the closure property?

no _ _ _

c) Does the operation O
yes

d) What

_____

have the associative property?

no _ _ _ _ _

(if any)

is the identity element of the operation Q 7

e) Pair the following elements with their inverse,
a

t

b______ /

c

f

d

APPENDIX

B

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 S
FORM A
FORM B

(PRE-TEST)

AND

(POST-TEST)

259

A TEST OF BASIC M A T H E MATICAL UNDERSTANDINGS
PREPARED BY:

Vtt. M <JLdn.td lejUULno. Vo66eXt
M I C H I G A N STATE UNIVERSITY
EAST LANSING,

MICHIGAN

1964

Directions:
This test is d e s i g n e d to measure your understanding of
mathematics.
Many of the items relate to the new content in p r e s e n t
programs of mathematics for elementary pupils.
Each of the fifty-five questions is of multiple-choice t ype and
includes four possible answers.
Read each question c a r efully and de c i d e
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 y o u change your
choice, erase your original mark and ci r c l e the correct one.
Sample Q u e s t i o n :
1.

Which of the following shows the deci m a l
a.

125

b.

12.5

c.

1.25

d.

.125

5

form of the fraction — ?

Answer Sheet:
1.

a

b

d

Since 1.25 is the correct answer, the letter

(c) is circled

260

FORM A (PRE-TEST)
1.

2.

3.

4.

When you write the numeral "S" you are writing
a.

the number S.

b.

a pictorial expression.

c.

a symbol that stands for an idea.

d.

a Hindu-BabyIonian symbol.

Bill discovered that > means "is greater than" and < means "is
leBS than."
in which of the following are these symbols not used
correctly?
a.

The number of states in the United States < the number of
United States Senators.

b.

The number of stateB in the United States > the number of
stripes in the flag.

c.

2 3 > 32

d.

3 + a < 5 + a

When two Roman numerals stand side by side in a symbol,
values are added
a.

always.

b.

sometimes.

c.

never.

d.

if the base is X.

their

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)

a, b, and d are correct.

261
A.
5.

6.

7.

8.

Zero may be used
a.

as a place holder.

b.

as a point of origin.

c.

to represent the absence of quantity.

d.

in all of the above different ways.

2,200.02

a.

2000 + 200 + 20.

b.

2000 + 20 + ^

c.

2000 + 200 + ■£—

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.

Which of the following does not show the meaning of 4 2 3 ^ ^ ?
a.

(4 x 100) + <2 x 10)

+ 3(1) - 423

b.

42 tens + 3 ones

- 423

c.

423 ones • 423

d.
9.

is shown by

4 hundreds + 42 tens + 23 ones “ 423

A numeral for the X's in this example can be written in many
different bases.
Which numerals are correct?
a.

1 00r
four

b.

14

c.

16ten
31

XX

X

XX

XX

X

X

X

X

XX

X

.
twelve
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.

X

262
A.
10.

11.

12.

13.

14.

15.

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 1S

d.

2 x 12*

In this addition example,

in what base are the numerals written?

a.

base two

120?

b.

base three

+10?

c.

base four

20Q?

d.

none of the above

About how many tens are there in 6542?
a.

6540

b.

654

c.

65 x

d.

6.5

2

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.

263
A.
16.

17.

18.

19.

20.

21.

The counting numbers are closed under the operations of
a.

addition and subtraction.

b.

addition and multiplication.

c.

addition«

subtraction, multiplication,

d.

addition,

subtraction,

If a and b are natural numbers,
a.

commutative property.

b.

associative property.

c.

distributive property.

d.

closure.

If a x b ' O

and division.

and multiplication.
then a * b = b + a

is an example of

then

a.

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, ho w 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

How would the product
29 above the 4306 and

in this example be affected if
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.

you put the

4306
x29
----

264
A.
22 .

An important mathematical principle can be helpful in Bolving the
following example.
28 + 659 + 72 - [

]

What principle will be of moat help?

23.

a.

the aasociative 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.

24.

25.

26.

27.

(300 x 50)
+

x

(6 + 7).

b.

(3 x 7 )

(5x7)

+ (6x7).

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 number?
a.

271

b.

277

c.

281

d.

282

Which of the following numbers is odd?

a.

IB 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

265
A.
28.

29.

30.

31.

32.

Look at the example at the right.
Why is the ''4"
in the third partial product m o v e d over two places
and written u nder the 2 of the multiplier?

157
x 246
942
628
314
38622

a.

If you p u t it d i r e c t l y uner the o ther partial
products, the answer w o u l d be wrong.

b.

You m u s t m o v e the t hird 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 m u s t come u nder the hundreds column.

d.

You are really mu l t i p l y i n g by 200.

Which of the fundamental prop e r t i e s of arithmetic would you employ
in proving that (a + b) + (a + c) ■ 2a + b + c?
a.

Associ a t i v e property.

b.

Commut a t i v e property.

c.

A s s o c iative and distributive properties.

d.

A s s o c iative and c o m m u tative properties.

If N represents an even number,
represented by
a.

N + 1

b.

N + 2

c.

N + N

d.

2 X N

the next larger even number can be

Every natural number has at least the following factors:

It
To

a.

zero and one.

b.

zero and

c.

one and itself.

d.

itself and two.

itself.

is said that the set of w h o l e numbers has
a natural order,
find the successor of a natu r a l 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.

266
A.
33.

34.

35.

The paper below has been divided into 6 pieces.

a.

sixths.

b.

thirds.

c.

halves.

d.

parts.

It shows

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,
ho w does the answer compare with the fraction divided?
a.

It will

be larger.

b.

It will

be smaller.

c.

It will

be twice as large.

d.

There will be no difference.

36.

Which algorithm 1 b illustrated by the following sketch?

37.

Another name for the inverse for multiplication of a rational
number is the
a.

reciprocal.

b.

opposite.

c.

reverse.

d.

zero.

267
A .
30.

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 —■ upside down.

b.

Multiplying always makes the answer larger.

c.

The divisor ~ is less than 1.

d.

39.

40.

41.

* 4

3

4
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.

we

b.

we multiply one term and divide the other term by that
same number.

c.

we subtract

d.

we multiply both terms by the same number.

add the same number to both terms.

the same amount from both terms.

The nearest to 45% is
a.

44 out of 100

b.

.435

c.

4.5

d.

.405

The principal of a school said that 27 per cent of the pupils went
to the museum.
Which statement beBt describes the expression
"27 per cent of the pupils went to the m u s e u m ”?
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 the museum.

d.

42.

5 , 3 ^ ^2

27 per cent is the same as
in per cent form.

2

.27— a decimal fraction written

Sally completed — of the story in 12 minutes.
At that rate how
long will it
take her to read the entire story?
a.

18 minutes

b.

12 minutes

c.

6 minutes

d.

24 minutes

268
A.
43.

There w e r e 400 students in the school.
One hund r e d per cent of the
children had lunch in the cafeteria on the first d a y of school.
On
the second d a y 2 boys were absent and 86 children went h o m e for
lunch.
W hich of the following equations ca n be used to find the
per cent of the school enrollment who went home for lunch?
a.

400 - 88 i

v
D,.

X
88
100 " 400

C.

d.
44.

X

88

■ 400

400 - 90 < X

What can be said about y in the following open sentence if x is

a natural number?
X + x + 1

45.

46.

a.

X < y

b.

X > y

c.

X - y

d.

X * y

which one of the
a.

1
2

b.

3
4

c.

5_
8

d.

6
11

Which of the following

□

a.

7 ♦ 2 -

b.

h - 5 - 9

c.

y - 3 0 - 6

d.

n - 3

269
A.
47.

48.

49.

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 conxnutative.

c.

the system has an identity element.

d.

all of the above are correct.

Measurement

is a process which

a.

compares an object with some known standard or accepted
unit.

b.

tries to find the exact amount.

c.

is never an exact measure.

d.

chooses a unit and then gives a number which tells how
m any of that unit it would take.
1)

a and b are correct.

2)

a and c are correct.

3)

a, b r and d are correct.

4)

a, c, and d are correct.

The set of points sketched below represents a
^

50.

51.

a.

line,

b.

ray.

c.

line segment.

d.

none of the above.

How many triangles does the figure contain?
a.

four

b.

six

c.

eight

d.

ten

The set of points on two rays wit h a common end-point is called
a.

a triangle.

b.

an angle.

c.

a vertex.

d.

a side of a triangle.

270
A.
52.

53.

54.

55.

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

The solution set of an open sentence ma y consist of
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-setB or groups

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.

271

FORM B (POST-TEST)
1.

2.

Which of the underlined words or
signs in the following sentences
refer to symbols rather than the things they represent?
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.

The number 1^6 is the same as the number 7 + 9.

When we use the * symbol between
two terms (as 2 + 2 * 4 ) we m ean
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.
d.

3.

4.

f5 + 2 ) x 3 * 7 x 3
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 numeration were a "place value system" with
the same value for the baBe 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.

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.

272
B.
5.

6.

7.

8.

In the numeral 7,843,
the value of the 87

how does the value of the 4 compare with

a.

2 times as great

b.

1/2 as g r e a t

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.

b.

1,000 t i m e s as great.

c.

the same since both are sixes.

d.

six times as much.

W h i c h of the following statements best tells w h y we write a zero
in the numeral 4,039 when we want it to r e p resent "four thousand
thirty-nine"?
a.

Writing the zero helps to fill a p lace which would
otherwise be empty and lead to misunderstanding.

b.

The numeral would represent "four hund r e d thirty-nine"
if we did not write the zero.

c.

Writing the zero tells us not to read the hundreds*

d.

Zero is used as a place-holder to s h o w that there is no
number to record in that place.
1)

a

and b are correct.

2}

a

and c are correct.

3)

a

and d are correct.

4}

a, b,

figure.

and d are correct.

B e l o w are four numerals written in expanded notation.
is not written correctly?
a.

4 (ten)2 + 9 ( t e n ) 1 + 3 (ones)

- 493.
ten

b.

3 (seven)3 + 6 (seven)1 + 1 (one) - 363

c.

4 (twelve)2 + 5 (twelve)1 + e(one)

d.

2 (five)2 + 2 (five)1 + 4 (one)

seven

» 45e^
,
twelve

- 224,.
f ive

Which one

273

9.

10.

If you arc permitted to use any or all of the sumbolB 0, 1, 2, 3,
4 and 5 for developing a system of numeration with a place value
system of numeration similar to ours, a list of all possible bases
would include:
a.

base one,

two, three,

b.

base two,

three, four,

c.

base two, three,

d.

base one, two, three,

12.

13.

five, and six.

five, and six.
and five.
four, and five.

About h o w m any hundreds are there in 34,870?
a.

11.

four,

four,

4

b.

35

c.

350

d.

3,500

In what base are the numerals in this multiplication example
written?
34 ?
23?

a.

base five

b.

base eight

c.

base eleven

d.

you c a n ’t tell

124?

?0?
1024 ?

Which of the following are correct?
a.

In the symbol 5 3, 5 is the base and 3 is the exponent.

b.

In the symbol 5 s, 3 is the base and 5 is the exponent.

c.

5S * 5 x 5 x 5

d.

5S « 3 x 3 x 3 x 3 x 3
1)

a and

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,
best applies to 19?
a.

nominal

b.

ordinal

c.

composite

d.

cardinal

18,

19, 20, 21,..., what term

274
B.
14.

Examine the following illustration:

Which of the following does the above best illustrate?

15.

16.

17.

a.

The idea of a cardinal number.

b.

The use of an ordinal n u m b e r .

c.

A means for determining the cardinal number of the set
by counting with ordinal numbers.

d.

None of the above.

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 Bolved this example by adding downj
work by adding up.
Add

34
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.

then he checked hie

2 75
B.
18.

The statement "the quotient obtained when zero is divided by
a number is zero" is expressed as

19.

When a natural number is divided by a natural number other than
1, how does the answer compare with the natural number divided?

20.

21.

22*

a.

larger

b.

smaller

c.

one-half as largo

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

Which one of the following methods could be used to find the
answer to this example?
171)612
a.

Multiply 17 by the quotient.

b.

Add 17 six hundred times.

c.

The answer would be the sum.

d.

Subtract 17 from 612 as many times as possible.
The
answer would be the number of times you were able to
su b t r a c t .

276
B.
23.

Which one of the following would give the correct answer to this
example?
2.1
X 21

24.

a.

The sum of 1

b.

The sum of 10 x 2.1 and 2 x

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

2.1.

correct answer to 439 x 563?

a.

Multiply 439
answer.

x 3, 439 x 60, 439 x 5and

b.

Multiply 563

x 9,

c.

Multiply 563
answer.

x 9, 563 x 39, 563 x439 and then add the

d.

25.

x 2.1 and 21 x 2.1.

then add the

563 x 3, 563 x 4and then

Multiply 439 x 3, 439 x 60,
answer.

add the

answer.

439 x 500 and then add the

Which of these numerals are names for prime numbers?
a. 3

b- I
c.

12,.
five

d.

9 - 2
1)
2)

a and c are correct.

3)

a, b, and d are correct.

4)

26.

a is correct.

a, b, c, and d are correct.

Let x represent an odd numberj
Then x + y must represent
a . an

even n u m b e r .

b.

a prime number.

c.

an

d.

odd number.

a composite number.

let y represent an even number.

277

B.
27.

28.

29.

30.

31.

The inverse operation for addition is

Tl

a.

addition.

b.

subtraction.

c.

multiplication.

d.

division.

least common multiple of 8,
a.

2 x 2.

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

12, and 20 is

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.

c.

Since the sum of the second column is 23
(which has two figures in i t ) , 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
p u t the hundreds in the next column.

The operations which are associative are
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 and d are correct.

Which of the following is an even number?
a.

(100)

b.

(100)

c.

(100)

d.

(200)

three
five
seven
.
five

251
161
252
271
935

278
B.
32.

33.

The fact that a + b
example of
a.

distributivity.

b.

commutativity.

c.

closure.

d.

associativity.

d e creases as the number of pieces

increases.

b.

increases as the number of pieces

decreases.

c.

increases as the number of pieces

increases.

decrease 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

m a y be used to represent the idea that

3 is to be divi d e d by 4.

b.

3 of the 4 equal

c.

3 objects are to be compared with

d.

36.

(c + b) + a is an

a.

a.

35.

is exactly equal to

Observe the drawing on the right.
When the circle
is cut into equal pieces, the size of each piece

d.

34.

(b + c)

parts are being considered.
4 objects.

all of the above.

When a w hole number is multi p l i e d 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.

T h e r e 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.
b.
c.
d.

1

1

2 + 3
2
3
3 + 4

2

1

3 + 4

w

+

279
B.
37.

38.

7_

3

We can change the denominator of the fraction ■
"1" without changing the values of the
r*
fraction by
5
a. adding — to the numerator and denominator.
b.

5
subtracting — from the numerator and the denominator.

c.

multiplying both the numerator and the denominator by

d.

5
dividing the numerator and the denominator by — .

What statement best tells why we "invert the divisor and
when dividing a fraction by a fraction?
a.
b.

39.

40.

to the number

.

multiply

It is an easy method of finding a common denominator and
arranging the numerators in multiplication form.
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 iB the same as multiplying by the
reciprocal of the fraction.
2

If the denominator of the fraction —
of the resulting fraction will be
a.

half as large.

b.

double in value.

c.

unchanged in value,

d.

a new symbol for the

same number.

may also be written as
a.

.45

b.

45/100

c.

45 x 100%

d.

.450

ismultiplied

1)

a and b are c o r r e c t .

2)

a and c are c o r r e c t .

3)

a and d are c o r r e c t .

4)

a, b. and d are correct.

by

2, the

value

280
B.
41.

.5 and ,27 are illustrations of "decimal fractions."
They could
1
27
be written as "common fractions" in the form of y and
,
respectively.

What is a decimal fraction?

a.

It is another 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

which has both a decimal and a fraction

as parts of it.
d.

42.

2

A number like -J=-r which is a fraction and has a decimal as
.56
either the numerator or denominator or both.

The ratio of x's in Cir c l e A to x's in Circle B can be shown by

a.

16
4
XX
XX

b- 7
C*
d.

43.

1
2
4
16

A

B

Sue paid 20C for 4 apples.
Which of the equations below could be
used to find the cost of 1 apple?
a*

4
1
20 “ x

b.

x + 4 - 20

c,

—X ■ in
20
4

d.

x - 4 - 20

0
44.

xxxx
xxxx
xxxx
xxxx

The decimal

for the numeral —

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.

281
B.
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, B t 12

b.

1,

2, 3,

4, 6 fi 12

c.

lr

2 1 3,

4 £ 6

d.

2,

3, 4,

6 & 12

Modular arithmetic is
a.

cumutative.

b.

associative.

c.

distributive with respect to mu l t i p l i c a t i o n over addition.

d.

all of the above.

Which of the following is an a pproximate measure?
a.

35 farms

b.

12 buttons

c.

7^

d.

15 beads

inches

Which of the following does the sketch below represent?
------------------------a.

line

b.

ray

c.

line segment

d.

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.

282
8.
50.

Which of these triangles are right triangles according to the
length of the sides given?

6"

10

4"

8"
6"

5"
5"

4"
51.

52.

53.

7"

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 round tablecloth that had a radius of 14
inches.
Which of the formulas can she use to determine the length
around the cloth?

2

a.

A ■ TTr

b.

C ™ TTd

c.

C “ 2Ttr

d.

A - C/d

Which of the following best defines a solution set?
a.

A solution set is a set w h i c h includes each
member that gives a true statement.

and every

b.

A solution set is a single sentence w h i c h identifies a
variable that w ill 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.

283
B.
54.

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),

c.

{3,

(-3),

d.

{l,

(-1),

5}
10}
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.

APPENDIX C
DU T T O N A R I T H M E T I C A T T I T U D E

I NV EN T OR Y

APPENDIX C

DUTTON ARITHMETIC ATTITUDE INVENTORY

Name

Student Number___________

Place a check (/) before those statement* w hich 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 good with 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 if 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 d o n ' t feel sure of myself in arithmetic.
10.
Sometimes I enjoy the challenge presented by an arithmetic
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 o ther
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

R

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 o n e ) .

18.

My average grades made in arithmetic were:

19.

List two things you like about arithmetic.

A D C D (circle one).

A.
B.
20.

List two things you dislike about arithmetic.

A.
B.

285

APPENDIX D
A T T I T U D E SCALES 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

287

A STUDY OP ATTITUDE OF PROSPECTIVE
ELEMENTARY SCHOOL TEACHERS

Dear Student!
We are attempting to evaluate the attitudes of prospective
elementary school teachers,
of mathematics,

such as yourself,

toward some aspects

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.

S t u d e n t 's Name

Student's Number

288

w
2
u
<

w
s
o

H
w

cc

LQ u
►I
D

A D
A

D

D

1. Most school work is memorizing of

information.

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.

A

D U

4. My mathematics teacher showed us different ways of
solving the same problum.

A

D U

5. Our teachers wanted us to do most of our learning from
the textbook which i3 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 to solve problems
only by the procedures ho taught.

A D D

8. We were expected to learn and discover many ideas for
ourselves.

A

9. We were expected to develop a thorough understanding of
ideas and not just to memorize information.

D U

A D D

10.

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 d i t 1 iculties.

A

A

D

Our teachers believed in strict discipline and each
student did exactly what he was told to do.

U

13,

In my mathematics classes, students who had original ideas
got better grades than did students wh o were most careful
and neat in their work.

D U

14.

Most of our classroom work was listening to the teacher.

A D D

15. My mathematics teacher required the students not only to
master the steps in solving problems, but also to under­
stand the reasoning involved.

289

z
w o o;
w

<

K

If)

O
c

H
z
a z

D

U

16.

My mathematics teacher encouraged us to try to find
several different methods of solving particular
problems.

D

U

17.

M y mathematics course required mor e thinking about
methods of solving problems than memorization of rules
and formulas.

D

U

18.

My mathematics teacher wanted us to discover m a t h e ­
matical principles and ideas for ourselves.

D

U

19. My mathematics teacher explained the basic ideasj we
were expected to develop the methods of solutions for
ourselves.

D

U

20.

We did not use just one textbook for most of our subjects.
Various sources and books from which we can learn were
suggested to us.

D

U

21.

Most of the problems m y mathematics teacher assigned are
to give us practice in using a particular rule or formula.

22.

Much of our classroom work was discussing ideas and
problems with the teacher and other pupils.

23.

In mathematics there is always a rule to follow an
solving problems.

D U

D

U

A

D

U

24.

I generally like my school work.

A

D

U

25.

It should be possible to eliminate war once and for all.

A

D

U

26.

A

D

U

27.

More of the most able people should be encouraged to
become mathematicians and mathematics teachers.

A

D

U

28.

Someday mos t of the mysteries of the world will be
revealed by science.

A

D

U

29.

A

D

u

30.

A

D

u

31.

By improving industrial and agricultural methods,
can be eliminated in the world.

poverty

290

W
2

P!
H
H
oi
w

(4 <
W

^
S
O
<

O

<
u
*-* V*

1/1

a

o

A

D

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.

A

D

U

34. Outside of science and engineering, there is little need
for mathematics (algebra, geometry, etc.) in most jobs.

A

13

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

0

U

37. Very few people can learn mathematics.

A

D U

A

D

U

39. Mathematics (algebra, geometry,
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

13

U

41.

1 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

4 3. Education can only help people develop their natural
abilitiesj it cannot change people in a fundamental way.

A

D

U

44.

1 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,
getting a good education.

A

D

U

47.

It is important to know mathematics
etc.) in order to get a good job.

A

D

U

48. Almost anyone can learn mathematics
to study.

38. Mathematics help one to think according to strict rules.
etc.)

is not useful

for

but 1 can see value in

(algebra,

geometry,

if he is willing

291

W M
w <
2 H
w
o
tu
<
w
2
LOu
o
M^
<
PD
A

D U

49.

Mathematics is a very g o o d field for creative people
to enter.

A

D U

50.

Unless one is planning to become a m a t h ematician or a
scientist the study of advanced m a t h e m a t i c s is not very
important.

A

D

U

51.

A n y person of average intelligence can learn to understand
a good d e a l of mathematics.

A

D U

52.

The most enjoyable part of my life is the time I B p e n d
in school.

A

D U

53.

Even complex ma t h e m a t i c s can be m a d e understandable and
useful to every high school student.

A

D U

54.

In the near future m o s t
advanced mathematics.

A

D U

55.

With h ard work anyone ca n succeed.

A

D U

56.

Almost every present h u m a n problem will be solved in
the future.

jobs will require a knowledge of

A

D

U

57.

Almost all pupils can learn complex mathematics
is properly taught.

if it

A

D

U

58.

I like all school subjects.

A

D

U

59.

There is little place for originality in mathematics.

A

D

U

60.

I enjoy m o s t of my school work and wan t to get as m uch
additional education as possible.

A

D

U

61.

Only pe o p l e with a v e r y special talent can learn
mathematics.

A

D U

62.

Ma t h e m a t i c s will change rapidly in the near future.

A

D U

63.

Although school is difficult,
as I can get.

A

A

D

U

D U

64.

65.

I w a n t as m uch education

In the study of mathematics, if the student misses a few
lessons it is difficult to catch up.
I find school interesting and challenging.

APPENDIX E
THE E X P E R I M E N T A L G R O U P E V A L U A T I O N OF
D I F F E R E N T A S P E C T S OF THE P R O G R A M

APPENDIX E

STUDENT EVALUATION OF THE COURSE

Your evaluation of this course will be helpful in the future
planning of similar courses in this program.
Consider each of the following statements separately and
indicate the extent to which you agree or disagree with it by circling
the appropriate symbol to the right of the statement.
The symbols used are:
sA--Strongly Agree
A — Agree in General
U — Undecided
D — Disagree in General
S D — Strongly Disagree
Please respond to all the items.
Responses made to any items
in these pages will have no bearing on your grade.
1.

2.

3.

4.

5.

6.

7.

There should be more activities using
manipulative materials in this course.

SA

A U

D SD

There should be more time spent on methods
of teaching elementary school mathematics.

SA

A U

D

SD

There should be more time spent on planning
of teaching strategies to be used at the
Allen Street School.

SA

A U

D

SD

There should be more time spent teaching
mathematics at the Allen Street School
Laboratory.

SA

A U

D

SD

There should be m ore lectures about
mathematical content.

SA

A U

D

SD

There should be m o r e lectures about the
methods of teaching mathematics.

SA

A U

D

SD

There should be more films or video-tapes
related to the teaching of elementary school
mathematics.

SA

293

A

U

D

SD

294

8.

9,

10.

There should be m o r e hours assigned to this
combined method- c o n t e n t course.

SA

A

U

D

SD

There should b e m o r e time spent on paper and
pencil problem solving activities.

SA

A

U

D

SD

There should be m o r e contacts with Allen Street
School teachers in the planning of strategies
for teaching m a t h e m a t i c s at the school.

SA

A

U

D

SD

What else would you suggest to improve the quality of this course in
terms of content, method, materials, activities, etc?
W rite your
comments below.
T hey will be grea t l y appreciated.

Thank you.

APPENDIX F
SPECIMEN O F STUDENT FI L E FOR T H E L E A R N I N G
UNIT ON NUMERATION SYSTEMS

296

Activities

Content Objectives

1.

2

.

□

Write a number in expanded
notation.

□

3.

Identify the place value
of a numeral.

4.

Interpret inequality of
numbers in bases other
than ten.

5.

Interpret equality of
numbers in bases other
than ten.

6.

Addition in other bases.

7.

Subtraction in other
bases.

8

.

9.

10

.

11.

I.

Interpret a numeration
system using different
symbols.

Arithmetic in base
twelve.
Odds and evens in other
bases.
Multiplication in other
bases.
Division in other bases

□

II.

III.

□
IV.

□
□
□
□
□
□
□

Make a set of beansticks
and illustrate how to
use them to explain some
arithmetic
problems. □
Play with and become
familiar with a number
of chip trading games.
Use the MA Blocks to
illustrate arithmetic
in different number
bases.

Qj

Explore other numeration
systems.
□

Activity
Numer a t i o n U n i t
Be a n Sticks

Hake some bean sticks for base ten with y o u r partner.

Partner A — Explain to partner B h o w to solve the p r o b l e m 2 3 - 1 7 - 7
using the bean sticks.

Was that satisfactory to you
Partner B?
_____________
(If yes, go on)

Partner B--Explain to partner A ho w to s olve the p r o b l e m 47 -• 3 w i t h
the bean sticks.

Was tha t O . K . , Partner A?

298

Optional Activity--Ia

Conduct a contest to see who can best guess the number of beans in a jar.

See E.M.I., Volume I.

Page II

299

Activity II
Numeration Unit
Chip Trading

Make a chip trading notebook

(see card 1-5).

Play at least one of the chip trading games from each of the sets 1-V.

Tell someone about your favorite chip trading game and show them how
to play it.

Page III

300

Activity III

Multi Base Arithmetic Blocks

Sketch h o w one hundred unit cubes look when represented with the m i n i m u m
number of pieces of wood.

Base 6

Base 5

Base 4

Base 3

Page IV

301

Optional Activities III

a.

Make your own set of MA Blocks with sugar cubes and Elmer's glue.
Spray them with plastic or they get sticky.

b.

Make a Bet of activity cardB for MA Blocks.

c.

Make a binary computer.

Page V

302

Activity IV
Numeration

List as many ways you can write 1972 in other systems of numeration?

l‘age VI

303

Optional Activities IV

a.

Explore a numeration system with a negative base.

b.

Invent a new numeration system and see if your friends can figure
it out.

c.

Read about and report on the Duodecimal Society.

Page VII

304

Numeration Vocabulary List

1.

abacus

2.

additive principle

3.

binary

4.

decimal

5.

digit

6.

duodecimal

7.

expanded notation

8.

exponent

9.

numeral

10 .

place value

11 .

power

12.

Roman numeral

13.

subtractive principle

Page VIII

305

Tasks
la □

Ha □

Ilia Q

Illb □

m e

□

iVa □

IVb □

IVc □

Having identified a group
of 5 to 10 students...
The TTT freshman student
will demonstrate his
ability to use his knowl­
edge about the "Tasks of
Teaching," by designing a
lesson which incorporates
assessment goals/objectives,
strategies and evaluation.
The academic content of
the instruction design
will be the topic of the
week.
The instructional
design will be evaluated
on the basis of:
1. Inclusion of an
assessment instrument
(pre-test).
2. Goals for the week as
developed by the four
member team.
3. Specific objectives for
the lesson including
terminal behavior,
conditions and
criteria.
4. Strategies and the
necessary materials
which are appropriate
for:
the readiness
and chronological level
of the child, the con­
tent to be taught, and
employ the use of con­
crete objects.
5. Inclusion of an
evaluation instrument
which tests specifically
for the lesson objectives,

Page IX

Completed

Assessment
Goals/Objectives
Strategies
Materials
Evaluation

□□□□□

INSTRUCTIONAL DESIGN

INSTRUCTIONAL DESIGN
TO BE USED WITH CHILDREN

ADDITIONAL LESSONS
DEVELOPED
(Simulation or No
Instruction)
Assessment
Goals/Objectives
Strategies
Materials
Evaluation

INSTRUCTIONAL FEEDBACK
Comments/Questions

□□□□□

OPTIONAL
ACTIVITIES

APPENDIX G
SCORES O F S T U D E N T S IN T H E E X P E R I M E N T A L G ROUP
ON THE P R E - AND P O S T - C R I T E R I O N
R E F E R E N C E D TESTS

SCORES OF STUDENTS IN THE EXPERIMENTAL GROUP Off THE PRE- AND POST-CRITERION REFERENCED TESTS

MEASURBtEKT
STU.
I.D.

PreTest

1
2
3
4
5
e
7
8
9
10

NUMERATION

SETS 5 SET
RELATIONS
PreTest

PostTest

WHOLE NUMBER
SYSTOf

PreTest

PostTest

PreTest

67
64
58
88
66
70
16
36
57
70

50
70
70
100
20
60
20
50
80
60

55
91
46
100
10
73
82
73
74
100

78
34
32
98
64
64
24
56
48
26

80
100
80
100
58
86
66
68
80
70

36
46
46
82
46
72
64
36
54
36

11
12
13
14
15
16
17
18
19
20

42
33
85
8S
57
55
50
92
71
43

30
40
80
80
70
90
70
80
80
60

76
76
82
100
IB
48
70
20
78
85
60
10
80
73
73
86
80
60
86
46

60
92
92
74
74
82
92
82
100
50

66
11
46
76
54
54
86
86
66
30

72
58
90
100
76
90
86
88
70
90

21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38

92
85
92
92
42
71
100
64
29
71
43
36
43
88
45
75
79
79

100
90
90
100
60
50
100
90
50
80
30
40
30
100
SO
70
70
60

80
74
33
100
33
80
93
93
53
86
27
60
40
100
60
40
64
39

82
92
100
100
46
100
100
100
75
100
30
64
46
82
40
64
70
74

100
88
86
78
72
72
77
58
17
83
68
60
34
70
46
26
88
44

100
100
98
100
90
90
90
90
76
90
88
64
6B
90
65
80
90
78

PostTest

PreTest

DECIMALS

RELATION fc
FUNCTION

PROBABILITY
& STATISTICS

MATHWATICAL
SYSTEMS

PostTest

PreTest

PostTest

PreTest

PostTest

PreTest

PostTest

PreTest

90
95
92
90
50
84
57
95
95
85
71
87
94
70
87
92
90
90
85
90

35
50
60
80
40
20
50
30
60
60

45
96
88
90
54
82
77
57
90
85

0
33
0
67
0
0
40
33
0
33

40
30
50
70
50
70
68
50
50
80

20
47
90
90
60
88
80
90
86
80

33
0
60
43
0
67
50
38
50
33

44
80
80
90
30
70
60
26
32
38
40
30
80
52
80
100
69
100
62
68

80
40
90
70
60
60
80
60
70
60
50
40
55
100
50
20
50
60

89
85
96
90
70
70
100
80
65
83
55
71
86
94
66
78
76
78

B3
50
28
B3
33
33
67
50
10
50

90
80
80
100
80
50
80
72
56
46

17
10
17
70
43
0
SO
0

50
52
80
80
50
60
78
76

64
64
55
82
45
45
36
82
70
73

73
91
82
100
45
91
82
55
91
82

64
64
64
100
73
45
70
64
91
82

B2
82
82
91
64
73
91
82
91
82

70
50
30
80
40
50
40
50
60
50

46
28
92
36
46
54
72
64
72
46

75
100
88
90
82
B4
77
61
66
61
60
44
90
89
62
88
89
88
90
89

55
36
82
64
55
82
45
73
73
73

82
55
82
100
55
82
91
91
82
91

55
55
82
82
64
73
82
82
64
55

73
55
82
82
91
100
100
82
100
73

50
30
40
40
50
30
50
60
50
40

72
54
46
92
54
46
92
46
46
64
46
36
64
82
36
54
54
54

99
88
89
91
79
81
100
77
57
99
56
42
76
99
43
98
75
75

91
73
45
73
82
73
82
36
27
64
64
18
73
73
18
64
82
64

91
91
82
91
82
100
100
82
55
100
82
36
82
91
54
82
91
82

64
91
73
91
64
54
100
B2
45
45

91
91
B2
100
100
82
100
91
55
82
64
55
65
91
36
100
81
74

60
50
55
70
60
70
90
40
20
90
40
40
30
80
40
40
50
50

36
18
64
91
36
91
81
64

89
90
60
97
80
97
87
100
47
100
42
62
97
84
67
92
70
87

PostTest

307

Post—
Test

FRACTIONS

APPENDIX H
SCORES O F S T U D E N T S IN TH E E X P E R I M E N T A L GROUP
ON THE 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
A N D D U T T O N A R I T H M E T I C A T T I T U D E INVENTORY

APPENDIX H
SCORES OF STUDENTS IN THE EXPERIMENTAL GROUP ON THE TEST OF MATHEMATICAL
UNDERSTANDINGS AND DUTTON ARITHMETIC ATTITUDE INVENTORY

STUDENT
I .D.

TEST OF MATHEMATICAL
UNDERSTANDINGS

DUTTON ATTITUDE
SCALE

Pre-Test

Post-Test

Pre-Test

Post-Test

35
40
35
49
33
35
39
34
35
42

41
46
43
50
36
46
45
41
38
45

56
63
54
78
24
19

57
75
63

33
33
45
34
38
35
33
40
48
38

38
35
47
43
46
47
41
47
47
47

57
56

23
24
25
26
27
28
29
30

41
41
37
42
37
37
48
38
33
39

48
47
49
51
43
45
49
44
42
48

74
74
54
71
74
57

31
32
33
34
35
36
37
38

41
34
34
46
32
39
44
32

45
39
40
48
33
45
47
41

26
54
26
74
63
58
63
49

1
2
3
4
5

6
7

8
9

10
11
12
13
14
15
16
17
18
19

20
21
22

21
29
48
54

68
48
69
54

68
90
54
65
81

68

21
54

309

86
42
71
78
52
79
74
74
26
82
56
74
79
73
91
80
78
90
84
74

86
67
79
84
78
27
79

FEELING TOWARD
A RITHMETIC
Pre-Test

Post-Test

8

8

9

9
9

6
10
6
5
5

11
8
6
8

10

9

7
7

6
8

6
8

7
4

9

10
6
8

6
9

6

7

9

8
11
8
10

11
10
7

11
9
9
9
7
3

11
11
8
10
8
9

9

11
11

1

4

5

8

57
54
31
81

8
6

4

68

6

71
80
71

9

6
2
10
6
8

8
6

9
7

11

3
9

APPENDIX I
T H E E X P E R I M E N T A L G R O U P HIGH SCHOOL B A C KGROUND
F A C T O R S A N D F I N A L G R A D E ON THE C O M B I N E D
C O N T E N T - M E T H O D S COURSE

APPENDIX I

THE EXPERIMENTAL GROUP HIGH SCHOOL BACKGROUND FACTORS AND
FINAL GRADE ON THE COMBINED CONTENT-METHODS COURSE

Student
I.D.

1
2
3
4
5

6

Number of
Mathematics Courses
Taken in High School
4
3
4
5

2

10

3
3
3
3
3

11
12

2

7

8
9

13
14
15
16
17
18
19

20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38

3
5
3
3

High School
Grade Point
Average

Final Grade on
the Combined
Content-Methods
Course

3.00
3.78
3. 50
3.78
3.44
3.11
3.00
2. 65
2. 84
2.80

3.0
4.0
4.0
4.0
2.5
4.0
2.5
3.5
3.0
3.0

3.57
3.29
3. 50
2.87

3.0
3.0
4.0
4.0
3.0
4.0
4.0
4.0
3.5
4.0

2.68

2

3.36
3. 28
3.71
2.90
3.72

4
4
4
3

6

4

3.67
3.52
3.71
4.00
3. 31
3.33
3.81
3.35

1

2.68

4

3.82

4.0
3.5
4.0
4.0
3.0
3.5
4.0
4.0
1.5
4.0

3

3.00
2. 73
2.80
3.68
3.OB
3. 55
3.33
3.29

3.0
3.0
2.5
4.0
3.0
4.0
4.0
4.0

4
4

6
3
4

6

2
3
5

2
3
4
3

311

APPENDIX J

C O R R E L A T I O N MATR I X

APPENDIX J

CORRELATION MATRIX

•The symbolic notations on the correlation matrix indicate the following:
X^ = Pre-Test on Measurement
Y^ = Post-Test on Measurement
X^ ° Pre-Test on Numeration Systems
Y 2 = Post-Test on Numeration Systems
X^ = Pre-Test on Sets and Set Relations
Y^ - Post-Test on Sets and Set Relations
X. = Pre-Test on Whole Numbers

4

Y

4

= Post-Test on Whole Numbers

X^ = Pre-Test on Fractions
<= Post-Test on Fractions
X- = Pre-Test on Decimals

t>

Y

6

•» Post-Test on Decimals

X ? = Pre-Test on Relations and Functions
Y^ = Post-Test on Relations and Functions
X
Y

a
O

= Pre-Test on Probability and Statistics
= Post-Test on Probability and Statistics

Xg = Pre-Test on Mathematical Systems
Y

9

= Post-Test on Mathematical Systems

BIG X = Test of Mathematical U n d e r s t a n d i n g — Pre-Test
BIG Y = Test of Mathematical Understanding--Post-Test
D V T X = Du t t o n Arithmetic Attitude S c a l e — Pre-Test

D U T Y = Dutton Arithmetic Attitude S c a l e — Post-Test
FEEL X =■ General Feeling Toward M a t h e m a t i c s — Pre-Test
FEEL Y = General Feeling Toward M a t h e m a t i c s — Post-Test
PROCESS «= Attitudes Toward Mathematics as a Process
DIFFIC = Att i t u d e s Toward Difficulties of Learning Mathematics
PLACE = Attitudes Toward Place of Mathematics in Society

313

314

SCH & LEAR = Attitudes Toward School and School Learning
ENVIRCN = Attitudes Toward Man and His Environment
METHODS = Attitudes Toward Different Methods of Teaching Mathematics
HS GPA = High School Grade Point Average
READ = MSU Reading Test
ARITH ** MSU Arithmetic Tost
M ATH = MSU Mathematics Test
COURSES = Number of Mathematics Courses Taken in High School
FIN GR * Final Grade Received in the Combined Content-Methods Course

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APPENDIX K
N U M B E R OF S TUDENTS

IN THE E X P E R I M E N T A L G R O U P

WHO A N S W E R E D T H E T E S T ITEMS C O R R E C T L Y ON
THE N I N E PRE- A N D P O S T - C R I T E R I O N
R E F E R E N C E D M E A SURES

NUMBER OF STUDENTS IN THE EXPERIMENT AX GROUP (N - 38) MHO ANSWERED THE TEST ITEMS CORRECTLY ON THE NINE PRE- AND POST­
CRITERION REFERENCED MEASURES

KEASURB4ENT
PostTest

I
2

33
29

3

28

32
26
29

4

30

S
6
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33
21
11

29
13

8
9
10

17
26
15

22
17

32
32

20

SETS AND SET
RELATIONS

WHOLE NUMBER
SYSTEM

FRACTIONS

Itea

PreTest

PostTest

Itea

PreTest

PostTest

Itea

PreTest

PostTest

Itea

Is
lb

34
33

23
23

Is
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31
32

36
35

1-1
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26
24

35
34

1
2

23
27

lc

25

31

1-3

21

33

Id

14

1-4

le
2

6
20

13
26
22

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31
32
27

3
4

15
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31
34
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17
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31

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23
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30
37

37

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12

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24

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33

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11

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31
32
34

14
17
17

5
6
7

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PreTest

24

1-8
1-9
1-10
2

21

25
15

DECIMALS

10
11

Itea

23

322

PreTest

Itea

NUMERATIONS

APPENDIX K— Continued

RELATION AND
FUNCTION

Item

PreTest

PostTest

PROBABILITY
AND STATISTICS

I ten

PreTest

PostTest

MATHEMATICAL
SYSTEMS

Item

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PostTest

8

29

1

26

29

1

12

24

12

20

2

24

26

2

14

33

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31

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10

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25

2

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27

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19

23

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24

3

14

26

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12

24

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23

15

4

33

36

4

8

25

Sd

16

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5

3

31

5

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6

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27

9a

20

35

10

26

35

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24

32

34

36

10

323

la
Ifc

APPENDIX L
R A W SCORES O F S T U DENTS

IN THE E X P E R I M E N T A L

G ROUP O F E N TRY C H A R A C T E R I S T I C S

325

RAH S COR ES OF STUDE NT S IN T HE EX PE R IM EN TA L G R O U P ON ENTRY CH AR AC T E R I S T I C S

STUDENT
I ,D .

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1

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ATTITU DE S TO W A R D M A T H E M A T I C S A S A PROCESS.

*ATT.

1

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2 - ATTITUDES T O W A R D D IF FI CU LT I ES OF LEARNING MATH EM AT IC S

CATT.

3 - A TT ITUDES T O W A R D PLACE OF M AT HEMATICS

dATT.

4 - ATTITUDES TO W A R D SCHOOL A N D SCHOOL LEARNING.

MB

®ARITH.

- MSU A R I T H M E T I C S TEST.

fCOURS.

-

IN SOCIETY.

NUMBER OF M A T H E M A T I C S C O U R S E S TAKEN IN HI G H SCHOOL.

APPENDIX M
T E S T S C O R E S OF S T U D E N T S

IN 3 2 5E ON 50 P E R C E N T

ITEM S A M P L E OF C R I T E R I O N - R E F E R E N C E D TESTS

TEST SCORES OF STUDENTS IN 325E ON 50 PERCENT ITEM SAMPLE OF CRITERION-REFERENCED TESTS

MEASUREMENT

1
2

PreTeat
4
3

3

PoetTest
2

4

PreTeat

S

20

3

5

5

5

1

1

1

4

S

3

3

4

4

3

2

1

1

2

21

3

5

3

31

S

5

4
3

6

1

5

4

5

23

4

4

S

S

24

4

4

3

PoatTeat

4

37

1

2

3

4

2

1

1

3B

1

1

1

0

1

0

3

39

4

3

4

4

3

3

PreTeat

FoatTeat

PreTeat

PoetTaat

5

40

5

5

4

5

4

2

4

41

4

2

4

S

3

2

25

S

3

4

S

3

4

42

1

1

0

1

4

26

S

5

4

S

5

5

43

5

5

5

4

5

5

2

5

S

4

4

27

1

1

1

1

1

44

5

3

4

3

4

4

4

3

3

2

2

IB

1

1

2

3

29

11

3

4

2

3

3

4

30

3

12

5

4

4

4

4

31

1

13

2

2

1

1

1

32

5

1
2

4

3
1

1

1

0

4

2

3

2

2

45

1

1

0

1

4

2

2

4

S

46

3

4

3

3

3

4

4

47

3

2

2

2

2

1

5

5

5

5

4

1

1

2

1

0

2

1

0

3
1

I

2

1

1

4a

5

2

4

3

2

2

49

1

0
3

3

5

33

3

5

S

3

SO

1

3

2

4

3

34

2

3

3

4

3

3

51

5

4

5

3

5

4

1

2

35

2

2

2

2

2

2

52

1

2

1

2

1

1

S

4

3
1

17

2

1

3

1

2

2

ie

1

2

2

1

0

1

19

5

5

S

4

S

5

36

4

3

5

4

S3

2

1

2

1

2

54

0

0

0

1

0

55

4

5

4

5

3

5

56

0

0

0

1

0

0

327

0

S

2

2

PreTaat

1

1

16

STD.
I.D.

3

1

3

4

PoatTeat

1

4

5

PreTeat

MATHEMATICAL
SYSTBtS

4

9

14

PoatT«*e

RELATIONS &
m e n cms

3

10

IS

PreTeat

PROBABILITY
t STATISTICS

DECIMALS

STD.
t.D.

4

4

trac t i o n s

PostTeat

3

4

5

2

PreTeet

3

5

e

2

Po*tT»lt

HH01X NUMBERS

2

4

7

PreTeat

SETS t SET
RELATIONS

1
« *
O «l
a h

STO.
I.D.

NUMERATION

APPENDIX N
SCORES OF THE "C O M P A R I S O N GROUPS"
O N ENTRY D A T A

SCORES OF THE FRESHMAN ELDtENTARY EDUCATION MAJORS (COMPARISON GROUP) ON ENTRY DATA
ATTITUDES TOWARD
CT LEARNING
MATHEMATICS

ATTITUDES TCMUID
place or
MATHEMATICS
IN SOCIETY

4
9
4
6
5
6
2
9
7
9

3
6
10
8
4
10
5
14
4
2

5
10
12
13
4
9
7
9
4
7

12
5
14
14
12
16
5
16
22
13

7.4
2.4
6.6
2.5
7.9
5.4
6.0
3.8
6.1
5.4

4
8
5
4
6
10
6
2
4
6

14
6
8
6
14
6
11
4
11
14

7
2
12
2
8
11
7
10
12
11

19
14
14
14
10
15
10
13
14
16

3
22
26
6
14
3
9
13
17
10

7.0
6.3
2.6
4.1
1.9
4.5
4.6
5.1
6.6
4.9

4
6
6
6
6
4
7
5
4
4

12
13
10
4
11
14
6
7
10
7

11
7
10
4
7
13
9
3
2
10

14
14
16
14
16
15
16
13
11
17

22
28
15
6
12
9

6.5
6.0
7.8
6.B
4.5
NT

3
10
8
10
6
NT

10
14
9
7
14
NT

10
10
8
13
12
NT

6
20
18
20
16
NT

m i d p l a c d o m t t e st

IN ARITfKEnC

1
2
3
4
5
6
7
e
9
10

23
37
26
34
21
36
17
27
24
36

10
16
17
15
3
21
7
13
9
13

6.9
7.3
7.4
6.4
5.2
7.4
2.3
6.0
7.4
8.6

11
12
13
14
15
16
17
IS
19
20

31
31
29
34
31
36
30
32
31
27

11
6
21
15
16
18
5
17
17
6

21
22
23
24
25
26
27
26
29
30

20
34
39
31
39
23
37
32
35
29

31
32
33
34
35
36

35
39
40
23
26
31

DUTTON
AKTTWETIC
ATTITUDE SCALE

ATTITUDES TOWARD
MATHEMATICS
AS A PROCESS

difficulties

ATTITUDES TOWAD
SCHOOL AMD
SCHOOL LEARNING

3 29

STU.
I.D.

HSU BASIC SKILLS
AND PLACEMENT TEST
IN MATHEMATICS

MSB BASIC SKILLS

SCORES OF THE FRESHMAN MATHEMATICS-SECONDARY EDUCATION MAJORS (COMPARISON GROUP) ON ENTRY DATA

MOT BASIC S K IL L S
AMD PLACEMENT TEST
a ar m m t m c

MSU BASIC S K IL L S
AMD PLACEMENT TEST
in m a th e m a tic s

W TTOH
ARITHMETIC
a t t i t u d e s c a le

ATTI TUDES TCMATO
MATHEMATICS
a s a p ro c e s s

ATTITUDES TC4BLRD
D im c U L T T E S
OP LEARNING
m a tk h m a tic s

ATTITUDES TOMATO
PLACE CT
MATHEMATICS
i n s o c ie t y

38
39
35
36
36
39
36
36
40
39

30
28
26
22
IB
27
22
23
IB
26

6 .B
7 .7
s .e
8 .4
9 .0
7 .1
5 .7
9 .0
7 .3
7 .3

7
IS
e
3
16
12
10
11
e
6

14
5
12
4
14
12
12
9
4
6

7
11
14
6
10
6
12
12
4
6

14
15
15
14

39
40
37
31
35
32
37
39
37
39

2B
30
15
23
12
21
24
24
21
26

B .l
7 .7
9 .3
7 .2
7 .2
8 .8
B. 8
S .7

14
14
8
12
3
13
10
0
4
12

B
16
10
10
8
8
12
10
9
9

16
20
14
16
16

7 .4
7 .8

16
12
14
7
4
10
9
10
a
12

38
30
39
32
40
36
31
36
40
38

16
22
24
20
26
IB
20
29
2B
28

6 .9
8 .5
0 .8
S .6
7 .7
8 .3
7 .6
7 .9
7 .1
S .3

S
11
9
3
3
10
5
0
8
4

B
12
B
5
10
12
14
10
5
8

15
11
12
10
6
14
14
B
12
10

33
36
31
36
36
38
34
37
39
30

23
26
21
16
23
22
17
IS
27
24

8 .2
8 .7
S .2
6 .9
6 .8
7 .9
7 .3
8 .3
8 .3
8 .3

12
6
8
4
9
12
5
B
12
11

6
6
9
7
13
5
11
4
12
12

7
6
9
3
10
12
13
6
9
10

37
36
35

30
27
25

B .8
6 .3
8 .6

9
6
4

12
12
10

6
10
6

IS

16
IS
IS

12
12

ID

IS
16

10
14

12

14
16
10

13
13
14
16

15
14
14

16
13
13
11

14
16
14

15

IB
15
17

12

SCORES OF THE FRESBHAN KATHQU71C5 KAJORS (CCKPARISCN CROUP) OH ENTRY DATA

JU .

.D.
1

:
3
4
5
€

7

fl

9
ID
11

12
13
14

21

22
23

24
25

26
27
26
39
30
31
32
33

34
35
36

37
36
39

40

ATTTTCCES TCVUtD
p tA c i c r
nm H A ncs

HSU BASIC SKILLS
U C FIACEMDTT TEST
i k P o ra E m n c s

WTTOH
w tm m n c
ATTITUDE SCALE

ATTTTOES TCMUO
HRTUMATICS
AS A PROCESS

39
39
36
37
33
'3
36
39
39
37

26
20
26
26
26
24
30
29
21
27

6 ,a
7 .6
0.1
ft.7
2 .6
7.B
7.4
2 .6
9 .0
4 .a

3
10
13
11
9
a
7
10
12
16

14
13
9
14
5
10
11
14
a
6

a
16
10
15
6
10
10
7
7
7

10
20
15
13
11
15
17
14
16
9

36
39
36
36
36
39
35
35
40
30

26
25
17
21
21
30
27
27
29
23

7 .4
a ,a
ft.f
t
6 .0
9.1
9.0
5.6
6 .6
9 .0
9 .a

4
16
9
9
11

10
u
13
a
6
6
10
12
a
12

9

16
12
16

40
39
36
36
36
36
36
37
24
34

30
2€
37
26
26
2ft
17
26
2
20

ft.a
7.1
ft.2
6 .0
6 .5
7. a
6.3
7 .4
9.3
a .a

16
12
12
7

37
39
36
36
37
40
39
39
37
34

30
IB
2B
23
25
29
29
23
23
27

7 .9
7.0
0.0
B.6
7 .4
7 .7
4 .5
9 .0

14

irr

rr

jrr

16
16
3
2
11
a
16
irr

wr

WT

TT

WT

9
4
14
13
10

8
10
3
12
5

7
S
5
3
7
10
15
12

h a t h m a t ic s

9

a
12
5
6
6
5
7
14
4
13
a
12
3
4
14
14

ir

so c ie ty

9
12
6
14
7
a
a
a
12

ATTITUDES TC W U
SCHOOL M©
SCHOOL LEAWIRG

fl
IB
10
16
14
10
10

9
14
12
6
6
9
10
a
9
7

20

a

16
16
14
13
14
16
14
16
Wt
WT

8
19
ia
3
12
16
16
ia
12

33 1

15
It
17
le
19
20

APT i n , IBS TTRBU©
DIFFICULT! IS
OF IXMRIHG

MED BASIC SKILLS
M D FlAOMC«T TEST
» X A m P ttT IC

APPENDIX O
PRE- A N D P O S T - T E S T SCORES OF STUDENTS
M E T H O D S CO U R S E

IN REGULAR

(325E) ON D U T T O N A T T I T U D E SCALE

A N D TES T OF M A T H E M A T I C A L UNDE RS TA ND IN GS

APPENDIX O
PRE- AND POST-TEST SCORES OF STUDENTS IN REGULAR METHODS COURSE (325E)
ON DUTTON ATTITUDE SCALE AND T EST OF MATHEMATICAL UNDERSTANDINGS
TEST OF MATHEMATICAL
UNDERSTANDINGS

DUTTON ARITHMETIC
ATTITUDE INVENTORY

STUDENT
I.D.

Pre-TeBt

Post-Test

Pre-Test

1

7.3

7.3

46

44

2

1.9

1.9

30

29

3

1.4

7.8

40

48

4

5.1

7.4

43

42

5

5.5

5.2

38

40

6

5.7

6.9

35

37

7

7.9

8.8

45

44

8

7.4

5.2

46

45

9

4.9

5.1

39

35

10

7.1

6.3

45

44

11

B. 2

7.9

36

41

12

7.4

8.4

48

47

13

2.4

2.4

37

39

14

6.4

8.2

44

40

IS

7.8

8.0

48

43

16

8.6

7.4

42

39

17

2.8

5.4

31

38

18

5.1

5.1

37

38

19

7.9

6.7

46

45

20

2.7

2.5

36

39

21

8.6

7.4

47

50

333

Post-Test

APPENDIX P
O N E - W A Y A N A L Y S I S OF V A R I A N C E REL AT IV E T O
TESTING DIFFERENCES

BE TWEEN THE

E X P E R I M E N T A L G R O U P A N D THE
" C O M P AR IS ON G R O U P S "

APPENDIX P

ONE-WAY ANALYSIS OF VARIANCE RELATIVE TO TESTING DIFFERENCES
BETWEEN THE EXPERIMENTAL GROUP AND THE "COMPARISON GROUPS"

Table 19
Summary of Analysis of Variance

Source of
Variance
Treatment
Error
Total

for MSU Arithmetic Test

Sum of
Squares

Mean
Square

F-Value

3

1075.65

358.55

21.93

152

2484.52

16. 35

155

3560.17

D.F.

F , s {3, 152) = 2.66

Table 20
Summary of Analysis of Variance for MSU Mathematics Test

Source of
Variance
Treatment
Error
Total

Sum of
Squares

Mean
Square

3

3946.47

1315.49

152

4578.37

30.12

155

8524.85

D.F.

F-Value
43.67

^9 5 *3 ' 152) = 2.66

335

336

Table 21
Summary of Analysis of Variance on the Scale of Attitudes
Toward Mathematics as a Process

Source of
Variance
Treatment
Error
Total

Sum of
Squares

Mean
Square

3

264.16

88.05

149

1339.72

0.99

152

1603.80

D.F.

F-Value
9.79

F s 5 (3, 149)

= 2.66

Table 22
Summary of Analysis of Variance on the Scale of Attitudes
Toward Difficulties of Learning Mathematics

Source of
Variance
Treatment
Error
Total

Sum of
Squares

Mean
Square

3

12.62

4 .20

149

1676.06

11.25

152

1680.68

D.F.

F 9 5 <3, 149)
*

F-Value
0.37

- 2.66

337

Table 23
Sumnary of Analysis of Variance on the Scale of Attitudes
Toward School and School Learning

Source of
Variance
Treatment
Error
Total

Sum of
Squares

Mean
Square

3

4.16

1.39

149

1540.69

10. 39

152

1552.84

D.F.

F

•* 5

F-Vaiue
0.13

(3, 149) ** 2.66

Table 24
Summary of Analysis of Variance on the Scale of Attitudes
Toward Place of Mathematics in Society

Source of
Variance

D.F.

Sum of
Squares

Mean
Square

Treatment

3

43.05

14.62

149

1484.83

9.97

152

1526.60

Error
Total

F a (3, 149)

♦95

F-Value
1 .47

« 2.66

338

Table 2 5
Summary of Analysis of Variance on the
Dutton Arithmetic Attitude Scale

Source of
Variance

D.F.

Treatment

3

162.32

54.11

150

406.34

2.72

Error
Total

153

Sum of
Squares

Mean
Square

F-Value
19.69

570.66
F i # 5 <3, 150)

** 2.66

APPENDIX Q
HO YT R E L I A B I L I T Y C O E F F I C I E N T FOR C R I T E R I O N REFERENCED ACHIEVEMENT MEASURES

APPENDIX Q

HOYT RELIABILITY COEFFICIENT FOR CRITERION-REFERENCED
ACHIEVEMENT MEASURES

Table 26
Summary of Analysis of Variance for Pre-Test in Measurement

Sum of
Squares

Mean
Square

F-Value

16

8.0842

0.4491

3.1942

Items

4

2.6737

0.6684

4.7539

Error

72

10.1263

0.1406

94

20.8842

Source of
Variance
Individuals

Total

D.F.

Hoyt Reliability Coefficient - 0.6869

Table 27
Summary of Analysis of Variance for Post-Test in Measurement

Source of
Variance

Sum of
Squares

Mean
Square

F-Value

18

7.9368

0.4409

2.7149

Items

4

5.9052

1.4763

9.0905

Error

72

11.6948

0.1624

94

25.5368

Individuals

Total

D.F.

Hoyt Reliability Coefficient “ 0.6317

340

341

Table 28
Summary of Analysis of Variance for Pre-Test in Numeration

Source of
Variance

Sum of
Squares

Mean
Square

F-Value

18

9.9579

0.5532

4.9526

Items

4

5.1579

1.2894

11.5434

Error

72

8.0420

0.1117

94

23.1578

Individuals

Total

D.F.

Hoyt Reliability Coefficient = 0.7981

Table 29
Summary of Analysis of Variance for Post-Test in Numeration

Source of
Variance

Sum of
Squares

Mean
Square

F-Value

18

8.7 368

0.4854

4.6097

Items

4

7 .2210

1.8053

17 .1443

Error

72

7.5790

0.1053

94

23.5368

Individuals

Total

D.F.

Hoyt Reliability Coefficient = 0.7831

342

Table 30
Summary of Analysi s of Variance for Pre-Test
l
of Sets and Set Relation

D.F

Sum of
Squares

Mean
Square

F-Value

18

9.7895

0.5439

5.0975

Itens

4

5.5158

1.3790

12.9241

Error

72

7.6842

0.1067

94

22.9895

Source of
Variance
Individuals

Total

Hoyt Reliability Coefficient = 0.8038

Table 31
S umm a r y of Analysi s of Variance for Post-Test
of Sets and Set Relation

Sum of
Squares

Mean
Square

F-Value

18

8.9053

0.4947

4 .8076

Items

4

5.7895

1.4474

14.0661

Error

J_2_

7.4105

0.1029

94

22.1053

Source of
Variance
Individuals

Total

D.F.

Hoyt Reliability Coefficient = 0.7920

343

Table 32
Summary of Analysis of Variance for Pre-Test in Whole Numbers

Sum of
Squares

Mean
Square

F-Value

16

6.7765

0.4235

3.2728

Items

4

3.7177

0.9294

7 .1824

Error

64

8.2823

0.1294

84

10.7765

Source of
Variance
Individuals

Total

D.F.

Hoyt Reliability Coefficient = 0.694 5

Table 33
Summary of Analysis of Variance for Post-Test in Whole Numbers

Source of
Variance

Sum of
Squares

Mean
Square

F-Value

16

6.6118

0.4132

3.5226

Items

4

5.2942

1.3236

11.2839

Error

64

7.5058

0.1173

84

19.4118

Individuals

Total

D.F.

Hoyt Reliability Coefficient * 0.6945

344

Table 34
Summary of Analysis of Variance for Pre-Test in Fractions

Sum of
Squares

Mean
Square

F-Value

16

7.9529

0.4971

3.5866

Items

4

3.1294

0.7824

5.6450

Error

64

8.8705

0.1386

64

19.9529

Source of
Variance
Individuals

Total

D.F.

Hoyt Reliability Coefficient = 0.7212

Table 3 5
Summary of Analysis of Variance for Post-Test in Fractions

Source of
Variance

Sum of
Squares

Mean
Square

F-Value

16

6.7059

0.4191

3.1654

Items

4

3.9294

0.9824

7.4199

Error

64

8.4706

0.1324

84

19.1059

Individuals

Total

D.F.

Hoyt Reliability Coefficient “ 0.7212

345

Table 36
Summary of Analysis of Variance for Pre-Test In Decimals

Sum of
Squares

Mean
Square

F-Value

16

7.2000

0.4500

4.3103

Items

4

6.5176

1.6294

15.6073

Error

64

6.6824

0.1044

84

20.4000

Eiurce of
Variance
Individuals

Total

D.F.

Hoyt Reliability Coefficient - 0.7572

Table 37
Summary of Analysis of Variance for Post-Test in Decimals

Source of
Variance

Sum of
Squares

Mean
Square

F-Value

16

6.8941

0.4309

4.0844

Items

4

6.0470

1.5118

14.3299

Error

64

6.7530

0.1055

84

19.6941

Individuals

Total

D.F.

Hoyt Reliability Coefficient - 0.7572

346

Table 30
Summary of Analysis of Variance for Pre-Test
in Relations and Functions

D.F.

Sum of
Squares

Mean
Square

F-Value

19

13.3600

0.7032

7.7021

Items

4

4.6600

1.1650

12.7601

Error

76

6.9400

0.0913

99

24.9600

Source of
Variance
Individuals

Total

Hoyt Reliability Coefficient *= 0.0702

Table 39
Summary of Analysis of Variance for Post-Test
in Relations and Functions

Source of
Variance

D.F.

Sum of
Squares

Mean
Square

F-Value

19

11.7900

0.6205

6.1550

Items

4

5.5400

1.3050

13.7401

Error

76

7.6600

0.1000

99

24.9900

Individuals

Total

Hoyt Reliability Coefficient “ 0.0376

347

Table 40
Sumnary of Analysis of Variance for Pre-Test
In Probability and Statistics

D.F.

Sum of
Squares

Mean
Square

F-Value

19

12.6000

0.6632

6.2982

Items

4

4.4000

1.1000

6.2982

Error

76

8.0000

0.1053

99

25.0000

Source of
Variance
Individuals

Total

Hoyt Reliability Coefficient = 0.8412

Table 41
Summary of Analysis of Variance for Post-Test
in Probability and Statistics

Source of
Variance

D.F

Sum of
Squares

Mean
Square

F-Value

19

10.5100

0.5532

5.1033

Items

4

6.1600

1.5400

14.2066

Error

76

8.2400

0.1084

99

24.9100

Individuals

Total

Hoyt Reliability Coefficient “ 0.8041

348

Table 42
Summary of Analysis of Variance for Pre-Test
in Mathematical Systems

D.F.

Sum of
Squares

Mean
Square

F-Value

19

11.5500

0.6079

6.5365

Items

4

5.5000

1.3750

14.7850

Error

76

7.0700

0.0930

99

24.7 500

Source of
Variance
Individuals

Total

Hoyt Reliability Coefficient = 0.8470

Table 43
Summary of Analysis of Variance for Post-Test
in Mathematical Systems

Source of
Variance

D.F.

Sum of
Squares

Mean
Square

F-Value

19

12.5100

0.6584

5.6956

Items

4

3.2100

0.8025

6.9420

Error

76

8.7900

0.1156

99

24.5100

Individuals

Total

Hoyt Reliability Coefficient = 0.8244