university of nigeria patrick... · work for the degree of doctor of philosophy in education...
TRANSCRIPT
University of Nigeria Research Publications
OGOAMAKA, Patrick M. C.
Aut
hor
PG-Ph.D-85-3364
Title
Projects In Junior Secondary School Mathematics: Implications For Teaching,
Learning And Continuous Assessment
Facu
lty
Education
Dep
artm
ent
Education
Dat
e
September, 1989
Sign
atur
e
CERTIFICATION
OGOMHKA, P a t l - i c k M a t t h i a s Chukwuaku, a P o s t g r a d u a t e - U___
s t u d e n t i n t h e Department of E d u c a t i o n and w i t h t h e
R e g i s t r a t i o n b!umber PG/Ph,D/85/3364 h a s s a t i s f a c t o r i l y
c o m p l e t e d t h e r e q u i r e m e n t s f o r course and r e s e a r c h
work f o r t h e d e g r e e o f Doctor of P h i l o s o p h y i n E d u c a t i o n
( R e s e a r c h and E v a l u a t i o n ) . The work embodied i n t h i s
t h e s i z / d i s s e r t a t i o n i s o r i g i n a l and h a s n o t been
s u b m i t t e d i n p a r t or f u l l f o r any o t h e r d ip loma or
d e g r e e o f t h i s or any o t h e r U n i v e r s i t y .
/
Pro f . Uga Onwuka Head o f - ~ e ~ a r + . l i e n t
p r o f . R.O. \ Ohuche
T I T L E PAGE
PROJECTS I N JUNIOR SECONDARY SCHOOL IWTHr; lUT1CS : IMPLICATLONS FOR TEACHING,
L E A H N I I ~ G HWU CONTINUOUS ASSESSMENT,
A T H E S I S PRLSENTED TO
THE DEPARTMENT O F EDUCATION, UNlVEHSlTY O F N I G E R I A , NSUKKA.
I N FULFILMENT O F THE REQUIREMLNTS FOR .THE DEGREE. O F DOCTOR OF PHILOSOFHY (Ph.lL) I N
EDUCATION (RESEARCH AND EVALUATION)
OGOMkKA , PATRICK MATTHIAS CHUKWUAKU
(REG. NO. PG/Ph.U/85/3364)
APPROVAL PAGE
T H I S THESLS HAS BEdN APPROVED, FOR THE Di;' PtifiTM'.;NT OF EDUCATION,
UNIVZRS1:Y O F N I G E R I A , NSUKKA
--- -.-a-
' POST-GRADUATE STUDIES ' COIYMITTEE/
INTERNAL EXAIvIINER
HL FACULTY
iii
DEDICATED TO MY PARENTS
Ogomaka Chukwuaku and
Mrso U junwa Agnes Chukwuaku,
ACKNOWLEDGEMENT
F o r t h e c o m p l e t i o n o f t h i s work , t h e r c - y e a r c h e r i s
d e e p l y i n d e b t e d t o P r o f . R.0 . Ohuche, h i s t h e s i s
s u p e r v i s o r who o f f e r e , him lmmense a c a d e m i c a s s i s t a n c e ,
a n d r e a d t h i s work a t a l l i t s v a r i o u s s t a g e s . H i s
a c h i e v e m e n t s a n d d i s p o s i t i o n are s o u r c e s of e n c o u r a g e m e n t ,
h o p e a n d i n s p i r a t i o n t o t h e r e s e a r c h e r ,
The r e s e a r c h e r a c k n o w l e d g e s w i t h immense g r a t i t u d e
t h e c o n s t r u c t i v e criticism a n d e n c o u r a g e m e n t g i v e n t o
h im by D r . (Mrs.) V.F. H a r b o r - P e t e r s e s p e c i a l l y d u r i n g
t h e p r o p o s a l s t a g e of t h i s work. S i m i l a r g r a t i t u d e .
g o e s to D r . T o Ama Nwachukwu, P r o f . A. A l i , P r o f . (~ r s .1
E. Maduewesi a n d D r . (Mrs.) EoHoCo Okeke fo r t h e i r w o r d s
o f a d v i c e a n d e n c o u r a g e m e n t and the resource materials
t h e y made avai lable to t h e r e s e a r c h e r . To P r o f . O.C.
Nwana who e x p o s e d t h e r e s e a r c h e r t o h i s w e a l t h of
knowledge , wisdom a n d e x p e r t i s e when h e t a u g h t t h e course
I I P r a c t i c u u m i n I n s t r u m e n t C o n s t r u c t i o n " , t h e r e s e a r c h e r
g ives t h a n k s . T h e d e v e l o p m e n t o f t h e i n s t r u m e n t s u s e d
i n t h i s s t u d y was f a c i l i t a t e d b y t h e e x p e r i e n c e s g a i n e d
i n t h e c o u r s e , A l s o t h e r e s e a r c h e r i s g r a t e f u l to:
P r o f . S.O. O l a i t a n , t h e Dean o f t h e F a c u l t y , P r o f . Uga
Onwuka, t h e h e a d o f d e p a r t m e n t o f e d u c a t i o n , a n d
Mr. K , I d i k a f o r t h e i r o p n n n s s and h e l p s ,
The researcher's g r a t i t u d e g o e s to ML, F, O k w a r d ,
M i s s E,U, Onwuarna, M r , A, Iwueze and M r , C, Agbo f o r
c a r e f u l l y f o l l o w i n g t 'le r e s e a r c h e r t s i n s t r u c t i o n s and
p a i n s t a k i n g l y l e a d i n g t h e i r s t u d e n t s t h r o u g h t h e
e x p e r i m e n t a l c o n d i t i o n s o f t h i s s t u d y i n t h e i r v a r i o u s
s c h o o l s .
F i n a l l y , t h e i n v a l u a b l e c o n t r i b u t i o n s o f t h e
r e s e a r c h e r ' s p a r e n t s , b r o t h e r s and sisters and M r .
B,N, Osuchukwu are acknowledged w i t h immense t h a n k s
of a p p r e c i a t i o n .
Depa r tmen t of E d u c a t i o n , U n i v e r s i t y of N i g e r i a , Nsukka,
A P R I L , 1989
TABLE OF CONTENTS
PAGE - TITLE PAGE o o o o o o ~ O O O O O O O O O O o O D o o O O O O O m O o O o O O O O D D O o i
APPROVAL PAGE , , o o o . . o . o o o . o o . o o . o o o a o o o o o o o o o o o o . m ii
DEDICATION , , o . . o . o r O ~ o O O O O O O O O O O O o O O o . o O O o D o o O o O m m iii
ACKNOWLEDGEMENT o . o o . O O O O O . O O O O O O O O O O O o O O O O O O O O O O O . i v TABLE OF CONTENTS - o e o o o o o o o o o o o o o o o o o o o P . o o o o o o o o o m vi
LIST OF TABLES o o o o o . o o o o o o o o o o o o o o o o o o o o o o o o o o o o ~ o X
LIST OF APPENDICES O , O O O O O O O O O O O 1 O O O O O O O O O e O O O O O O O O ~ i i
PROJECT ABSTRACT , . . ~ O O O O O . O O O O O o ~ O O O O O O O O O m O O O O m , , X ~ V
CHAPTER ONE: INTRODUCTION ,,.o.oo~oo,oo,o,o~oOoO., 1
1.1 Background ~ ~ m ~ o o o o o o m o o o o o o m o o o o o a m 1
1 - 2 S t a t e m e n t of Problem , , . o o . o . o . . o o . o , 9
1 - 3 Purpose of t h i s Study ., . . , , ., . . . . , ,, 1 0
1 - 4 Need f o r T h i s Study , o . o o , o o . , o . o , , . . 1 2
1-5 D e l i m i t a t i o n of t h e Study.. , . . . . . . . , , 1 6
1.6 Research Q u e s t i o n s and Hypotheses.,.. 1 9
1.6.1 Research Q u e s t i o n s . . . O o I O O O O O O o . O O O . 1 9
l 0 6 , 2 Research Hypotheses . O O O O O o o . O . . . m O ~ 21
CHAPTER TWO: LITERHTUAXL R E V I E W . . o o o o o . o o o o o a o o m ~ ~ 22
2 - 0 I n t r o d u c t i o n a . o a o o o o o o o o o o o o o o o O O O o m 22
2 - 1 Teaching and Learning , O O O , O a O O O . . . O , 22
2 , l . l Teaching and Learning Mathematics, , , , 29
2-1-2 Teaching and Learning Mathematics i n Nigerian,Seconda~y.Schools . m m o ~ o o 32
2,2 E v a l u a t i o n of S tuden t s1 Achievement i n Mathematics , o . . o o o o o . o . o . . o o . o o m o 35
2,3 Cont inuous Assessment i n N i g e r i a n Secondary Schoo l s , O O O . O O O O O O O O O O O O . e 4 2
v i i
TABLE OF CONTENTS (con,t;d21
CHAPTER TWO (Con td , ) PAGE -- 2-3 Expec ted L e a ~ n i n g Outcomes of
Mathema t i c s E d u c a t i o n r r o o e o o o ~ o o e e o ~ r r 52
2.5 I n s t r u m e n t s f o r E v a l u a t i n g t h e L e a r n i n g Outcomes r s f Matnemat ics E d u c a t i o n ...,, 65
2.6 V a l i d a t i o ~ of I n s t r u m e n t s ~ ~ ~ o o o o . o e o ~ o 74
2,7 Summary o f I s s u e s Discussed,..,,..,.,.. 81
CHAPTER THREE: RESEARCH METHODOLOGY ~ 8 0 ~ ~ o o o o o o ~ o ~ 86
3 - 0 I n t r o d u c t i o n . ~ O ~ . O . O o . O O O O . o o O O O O O . O . m 86
3.1 R e s e a r c h Des ign , , , . ~ , o o . . o . . . o o . o o m O O . . 86
3.2 P o p u l a t i o n o f t h e S tudy ...m.,..m,,.,~, 88
3-3 Sample and Sampling Techniques . , , . , , . . . 89
3-4 I n s t r u m e n t s f o r Data C o l l e c t i o n ,.,.... 92
3.5 Development and V a l i d a t i o n o f I n s t r u m e n t s .o.,o.o.o....,..~..Oo.Oooom 96
3 - 6 P r o c e d u r e s f o r T r e a t m e n t s and Pos t - t e s t s . . l 02
3.7 Methods o f Data A n a l y s e s . . , , o o . o o , , . . , 1 0 6
CHAPTER FOUR: DATA ANALYSES , o . . o . o . . o , o . . o . . o o o , . 1 1 0
4.0 I n t r o d u c t i o n ,OOO...OOOOO...O....OOOeOO 1 1 0
4.1 A n a l y s e s and R e s u l t s R e l a t e d t o R e s e a r c h Q u e s t i o n One , , o o o o . . . . o . o . o m , 110
4.2 A n a l y s e s and R e s u l t s R e l a t e d to R e s e a r c h Q u e s t i o n Two , . . . , . o o , o , . . o o , o 1 1 3
4.3 A n a l y s e s and R e s u l t s With R e s p e c t t o R e s e a r c h Q u e s t i o n T h r e e 115
4.4 R e s u l t s of A n a l y s e s w i t h Respeck to R e s e a r c h Q u e s t i o n Four , , . . . . . . .. . . . . . . 1 1 6
4.5 R e s u l t s of A n a l y s e s w i t h R e s p e c t t o R e s e a r c h Q u e s t i o n F ive , O . O ~ O . . . . . ~ O . ~ . 1 1 7
4.6 R e s u l t s o f A n a l y s e s and T e s t s w i t h R e s p e c t t o H y p o t h e s i s One ..~,......... 1 2 1
viii
TABLE OF CONTENTS Q Contd, )
CHAPTER. FOUR ( C o n t d d PAGE - 4 0 7 R e s u l t s of A n a l y s e s and T e s t s
R e l a t e d t o H y p o t h e s i s Two , ,, . ,. ,. . . 1 2 8
4.8 ~ e s u l t s P€ Anal.yses and T e s t s R e l a t e d .=, Hypo theses T h r e e .. 1 3 1
CHAPTER FIVE: INTEHPRLTATION AND DISCUSSION - OF RESULTS O O O O O O . O O O O O . . . . . O O O O D D O O 1 3 6
5.1 I n t e r p r e t a t i o n of R e s u l t s o o o o o o . o o o , . 1 3 6
5 1 1 Re levance and S u i t a b i l i t y o f t h e P r o j e c t s t o and f o r Teach ing and L e a r n i n g JSS Mathem3t ics , o o o o o o o , o o o . 1 3 7
5,1.2 V a l i d i t y and U s a b i l i t y o f SAPIM and SMALOMP a s Asses smen t I n s t r u m e n t s , o o o o o o o o o , o . o o . o o . o o o o ~ , o . 1 3 9
5,1,3 A c c e p t a b i l i t y and S u i t a b i l i t y of t h e Marking Schemes f o r S c o r i n g Responses t o t h e Projects , . , . . , , , , . , , 1 4 1
5.1-4 A b i l i t y of T e a c h e r s i n Sco.r ing S t u d e n t s g C o g n i t i v e and P s ~ c h o m o t o r Responses to t h e Projects,, , , , , . . , . . .. 1 4 2
5.1.5 E f f e c t s of Use of P r o j e c t s on S t u d e n t s l A f f e c t i v e Behaviour Toward Mathematicso..o.oooODOOOOOOOOoo 1 4 3
5.1-6 E x t e n t o f C o r r e l a t i o n s Among S t u d e n t s 1 C o g n i t i v e , Psychomotor and A f f e c t i v e Pe r fo rmance S c o r e s as Moni tored by t h e P r o j e c t s o . o o e e o e , 14 6
5 - 1 - 7 L i n e a r R e g r e s s i o n C c e f f i c i e n t s Among S t u d e n t s ' C o g n i t i v e , Psychomotor and A f f e c t i v e Per formance S c o r e s i n Mathe~natics,,,.,~.,~~~~~~,. 14 7
5.1.8 R e a c t i o n s of t h e S c h o o l s t o t h e Use of t h e Ma thema t i c s Projects . , , , . 1 4 9
CHAPTER FIVE (Con td .
15,1.9 A l t e r n a t e Ranks t Grouping and Randomised Grouping as Rega rds t h e Use of ,iiVCOVP , O . O O O . O O O . ~ . . . s O . O O O O
5 - 2 E d u c a t i o n a l I m p l i c a t i o n s o f t h e F i n d i n g s of T h i s S t u d y . O . O O , O O O O O o ,
5.2.1 I m p l i c a t i o n s for L e a r n i n g Mathemat ics i n Secondary S c h o o l s O O Q o O O o O O O . O O O ~
5.2.2 I m p l i c a t i o n s for Teaching Mathemat i c s i n Seconda ry S c h o o l s o o ..
5.2-3 I m p l i c a t i o n s f o r C o n t i n u o u s Asses smen t i n Seconda ry S c h o o l s o o , ,
5-2-4 I m p l i c a t i o n s f o r T e a c h e r E d u c a t i o n and R e s e a r c h i n E d u c a t i o n O . O O O O O O O O O O O O O O O ~ O O O O O O O
5.3 L i m i t a t i o n s of T h i s S tudy o o o o o o o o ~
5 - 5 S u g g e s t i o n s For F u r t h e r S tud ie s . . , . 5-6 Summary and C o n c l u s i o n o o o o o o o o o ~ o o o
BIBLIOGRAPHY o . . o . e o O O O O Q O O O O O O O O O ~ ~ O O o O O O o O D o O O O
LIST OF TABLES -- TABLES - ?AGE
3.1 - T a b u l a r R e p r e s e n t a t i o n o f t h e Grouping of t h e Sample of T h i s Study, , , 9 1
4.3.1 - Marker R e l i a b i l i t y C o e f f i c i e n t s of t h e Ma thema t i c s T e a c h e r s r r . o e o o ~ o o o o o o 1 1 5
4.4.1 - Two d a y ANOVA S t a t i s t i c s Based On S c o r e s Awarded to t h e Randomly S e l e c t e d Responses by t h e Ten Marke r s . , ooo , . . o .o . 1 1 6
4 ,5 ,1 - Mean;, S t a n d a r d D e v i a t i o n s , and Number of Cases of V a r i a b l e s X, Y and Z,, , , . ., 118
4.5.2 - C o r r e l a t i o n C o e f f i c i e n t s Between P a i r s of t h e Variables X, Y and Z , , . , , , 118
4,5.3 - Simple R e g r e s s i o n C o e f f i c i e n t s Between Variables and T h e i r C o r r e s p o n d i n g C o n s t a n t s , . o , . o . . o . , , o . o o , , , . ~ o , O , o O m , 1 1 9
4,5,4 - Mu1 t i p l e L i n e a r R e g r e s s i o n C o e f f i c i e n t s Among V a r i a b l e s X, Y and 2,. 1 1 9
4.6.1 - Pre-test and Post-test Mean S c o r e s and S t a n d a r d D e v i a t i o n s on SAPIM a c c o r d i n g t o Subgroups and Groups, , . , . . . O . . .. . ... 1 2 2
4,642 - HNCOVA TABLE - A l l E x p e r i m e n t a l Groups V e r s u s ~ l l C o n t r o l group^.,,^^., 1 2 5
4.6-3 - ANCOVA TABLE f o r ~ l t e r n a t e Ranks Group ings , . . ~ o o ~ o o o o o o o o o o o ~ U ~ O O D O O ~ O m 1 2 6
4.6.4 - ANCOVA T a b l e fo r Randomised Groupings. , 1 2 7
4 ,7 ,1 , R e s u l t s of t-tests S t a t i s t i c s on Marker R e l i a b i l i t y C o e f f i c i e n t s , , , , . - - , 1 2 9
LIST OF TP dLES ( ~ o n t d , ! - ._--- _I----..-
TABLES PAGE --- 4.8.1 - R e s u l t s of t-tests and F - r a t i o s af
C o r r e l a t i o n C o e f f i c i e n t s and R e g r e s s i o n C o e f f i c i e n t s R e s p e c t i v e l y Between Variables ....,...~,.,,~,,,,~~ 133
4,8 ,2 - R e s u r t s of F - t e s t s on M u l t i p l e L i n e a r R e g r e s s i o n and C o r r e l a t i o n C o e f f i c i e n t s Among X, Y and 2.. . . . . . . . 134
APPENDICES
LIST O F --- APPENDICES
IA - An 1nst .rumeni: foi- C a r r y i n g O u t Some V a l i da'i.i.on of SWIM . . . . . . . . 1 8 7
IB - An I ~ L Lruinent fo r C a r r y i n g O u t Some h.,pcc t O L !ri?.Lllai.:ion at SMHLOMP . ~ o ~ . O O O E C O ~ O O O O O O O O O o O O O ~ 1 8 9
I1 - SAPIM for S t u d e n t s o o r o e s o o o o D v o o o 191
I11 - Some P r o j e c t s Based On JSS M a t h e m a t i c s C u r r i c u l u m ..o,,,o,o~a 1 9 3
IVA - SMALOMP F o r S t u d e n t s . . . o o o o o o o o o e 1 9 7
I V C1 - C o g n i t i v e A s p e c t s P r o j e c t 1 . , . . . , 1 9 8
I V P l - P s y c h o m o t o r A s p e c t s P r o j e c t 1 ,,,. 1 9 9
I V C2 - C o g n i t i v e A s p e c t s P r o j e c t 2 ,,.,,, 200
I V P2 - P s y c h o m o t o r Outcomes 2,. , , . , . , , , , , 2 0 3
I V C3 - C o g n i t i v e A s p e c t s P r o j e c t 3 ,,, . , , 204
I V P3 - P s y c h o m o t o r Outcomes 3 ,,,ooo,,.. 206
V - V a l i d a t i o n I n s t r u m e n t f o r t h e P r o j e c t s by E x p e r t s mo....oo.o.o.~ 208
V I - R e s u l t s of Some A s p e c t s of V a l i d a t i o n of P r o j e c t s by E x p e r t s a n d T e a c h e r s ~ o . o e o ~ o o ~ ~ ~ ~ o ~ ~ o o a o ~ 217
V I I - R e s u l t s of D e t e r m i n a t i o n of W e i g h t s for Items of SAPIM a n d SIWLOMP m e o o , o o o . . . o m o o e o o m o e e e e - o 220
V I I I - A L e t t e r to E x p e r t M a t h e m a t i c s E x a m i n e r s (wAsc/G,C .Ea ) t o C r i t i q u e t h e Mark ing Schemes, a n d C o n s e q u e n t R e s u l t s 222
LIST O F APPENDICES LContd,)
APPENDICES PAGE - I X - Whole Number A p p r o x i m a t i o n s
o f S t u d e n t s ' S c o r e s on SAFIM, ,, ,, . 226
- df tbo - T a b l e 2 . . , o , , , , . , , o . o , o o 227
X - P e r c e n t a g e S c o r e s of t h e S t u d e n t s on t h e P r o j e c t s i n terms of t h e T h r e e Domains, , , , , , , . 228
- d i t b o - T a b l e 2 . O . O O O O O 229
- d i t t o - T a b l e 3 . , , o , , , o , , o o , , 230
- d i t t o 0 T a b l e 4 , O e , . , , O O , O . O , 231
X I - S c o r e s Awarded to t h e C o g n i t i v e and Psychomotor Responses t o Each S e t of Randomly S e l e c t e d Ten o f S t u d e n t s ' S u b m i s s i o n s by Ten Randomly S e l e c t e d Mathema t i c s T e a c h e r s O , O O O O , O O O O O O . O O O . . O ~ O O O O 232
- d i t t o - T a b l e 2 ,,,,,,.,,,.,, 233
- d i t t o - T a b l e 3 .,,,,,,.,,,,, 234
X I 1 - A D r a f t o f SAPIM C r i t i q u e d By E x p e r t E d u c a t o r s . o , , , e o . a o o o , 235
X I 1 1 - U n i t s of Nork and Lesson T o p i c s Taugh t and t h e Approaches / S t r a t e g i e s Adopted i n Teach ing t h e S t u d e h t s I n v o l v e d i n t h e s t u d y , O ~ O ~ ~ O o ~ O o O O O o O ~ o O O e a ~ ~ ~ ~ ~ ~ 237
ABSTRACT
PROJECTS I N J U N I O R SECONDHRY SCHOOL MATHEiWTICS : IMPLICATIONS FOR TEACHIP'G ,
LEARNING AND COldTINUOUS ASSESSMENT,
P r o j e c t s a s s i g n e d to s ' - u d e n t s are n e c e s s a r y for
u t i l i t a r i a n Ma thema t i c s e d u c a t i o n e s p e c i a l l y w i t h t n e
i n t r o d u c t i o n of C o n t i n u o u s Asses smen t ( C A I and t h e
ernphasy on v o c a t i o n a l e d u c a t i o n . But s e c o n d a r y s c h o o l
t e a c h e r s a r e c o n f u s e d a b o u t w h a t M a t h e m a t i c a l p r o j e c t s
a r e , w h a t t h e p r o j e c t s c a n d o and how t h e y c a n be u s e d
e s p e c i a l l y a t t h e J u n i o r Seconda ry S c h o o l ( J S S ) l e v e l .
T h i s s t u d y t h e r e f o r e :
C o n s t r u c t e d some Ma t h e m a t i c a l p r o j e c t s fo r JSS
u s e and d e v e l o p e d t h r e e of t h e m i n t o e v a l u a t i o n
i n s t r u m e n t s ;
d e v e l o p e d ; a scale fo r A f f e c t i v e pe r fo rmance i n
Ma thema t i c s (SAPIM), and a s c a l e fo r M o n i t o r i n g
A f f e c t i v e L e a r n i n g Outcomes i n Ma thema t i ca l
p r o j e c t s (SMALOMP).
i n v e s t i g a t e d t e a c h e r s t marke r ( r a t e r ) r e l i a b i l i t y
and d i s c r i m i n a t o r y a b i l i t i e s i n s c o r i n g s t u d e n t s '
c o g n i t i v e and psychomotor p e r f o r m a n c e s i n t h e
Ma thema t i c s p r o j e c t s u sed as e v a l u a t i o n
i n s t r u m e n t s ;
examined t h e i n t e r - l i n e a r r e l a t i o n s h i p s among
s t u d e n t s t a f f e c t i v e , c o g n i t i v e and psychomotor
p e r f o r m a n c e s i n t h e projects used as e v a l u a t i o n
i n s t r u m e n t s and
( v ) n o t e d t h e s t u d e n t s : , t e a c h e r s ' a n d s c h o o l s '
r e a c t i o n s to u s i n g M a t h e m a t i c a l p r o j e c t s i n JSS
Mathema t i c s t e a c h i n g a d l e a r n i n g .
E x p e r t e d u c a t o r s and t e a c h e r s c r i t i q u e d and
v a l i d a t e d t h e d r a f t s of t h e i n s t r u m e n t s .
A l t e r n a t i v e l y sampled s t u d e n t s f rom e a c h s c h o o l
were p u t i n t o t w o g r o u p s ; random g r o u p s or 'a1 t e r n a t e
r a n k s 1 g r o u p s based on t h e i r p r e t e s t scores. I n a l l
240 s t u d e n t s ( 6 G y e a r I11 JSS s t u d e n t s p e r s c h o o l )
f rom f o u r s c h o o l s were i n v o l v e d , Each s c h o o l ' s t w o
g r o u p s w e r e randomly a s s i g n e d , o n e to t r e a t m e n t
( e x p o s u r e t o u s e o f p r o j e c t s ) and one t o c o n t r o l (non-
e x p o s u r e to t h e u s e of p r o j e c t s ) , SAPIM s e r v e d as
p r e t e s t and p o s t - t e s t on a l l t h e g r o u p s b u t SPL4LOMP
was a d m i n i s t e r e d on t h e e x p e b i m e n t a l ( t r e a t m e n t ) g r o u p s ,
a f t e r t h e g r o u p members e x e c u t e d e a c h of t h e t h r e e
p r o j e c t s . Ten randomly selected s t u d e n t s ' r e s p o n s e s t o
e a c h of t h e t h r e e p r o j e c t s were v a r i o u s l y s c o r e d by t e n
Mathema t i c s t e a c h e r s t e a c h i n g JSS s t u d e n t s a t t h e t i m e .
Data a n a l y s e s i n v o l v e d t h e bse of:
( i) c h i - s q u a r e s t a t i s t i c for a n a l y s i n g f r e q u e n c i e s of
t h e o p i n i o n o f t h e i n s t r u m e n t s ' v a l i d a t o r s ;
fii) ANCOVA to examine and test t h e s i g n i f i c a n c e or o t h e r w i s e of t e a c h s r s ' marke r ( r a t e r )
r e l i z b i l i t y and d i s c r i m i n a t o r y a b i l i t i e s ;
(iii) ANCOVA to examine t h e e f f e c t s of t h e u s e o f
Ma thema t i ca l p r o j e c t s on JSS s t u d e n t s 1 a f f e c t i v e
pe r fo rmance i n Mathemat ics ; and
( i v ) r e g r e s s i o n a l a n a l y s e s t o examine and tes t t h e
s i g n i f i c a n c e or o t h e r w i s e o f the i n t e r - l i n e a r
r e l a t i o n s h i p s among s t u d e n t s 1 a f f e c t i v e ,
c o g n i t i v e and psychomotor p e r f o r m a n c e s as
m o n i t o r e d by t h e t h r e e p r o j e c t s .
The r e s u l t s of t h e s t u d y are p o s i t i v e i n a number
o f ways, The r e l i a b i l i t y c o e f f i c i e n t s of SAPIM
c a l c u l a t e d from t h e s p l i t - h a l f and test-re-test t e c h n i q u e s
are 0,84 and 0.92 r e s p e c t i v e l y . Through t h e s ~ l i t - h a l f
a p p r o a c h t h e r e l i a b i l i t y c o e f f i c i e n t of SMLOMP i s
found to b e 0,796. The v a l i d a t o r s o f t h e t h r e e p r o j e c t s
a g r e e t o
( i) t h e s u i t a b i l i t y of t h e i n s t r u m e n t s i n terms o f
l a n g u a g e e x p r e s s i o n s and l e v e l o f t a s k s r e q u i r e d ;
and
( i i ) t h e r e l e v a n c e of t h e i n s t r u m e n t s i n terms of t h e i r
b e i n g , based on t h e J u n i o r Seconda ry Schoo l
Ma thema t i c s C u r r i c u l u m ( JsSMC and r e l a t e d t o
t r u e l i f e s i t u a t i o n s .
T e a c h e r s ' rater r e l i a b i l i t y and d i s c r i m i n a t o r y a b i l +l r
i n d i c e s a re b o t h s i g n i f i c a n t beyond t h e 0,01 and -
x v i i
he mean g a i n i n a f f e c t i v e pe r fo rmance i n Mathemat ics
o f s t u d e n t s exposed t o p r o j e c t s i s s i g n i f i c a n t l y
g r e a t e r (beyond t h e - l e v e l of 0.01) t h a n t h e mean J'
g a i n i n a f f e c t i v e pe r fo rmance i n Ma thema t i c s o f
s t u d e n t s n o t exposed t o t h e u s e of p r o j e c t s , A l l t h e
s i m p l e a n d m u l t i p l e c o r r e l a t i o n and l i n e a r r e g r e s s i o n
c o e f f i c i e n t s among s t u d e n t s ' a f f e c t i v e , c o g n i t i v e and
psychomotor p e r f o r m a n c e s as m o n i t o r e d by t h e t h r e e
p r o j e c t s a r e s i g n i f i c a n t ( p L 0,05) , S t u d e n t s r e sponded
f a v o u r a b l y to t h e u s e of p r o j e c t s . However t h e t e a c h e r s
who c o n d u c t e d t h e e x p e r i m e n t f i n d them t a s k i n g ,
T h e s e r e s u l t s have f a r r e a c h i n g i m p l i c a t i o n s f o r :
t e a c h i n g - l e a r n i n g Mathema t i c s a t the JSS l e v e l ;
C o n t i n u o u s Asses smen t i n s c h o o l s ; Ma thema t i c s t e a c h e r
p r e p a r a t i o n and t h e s c h o o l s ~ a u t h o r i t i e s , C e r t a i n l y
t h e r e s u l t s of t h i s work w a r r a n t s t h e c a l l t h a t
Ma thema t i c s t e a c h e r s ( a t t h e JSS l e v e l a t l e a s t )
s h o u l d be made a n d encouraged t o u s e p r o j e c t s ,
CHAPTER ONE
INTRODUCTION
1-0 T h i s c h a p t e r c o n t a i n s : t h e background t o t h i s
s t u d y ; t h e s t a t e m e n t o prok,lem; p u r p o s e of t h e s t u d y ;
and t h e need f o r t h e s t u d y , I n c l u d e d i n t h i s c h a p t e r ,
also are: t h e d e l i m i t a t i o n o f t h e s t u d y ; t h e r e s e a r c h
q u e s t i o n s , and t h e research hypo theses .
1.1 BACKGROUND
Most e d u c a t n r s a g r e e t h a t e d u c a t i o n as a p r o c e s s
i n v o l v e s p r e p a r i n g t h e l e a r n e r f o r h l s p r e s e n t and f u t u r e
roles as a p e r s o n and a s a member o f a g i v e n s o c i e t y .
I n modern societies, much o f t h e e d u c a t i o n i n d i v i d u a l s
r e c e i v e , t a k e p l a c e i n s c h o o l s , It i s s c h o o l s t h a t are
e x p e c t e d t o make i n d i v i d u a l s a c q u i r e most o f t h e
c a p a b i l i t i e s , basic and f u n d a m e n t a l s k i l l s ( a f f e c t i v e ,
c o g n i t i v e and psychomotor ) n e c e s s a r y fo r t h e roles
e x p e c t e d of them. Under ly ing t h e a b o v e e x p e c t a t i o n i s
t h e a s s u m p t i o n t h a t t h i n g s l e a r n t w i t h i n s c h o o l s e t t i n g s
are t r a n s f e r a b l e t o o t h e r s i t u a t i o n s i n s i d e or o u t s i d e
t h e s c h o o l , S c h o o l s , t h e r e f o r e are p r i m a r i l y e s t a b l i s h e d
to t e a c h f o r t r a n s f e r , O f c o u r s e , t h e human s p e c i e i s
c a p a b l e of t r a n s f e r r i n g much of t h e l e a r n i n g i t a c q u i r e s
i f s u i t a b l e c o n d i t i o n s or e n v i r o n m e n t s p r e v a i l .
I ndeed , p s y c h o l o g i s t s have l a b o u r e d to i d e n t i f y
a n d s p e c i f y s u c h c o n d i t i o n s or e n v i r o n m e n t s t h a t are
n e c e s s a r y f o r t r a n s f e r o f l e a r n i n g t o o c c u r , But t h e i r
s p e c i f i c a t i o n s seem o d i f f e r as shown by t h e i r v a r i o u s
s c h o o l s and t h e o r i e s , Accord ing to Bigge and Hunt
(1968) and Chauhan (19781, t h e s e t h e o r i e s o f t r a n s f e r
of l e a r n i n g / t r a i n i n g a r e embodied i n :
( i ) t h c t h e o r y o f m e n t a l d i s c i p l i n e ;
(ii) t h e t h e o r y o f i d e n t i c a l e l e m e n t s ;
(iii) t h e t h e o r y o f g e n e r a l i z a t i o ~ ;
( i v ) t h e t h e o r y o i t r a n s p o s i t i o n ; and
(u) t h e t h e o r y of l e a r n i n g to l e a r n .
Whi l e t h e t h e o r y of m e n t a l d i s c i p l i n e i s v e r y much
de-emphasized nowadays, t h e o t h c r f o u r s t i l l i n f l u e n c e
o u r e d u c a t i o n a l p r a c t i c e s t o v a r y i n g d e g r e e s . The
t h e o r i ~ s v a r y i n c o n c e p t i o n and s t i p u l a t i o n s b u t a n
eclectic p o s i t i o n , based on t h e i r s t i p u l a t i o n s , i s
p o s s i b l e .
Such a p o s i t i o n may recommend t h a t f o r t r a n s f e r o f
l e a r n i n g t o o c c u r :
( i ) t h e s i m i l a r i t y between t e a c h i n g / l e a r n i n g a n d t a s k
s i t u a t i o n s s h o u l d be maximal;
a d e q u a t e e x p e r i e n c e w i t h or e x p o s u r e t o t h e
e x p e c t e d r ea l l i f e s i t u a t i o n s h o u l d be p r o v i d e d ;
a m p l e e x a m p l e s or s i t u a t i o n s i l l u s t r a t i n g
d i f f e r e n t modes of t h e t a s k s t o b e u n d e r t a k e n
i n f u t u r e s h o l : l d be p r o v i d e d ;
t h e i m p o r t a n t f e a t u r e s a n d l i n k s o f a n d t a s k
b e i n g t r a i n e d f o r , s h o u l d be i d e n t i f i e d for
t h e l e a r n e r s a n d s t r e s s e d ; a n d
t h e u n d e r l y i n g a n d g e n e r a l p r i n c i p l e s i n v o l v e d
i n t a s k s b e i n g t r a i n e d fo r , s h o u l d b e t h o r o u g h l y
e x p l a i n e d .
B u t how f a r t h e t e a c h i n g a n d l e a r n i n g d o n e i n o u r
s c h o o l s , e n s u r e / p r o v i d e t h e s e f a v o u r a b l e e n v i r o n m e n t s
o r c o n d i t i o n s n e c e s s a r y f o r t r a n s f e r of l e a r n i n g i s
d o u b t i u l .
The a b o v e d o u b t stems f r o m s p e c u l a t i o n s t h a t many
o f t h e s k i l l s , k n o w l e d g e a n d frames of mind w h i c h
s t u d e n t s a c q u i r e from s e c o n d a r y s c h o o l s , d o n o t e n a b l e
them s o l v e some of t h e i r i n d i v i d u a l a n d societal
p r o b l e m s . T h e s e o b s e r v a t i o n s a re more so w i t h w h a t
a r e t a u g h t a n d l e a r n t i n M a t h e m a t i c s , s i n c e i t i s almost
a g l o b a l c o m p l a i n t ( b y s t u d e n t s a n d p a r e n t s ) t h a t t h e r e
i s a m i n i m a l c a r r y - o v e r of w h a t i s l e a r n t from s e c o n d a r y
s c h o o l Ma thema t i c s t o l a t e r l i f e s i t u a t i o n s , (Chauhan,
1 9 7 8 ) , T h i s d i s s a t i s f y i n g s i t u a t i o n i n t h e t e a c h i n g and
l e a r n i n g o f Ma thema t i c s i s c e r t a i n l y worse i n t h e deve lop -
i n g c o u n t r i e s ( i n c l u d i n g ~ i g e r i a ) t h a n i t i s i n t h e
advanced c o u n t r i e s o f t h e wor ld . I n N i g e r i a , some
r e s e a r c h e r s have o b s e r v e d t h a t t h e t e a c h i n g and l e a r n i n g
o f Mathemat ics f a l l f a r below e x p e c t a t i o n s . ( L a s s a ,
1978; Igboko, 1576; Ohuche and Obioma, 1983 ; Hassan,
1981; Osibodu, 1978; and A l i , 1988) . No wonder why t h e
a c h i e v e m e n t or ~ p e c i f i c a l l y t h e p e r f o r m a n c e s of o u r
s e c o n d a r y s c h o o l s t u d e n t s i n Mathemti t ics have been l o w
and t h e i r a t t i t u d e s t o Mathema t i c s h a v e been , on t h e
a v e r a g e , n e g a t i v e ,
How c a n t h e poor s t a t e o f a f f a i r s d e s c r i b e d above
and t h e a b s t r a c t n e s s i n t h e t e a c h i n g and l e a r n i n g o f
Ma t h e m a t i c s i n o u r s e c o n d a r y s c h o o l s bc overcome or
min imised? Could t h e u s e o f p r o j e c t s i n Ma thema t i c s
( t h a t a r e r e l e v a n t t o t h e s o l u t i o n s o f some rea l l i f e
p rob lems) i n t h e t e a c h i n g and l e a r n i n g of Mathemat ics i n
o u r s e c o n d a r y s c h o o l s b e o n e o f t h e ways? Indeed u s i n g
p r o j e c t s i n t h e t e a c h i n g and l e a r n i n g of M a t h e m a t i c s i n
o u r s e c o n d a r y s c h o o l i s v e r y l i k e l y t o lead fas te r t o t h e
a t t a i n m e n t o f g o a l s o f e f f e c t i v e Ma thema t i c s c d u c a t i o n ,
The above p o s t u l a t e i s t h e c a s e s i n c e by u s i n g t h e
p r o j e c t s i n t e a c h i n g and l e a r n i n g Mathemat ics ,
( e s p e c i a l l y i n t h e J u n i o r Seconda ry S c h o o l s ) , M a t h e m a t i c a l
c o n c e p t s , r u l e s and s k i l l s w i l l be p r e s e n t e d t o t h e
l e a r n e r t h r o u g h t h e ti.ree modes t h r o u g h wh ich i n d i v i d u a l s
l e a r n and i n t e r a c t w i t h t h e i r env i ronmen t ( ~ r u n e r , e t a1
1 9 6 6 ) . The t h r e e modes i n c l u d e : ( i ) t h e e n a c t i v e mode
which i m p l i e s p r e s e n t i n g l e a r n i n g e x p e r i e n c e s or g a i n i n g
e x p e r i e n c e s t h r o u g h b o d i l y a c t i o n , d o i n g by t h e movement
o f t h e body or t h e l imbs; (ii) t h e i k o n i c mode e n t a i l s
l e a r n i n g o r t e a c h i n g t h r o u g h o r by t h e u s e o f imagery ,
fo rming s e n s o r y images , mold ing , c a r v i n g a n d d rawing ;
(iii) t h e symbo l i c mode e n t a i l s g a i n i n g knowledge or
p r e s e n t i n g l e a r n i n g e x p e r i e n c e s t h r o u g h symbols s u c h a s
l anguage , m a t h e m a t i c a l e x p r e s s i o n s and e q u a t i o n s and
s i g n a l s . I n J u n i o r Seconda ry S c h o o l s t h e u s e o f p r o j e c t s
i n t e a c h i n g and l e a r n i n g Mathemat ics ( a s h a s been done
i n t h i s s t u d y ) f a l l s i n l i n e w i t h t h e P i a g e t i a n p o s i t i o n ,
T h i s i s s e q u e l t o t h e f a c t t h a t some JSS s t u d e n t s may
w e l l be o p e r a t i n g a t t h e c o n c r e t e o r semi-formal
o p e r a t i o n a l s t a g e . Even t h o s e JSS s t u d e n t s who have
a t t a i n e d t h e f o r m a l o p e r a t i o n a l s t a g e , c o u l d now and t h e n
be h e l p e d o u t i n d i f f i c u l t s i t u a t i o n s , s i n c e when f a c e d
w i t h n o v e l s i t u a t i o n s or d i f f i c u l t p rob lems i n d i v i d u a l s
may r e g r e s s t e m p o r a r i l y t o t h e c o n c r e t e o p e r a t i o n a l
s t a g e ( P i a g e t , l W O ) , T h e s e c o n s i d e r a t i o n s i n d i c a t e
t h a t t h e u s e o f p r o j e c t s i n t e a c h i n g and l e a r n i n g
Mathema t i c s i s l i k e l y t o enhance s t u d e n t s t a c h i e v e m e n t
i n Mathemat ics . T h i s i s t h e case s i n c e p r o j e c t s a r e
v e r y much l i k e l y t o accommodate t h e l e v e l s o f c o g n i t i v e
m a t u r i t y of t h e s t u d e n t s o r t h e v a r i o u s modes o f
r e p r e s e n t a t i o n t h r o u g h w h i c h t h e s t u d e n t s l e a r n . The
u s e o f p r o j e c t s may a l s o e n s u r e , t o a g r c a t e x t e n t , t h e
t r a n s f e r o f knowledge and s k i l l s a c q u i r e d i n Ma thema t i c s
t o t r u e l i f e s i t u a t i o n s , s i n c e p r o j e c t s , u n d e r t a k e n i n
s c h o o l Ma thema t i c s a r e p r o t o - t y p e s o r mode l s cf t r u e
l i f e t a s k s .
B e s i d e s b e i n g a p o s s i b l e a p p r o a c h toward improv ing
t h e t e a c h i n g and l e a r n i n g of Mathematics a t t h e s e c o n d a r y
school l e v e l , p r o j e c t s c o u l d be used as v e r s a t i l e o r
c o m p r e h e n s i v e i n s t r u m e n t s f o r e d u c a t i o n a l e v a l u a t i o n .
T h i s i s p r o b a b l y t h a r e a s o n why t h e F e d e r a l M i n i s t r y of
E d u c a t i o n ' s Handbook on C o n t i n u o u s Asses smen t (CA)
s t i p u l a t e s t h a t p r o j e c t s s h o u l d be p a r t of t h e i n s t r u -
m e n t s t o be u s e d for t h e e v a l u a t i o n of t h e a c h i e v e m e n t s
of o u r s e c o n d a r y school s t u d e n t s (Yo loye et a l , 1980) .
U r i l i k e t h e common paper and p e n / p e n c i l tests which
almost e x c l u s i v e l y a p p r a i s e t h e c o g n i k i t e l e a r n i n g
ou tcomes , p r o j e c t s c o u l d v a r i o u s l y o r c o l l e c t i v e l y
a p p r a i s e t h e a f f c c t i v e , c o g n i t i v e and psychomotor
l e a r n i n g outcomes of s t u d e n t s c s p c c i a l l y i n Mathemat ics ,
T h e r e f o r e t h e u s e o f m e a n i n g f u l and we l l - conce ived
p r o j e c t s i s i n d i s p e n s i b l e to t h e i m p l e m e n t a t i o n o f
C o n t i n u o u s Assessment (CAI. Indeed t h e u s e o f p r o j e c t s
a s t o o l s f o r e v a l u a t i n g s t u d e n t s f a c h i e v e m e n t s i s
n e c e s s a r y a t t h e JSS l e v e l . T h i s i s so, s i n c e t h e
p r o d u c t s o f t h i s l e v e l o f o u r e d u c a t i o n sys t em (as
s t i p u l a t e d by t h e N n t i o n a l P o l i c y on E d u c a t i o n 1981)
may g o i n t o o n e o f t h e f o l l o w i n g :
( i ) an a p p r e n t i c e s h i p sys t em cr o t h e r scheme f o r
out-of s c h o o l v o c a t i o n a l t r a i n i n g ;
( i i j a t h r e e - y e a r S e n i o r Seconda ry S c h o o l ( S S S )
s y s t e m t r a i n i n g s t u d e n t s who a re e i t h e r a r t s ,
s c i e n c e o r v n c a t i o n a l b u s i n e s s / t e c h n i c a l
i n c l i n e d ; and
(iii) a s e n i o r s e c o n d a r y t e a c h e r e d u c a t i o n programme.
Of c o u r s e t o b e t t e r a d v i c e and c h a n n e l any p r o d u c t o f
t h e JSS i n t o o n e o f t h e a b o v e tiers o f t h e post-JSS
e d u c a t i o n programmes, (where h e h a s a c o m p c r a t i v e
a d v a n t a g e o v e r o t h e r s ) , a v a l i d and reliable e s t i m a t e
o f h i s a b i l i t i e s and a t t i t u d e s s h o u l d be a v a i l a b l e ,
I t i s mos t p r o b a b l e t h a t p r o j e c t s b a s c d o n t h e J u n i o r
S e c o n d a r y S c h o o l Ma thema t i c s C u r r i c u l u m (JSSMC) w i l l
g o a l o n g way t o p r c b r i d i n g some d e p e n d a b l e estimates
of s t u d e n t s ' a b i l i t i e s a n d s k i l l s i n t h e a f f e c t i v e ,
c o g n i t i v e and psychomotor domains o f e d u c a t i o n o b j e c t i v e s ,
But upon t h e s e p r o m i s e s of q u a l i t y e d u c a t i o n which
t h e u s e of m e a n i n g f u l p r o j e c t s h o l d s , o u r s e c o n d a r y
s c h o o l t z a c h e r s are c o n f u s e d a b o u t w h a t p r o j e c t s are,
T h i s i s t h e case s i n c e H a r b o r - I b e a j a (1983) n o t e d t h a t
m a j o r i t y o f o u r s e c o n d a r y s c h o o l t e a c h e r s c o n c e i v e of
p r o j e c t s a s a s s e s s m e n t i n s t r u m e n t s t h a t c o u l d / s h o u l d be
a d m i n i s t e r e d o n s t u d e n t s o n c e a week, Even now t h a t t h e
e r r o n e o u s o n e week i n t e r v a l f o r t h e a d m i n i s t r a t i o n of
p r o j e c t s i s a p p a r e n t l y c o r r e c t e d , most of o u r s e c o n d a r y
s c h o o l t e a c h e r s d o n o t s t i l l know t h e a c t u a l n a t u r e and
form and a l so u s e s of p r o j e c t s ( e s p e c i a l l y i n ~ a t h e m a t i c s ) ,
From t h i s r e s e a r c h e r ' s many c o n t a c t s w i t h o u r s e c o n d a r y
s c h o o l t e a c h e r s , @hrough s ~ n d w i c h l e c t u r e scc t i o n s ,
t e a c h i n g p r a c t i c e s u p e r v i s i o n exercises, workshops
( n a t i o n a l , states, and l o c a l ) and z o n a l c o n t i n u o u s
a s s e s s m e n t committee m e e t i n g s and s u p e r v i s i o n of
c o ~ t i n u o u s a s s e s s m e n t e x e r c i s e s i n s c h o o l g wha t t e a c h e r s
p r e s e n t as p r o j e c t s f a l l s h o r t o f d h a t p r o j e c t s o u g h t t o
be. A l s o most o f t h e t e x h e r s score o n l y t h e c o g n i t i v e
a s p e c t s o f t h e s t u d e n t s ' r e s p o n s e s s u c h i n s t r u m e n t s ,
Even a t t h a t , i n most cases t h e t e a c h e r s d o n o t u s e
w e l l p r e p a r e d and w r i t t e n mark ing schemes f o r s c o r i n g
t h e s t u d e n t s ' r e s p o n s e s . T h e s e are n o t s u r p r i s i n g
s i n c e i t h a s bee? g e n e r a l l y o b s e r v e d t h a t o u r s e c o n d a r y
s c h o o l t e a c h e r s l a c k some o f t h e c o m p e t e n c i e s f o r scor-
i n g and g r a d i n g s t u d e n t s ' r e s p o n s e s t o e v a l u a t i o n
i n s t r u m e n t s (Yoloye e t a l , 1980 ; Ipa,ye, 1982 ; Harbor-
I b e a j a , 1983; and Ohuche, 1 9 8 8 ) .
1.2 STATEMENT OF PROBLEM
P . r o j e c t s i n JSS Mathema t i c s are v e r s a t i l e means o f
t e a c h i n g and l e a r n i n g Mathema t i c s and t h e i r u s e c a n :
( i ) improve t h e t e a c h i n g a n d l e a r n i n g of JSS
Mathemat ics ;
(ii) e n h a n c e t h e t r a n s f e r o f s k i l l s and t r a i n i n g i n
Ma thema t i c s t o s i t u a t i o n s o t h e r t h a n where t h e
s k i l l s / t r a i n i n g were a c q u i r e d ; and
(iii) p r o v i d e d e p e n d a b l e d a t a f o r C o n t i n u o u s Asscssment .
Could some p r o j e c t s a c c e p t a b l e t o e x p e r t s and t e a c h e r s
be d e v e l o p e d based on JSSMC? Could t p e u s e of such
p r o j e c t s f o r t e a c h i n g and e v a l u a t i n g s t u d e n t s , e n h a n c e
t h e s t u d e n t s ' l e a r n i n q outcomes i n Ma thema t i c s?
S i n c e t h e s c o r i i , g and g r a d i n g o f s t u d e n t s i
r e s p o n s e s t o e v a l u a t i o n i n s t r u m e n t s have been i d e n t i f i e d
a s problem a r e a s ; (Yoloye e t a l , 1980; I p a y e , 1952;
H a r b o r - I b e a j a , 1983 ; Ohuche 1 9 8 8 ) c o u l d a mark ing scheme
a c c e p t a b l e t o e x p e r t s , be drawn u p f o r e a c h of t h e
p r o j e c t s ? By u s i n g s u c h mark ing schemes, c o u l d t h e
r a t e r r e l i a b i l i t y and d i s c r i m i n a t o r y a b i l i t y of t e a c h e r s
i n s c o r i n g s t u d e n t s : r e s p o n s e s t o t h e p r o j e c t s be
s t a t i s t i c a l l y s i g n i f i c a n t ? Could e x p e r t s a g r e c t o t h e
c a t e g o r i z a t i o n of e x p e c t e d s t u d ~ n t s ' r e s p o n s e s i n t o
d i s t i r ~ c t i v e c o g n i t i v e and psychomotor l e a r n i n g outcomes?
T h e s e p o s t u l a t i o n s , u n c e r t a i n t i e s or q u e s t i o n s
c o n s t i t u t e t h e problem o f t h i s s tudy .
1.3 PUHPOSL OF THIS STUDY I -
T h i s s t u d y c o n c e r n e d i tself w i t h t h e deve lopmen t
and v a l i d a t i o n o f t h r e e p r o j e c t s based on JSSMC. The
d e v e l o p e d and v a l i d a t e d p r o j e c t s were u s e d , i n p a r t as
i n s t r u m e n t s fo r t e a c h i n g and l e a r n i n g o f some u n i t s o f
. work i n JSSMC, as w e l l as fo r e v a l u a t i n g s t u d e n t s '
a c h i e v e m e n t s a r i s i n g f r o m the t e a c h i n g and l e a r n i n g
a c t i v i t i e s , S p e c i f i c a l l y t h i s work i r lvo lved:
(i) t h e i n v e s t i g a t i o n o f t h e e f f e c t s o f t h e u s e o f
p r o j e c t s ( a s i n s t r u m e n t s f o r t e a c h i n g and l e a r n -
JYS Mathema t i c s and e v a l u a t i n g a c h i e v e m e n t t h , -re-
f rom) on JSS s t u d e n t s g a f f e c t i v e b e h a v i o u r toward
Mathema t i c s ;
(ii) t h e c o n s t r u c t i o n o f some p r o j e c t s based on JSSMC
and t h e u s e of some of t h e c o n s t r u c t e d p r o j e c t s
i n t e a c h i n g some s t u d e n t s u n i t s of work i n JSSMC;
( i i i ) t h e deve lopmen t and v a l i d a t i o n of three o f t h e
p r o j e c t s a s e v a l u a t i o n i n s t r u m e n t s and t h e draw-
i n g u p of t h r e e c o r r e s p o n d i n g mark ing schemes;
( i v ) t h e i d e n t i f i c a t i o n and c a t e g o r i z a t i o n of t h e
e x p e c t e d s t u d e n t s 1 r e s p o n s e s ( w i t h i n t h e mark ing
schemes ) i n t o c o g n i t i v e and psychomotor s k i l l s ;
(v) t h e deve lopmen t and p a r t i a l v a l i d a t i o n o f t w o
scales; o n e f o r measu r ing JSS s t u d e n t s f a f f e c t i v e
pe r fo rmance i n Mathemat ics (SAPIM) and t h e o t h e r
f u r m o n i t o r i n g JSS s t u d e n t s ' a f f e c t i v e l e a r n i n g
outcomes from Mathema t i ca l p r o j e c t s (SMALOMP) :
( v i ) t h e i n v e s t i g a t i o n of t h e e x t e n t t e a c h e r s ' s c o r i n g
of s t u d e n t s ' r e s p o n s e s t o t h e t h r e e p r o j e c t s
12
d i s c r i m i n a t e d among s t u d e n tsl l e v e l s of perform-
a n c e s i n t h e c o g n i t i v e and psychomotor domains ;
t h e i n v e s t i g a t i o n of t h e r a t e r r e l i a b i l i t i e s o f
t e a c h e r s , f rom t h e s c o r e s t h e y awarded t o
s t u d e n t s 1 COG i t i v e and psychomotor r e s p o n s e s
t o t h e p r o j e c t s ; and
t h e i n v e s t i g a t i o n of t h e i n t e r l i n e a r and
c o r r e l a t i o n a l r e l a t i o n s h i p s among s t u d e n tst
a f f e c t i v c , c o g n i t i v e and psychomotor p e r f o r m a n c e s
on t h e p r o j e c t s .
2.4 NEED FOR THIS STUDY - The e x i s t i n g p r o c e d u r e s and t e c h n i q u e s of t e a c h i n g
and l e a r n i n g Ma thema t i c s i n o u r s e c o n d a r y s c h o o l s h a v e
n o t p roduced s a t i s f a c t o r y r e s u l t s i n terms o f s t u d e n t s '
a t t i t u d e toward and p e r f o r m a n c e i n Ma thema t i c s . It i s
a l s o s p e c u l a t e d t h a t t h e s y s t e m h a s n o t made s t u d e n t s
t r a n s f e r a p p r e c i a b l y t h e s k i l l s t h e y a c q u i r e d i n
Ma thema t i c s t o t r u e l i f e s i t u a t i o n s . It i s t h e r e f o r e
n e c e s s a r y t o e x p l o r e new p r o c e d u r e s / t e c h n i q u e s t h a t
e n h a n c e s t u d e n t s l p e r f o r m a n c e s i n Ma thema t i c s and
e n s u r e g r e a t e r t r a n s f e r o f s k i l l s . One of s u c h
p r o c e d u r e s / t e c h n i q u e s i s t h e u s e of p r o j e c t s for
t e a c h i n g and l e a r n i n g P la themat ics h e n c e t h i s s t u d y .
A l s o t h i s s t u d y i s n e c e s s a r y b e c a u s e of t h e
demands o f C o n t i n u o u s Assessment . A s s t i p u l a t e d i n t h e
F e d e r a l M i n i s t r y of ducati ion's Handbook on C o n t i n u o u s
Assessment , p r o j e c t s s h o u l d form p a r t of t h e i n s t r u m e n t s
f o r e v a l u a t i n g s t u d e n t s ' a ch i evemen t , But i t h a s been
o b s e r v e d t h a t many t e a c h e r s ( i n c l u d i n g t h o s e of
Mathema t i c s ) i n N i g e r i a n Secondary S c h o o l s d o n o t know
w h a t p r o j e c t s are , a n d d o n o t have t h e compe tence t o
c o n s t r u c t s u c h i n s t r u m e n t s ( O k u n r o t i f a , 1977; Epunam,
1983 ; Harbor-Ibeaja and Nworgu, 19P5) . T h e r e i s t h e r e -
f o r e t h e need t o c o n s t r u c t model Mathematical p r o j e c t s
t o l e t t e a c h e r s (of Mathema t i c s s p e c i f i c a l l y ) i n o u r
s e c o n d a r y schools know what p r o j e c t s are and i f p o s s i b l e ,
l e a r n how t o c o n s t r u c t p r o j e c t s . Given t h a t Ma thema t i c s
t e a c h e r s i n o u r s e c o n d a r y s c h o o l s l e a r n t o c o n s t r u c t
p r o j e c t s and u s e same as e v a l u a t i o n and t e a c h i n g
i n s t r u m e n t s , i t may s t i l l be d i f f i c u l t f o r t h e teachers
t o know w h a t s k i l l s t o l o o k f o r and score, The above
d i f f i c u l t y may be r e d u c e d i f t h e p r o j e c t s s o
c o n s t r u c t e d / d e v e l o p e d a s e v a l u a t i o n i n s t r u m e n t s a r e
accompanied by detailed marking schemes t o s e r v e as
models f o r t e a c h e r s , T h i s s t u d y b e s i d e s mee t ing t h e
need t o c o n s t r u c t /devalop model at he ma tics based
p r o j e c t s , meets t h e need f o r d rawing u p d e t a i l e d
mark ing schemes fo r t h e p r o j e c t s a s a s s e s s m e n t
i n s t r u m e n t s .
Ano the r s i d e t o t h e s i g n i f i c a n c e of t h i s work i s
t h e p r o v i s i o n of some b a s e s fo r t h e c o m p a r a b i l i t y o f
r e s u l t s and s t a n d a r d s a c r o s s s c h o o l s and y e a r s . T h i s
i s so s i n c e t h e s e p r o j e c t s and t h e i r c o r r e s p o n d i n g
mark ing schemes (o r o t h e r s p a r a l l e l t o them) c o u l d be
used across s c h o o l s and y e a r s w i t h o u t c o a c h i n g and
p r a c t i c e d e t r a c t i n g from t h e i r e f f e c t s .
F u r t h e r m o r e , t h i s s t u d y i s e x p e c t e d to make
s i g n i f i c a n t c o n t r i b u t i o n s t o g u i d a n c e and c o u ~ s e l l i n g
( i n t h i s i n s t a n c e c a r e e r ) i n our- s e c o n d a r y s c h o o l s .
S i n c e f o r p r o p e r c o u n s e l l i n g o f any s t u d e n t , a g u i d a n c e /
t e a c h e r c o u n s e l l o r n e e d s t o know reliable estimates o f
t h e s t u d e n t s ' c a p a b i l i t i e s and s k i l l s ( a f f e c t i v e
c o g n i t i v e a n d psychomotor ) a s t h e s e a t t r i b u t e s c o n c e r n
s c h o o l and s c h o o l i n g . But t h e s e m e a s u r e s / e s t i m a t e s c a n
es a t p r e s e n t , o n l y be g o t from t h e u s e of w e l l c o n c e i v e d
and w e l l c o n s t r u c t e d e v a l u a t i o n i n s t r u m e n t s . A l m o s t a l l
o u r tests, a s s i g n m e n t s , c lass works and e x a m i n a t i o n s ,
as p r e s e n t l y c o n s t r u c t e d and used , o n l y f u r n i s h u s w i t h
estimates of s t u d e n t s ' cognitive achievements, B u t
p r o j e c t s a s i s u s e d i n t h i s s t u d y s u p p l y u s estimates
o f t h e a f f e c t i v e , c o g n i 4 A v e and psychorno to r ( a c h i e v e m e n t s )
s k i l l s a n d c a p a b i l i t i e s o f s t u d e n t s . Armed w i t h t .hese
estimates of s t u d e n t s ' s k i l l s a n d c a p a b i l i t i e s a
g u i d a n c e / t e a c h e r c o u n s e l l o r w i l l be i n a better p o s i t i o n ,
t h a n o t h e r w i s e , t o h e l p t h e s t u d e n t s c h o o s e a n d p u r s u e
s u i t a b l e careers. F o r i n s t a n c e , n g u i d a n c e / t e a c h e r
c o u n s e l l o r may, o n t h e bases of a v a i l a b l e estimates,
d i s u a d e a l l b r i l l i a n t l f b u t I1carelessii ( i n h a n d l i n g
o b j e c t s a n d t o o l s ) s t u d e n t from p u r s u i n g c o u r s e i n some
a s p e c t s o f m e d i c i n e or e l e c t r o n i c e n g i n e e r i n g for
m e n t i o n , s i n c e s u c h areas i n v o l v e t h e h a n d l i n q of
de l ica te objects o r s e n s i t i v e i r , s t r u m e n t s . It i s t h e r e -
fore i m p o r t a n t t o c o n s t r u c t / d e v e l o p i n s t r u m e n t s t h a t
c a n m o n i t o r t h e s e c a p a b i l i t i e s a n d s k i l l s of s t u d e n t s
i n o u r s e c o n d a r y s c h o o l s , h e n c e t h i s work.
B e s i d e s p r o v i d i n g some data for c o u n s e l l i n g
s t u d e n t s , p r o j e c t s are b a s e s / t e c h n i q u e s f o r i n t e g r a t i n g
v a r i o u s a s p e c t s of M a t h e m a t i c s a n d d e m o n s t r a t i n g u s e s
o f M a t h e m a t i c s i n a l l o t h e r s u b j e c t s e s p e c i a l l y t h e
s c i e n c e a n d v o c a t i o n a l s u b j e c t s , A l s o t h e u s e of
p r o j e c t s as t o o l s / t e c h n i q u e s of t e a c h i n g and l e a r n i n g
Mathemat i c s i n o u r secondary s c h o o l s i s v e r v i m p o r t a n t
e s p e c i a l l y r~ow when o u r c o u n t r y i s s t r e s s i n g the
deve lopmen t o f t e c h n o l o g y b e i t i n d i g e n o u s o r c o p i e d ,
The knowledge, s k i l l s and e x p e r i e n c e s w h i c h s t u d e n t s
g a i n w h i l e e x e c u t i n g p r o j e c t s i n Mathemat ics (as i s
e x a m p l i f i e d i n t h i s s t u d y ) may g o a l o n g way t o h e l p i n g
JSS d r o p - o u t s d u r i n g t h e i r a p p r e n t i c e s h i p and working
l i f e i n s u c h ou t -o f - schoo l v o c a t i o n a l e n d e a v o u r s as
c a r p e n t a r y , w e l d i n g , me ta lworks , b r i c k l a y i n g etc.
Such g a i n s w i l l , t o a g r e a t e x t e n t , a l so h e l p s t u d e n t s
who c o n t i n u e i n t h e i r f o r m a l c 4 u c a t i o n p u r s u i t s . But
f o r s u c h e x p o s u r e s and e x p e r i e n c e s t o b c w o r t h w h i l e ,
t h e p r o j e c t s u n d e r t a k e n s h o u l d b e w e l l t h o u g h t o u t ,
m e a n i n g f u l , r e l e v a n t and t r i e d o u t , Hence t h i s work
a s a f o r e - r u n n e r o f o t h e r similar and complementary
s t u d i e s , i s t i m e l y a n d c o g e n t .
1 e 5 DELIMXTATION O F THE STUDY ----- The number o f p r o j e c t s t h a t c o u l d be drawn from
o u r p r e s e n t s e c o n d a r y school. Ma thema t i c s C o r r i c u l a i s
u n d o u b t e d l y many. So t o d e v e l o p and t r y o u t model
p r o j e c t s , r e p r e s e n t a t i v e o f t h e many p o s s i b l e p r o j e c t s
are t a s k s too demanding i n terms of t i m e a n d r e s o u r c e s .
' T h i s s t u d y i s t h e r e f o r e f o c u s s e d on t h e J u n i o r Seconda ry
Schoo l Ma thema t i c s C u r r i c u l u m (JSSMC). C e r t a i n l y t h e r e
a r e a l s o many p r o j e c t s t h a t c o u l d be c o n s t r u c t e d based
on t h e JSSIC, t h i s s t u d y c o n c e r n s itself w i t h : l i s t i n g
a l a r g c number o f t k . 3 p r o j e c t s ; u s i n g t h r e e o f t h e
p r o j e c t s i n t h e t e a c h i n g and l e a r n i n g a c t i v i t i e s o f
some JSS y e a r I11 s t u d e n t s ; d e v e l o p i n g t h r e e o f t h e
p r o j e c t s a s e v a l u a t i o n i n s t r u m e n t s ; d rawing u p mark ing
schemes t o score s t u d e n t s 1 c o g n i t i v e a n d psychomotor
r e s p o n s e s to t h e t h r e e p r o j e c t s ; t r y i n g o u t t h e
i n s t r u m e n t s on s t u d e n t s ; and s u b m i t t i n g t h e s t u d e n t s 1
r e s p o n s e s t o a set o f t e a c h e r s t o s c o r e ,
The above a c t i v i t i e s c o u l d l e a d t o many i n v e s t i g a -
t i o n s , b u t t h i s s t u d y i s l i m i t e d to i n v e s t i g a t i n g t h e :
( i) ra te r r e l i a b i l i t y and d i s c r i m i n a t o r y a b i l i t i e s of
Mathema t i c s t e a c h e r s i n s c o r i n g s t u d e n t s 1
r e s p o n s e s t o t h e p r o j e c t s ;
( i i ) e x t e n t e x p e r t e d u c a t o r s a n d Mathema t i c s t e a c h e r s
a g r e e t o t h e s u i t a b i l i t y and r e l e v a n c e o f t h e
d e v e l o p e d p r o j e c t s and t h e i r c o r r e s p o n d i n g
mark ing schemes t o t h e t e a c h i n g a n d l e a r n i n g o f
JSS Mathema t i c s ;
( i i i ) e f f e c t s of t h e u s e of p r o j e c t s on s t u d e n t s 1
a f f e c t i v e pe r fo rmance toward Mathema-t ics ; and
' ( i v ) l i n e a r and c o r r e l a t i o n a l r e l a t i o n s h i p s between/
among s t u d e n t s l a f f e c t i v e , c o g n i t i v e and
p s y c ~ m m o t o r pe r fo rmances on t h e p r o j e c t s ,
The r e s e a r c h d i d n o t i n v e s t i g a t e t h e e f f e c t s o f
e x p o s u r e t o p r o j e c t 3 on s t a d e n t s ' c o g n i t i v e and
psychomotor p e r f o r m a n c e s f o r a number o f r e a s o n s , The
e x e c u t i o n o f t h e p r o j e c t s a f f o r d e d t h e s t u d e n t s of t h e
e x p e r i m e n t a l g r o u p s some direct and p u r p o s e f u l t r a i n i n g
i n some psychomotor s k i l l s . But t h e a p p r o a c h e s u sed
i n t e a c h i n g t h e s t u d e n t s of t h e c o n t r o l g r o u p s , d i d
n o t afford t h e s t u d e n t s a n y direct a n d p u r p o s e f u l
t r a i n i n g i n psychomotor s k i l l s . I t i s n o t e a s y i n
terms of t h e r e s o u r c e s a v a i l a b l e t o t h e r e s e a r c h e r to
c o n s t r u c t and v a l i d a t e a c o g n i t i v e tes t i n s t r u m e n t
t i n t h e Ma thema t i c s u n i t s used f o r t h i s s t u d y ) t h a t
i t s items a r e u n b a i s e d r e l a t i v e t o t h e modes o f
t e a c h i n g and l e a r n i n g o f t h e u n i t s o f work a s c a r r i e d
o u t i n t h e e x p e r i m e n t a l and c o n t r o l g r o u p s , The
r e s e a r c h e r c o n s i d e r e d t h e s t u d e n t s a f f e c t i v e pe r fo rmance
toward Mathema t i c s most i m p o r t a n t t o i n v e s t i g a t e , s i n c e
t h e s t u d e n t s mus t f i rs t a n d f o r e m o s t r e s p o n d / a t t e n d t o
Mathema t i c s before t h e s t u d e n t s c o u l d a t t a i n any l e v e l
of c o g n i t i v e or psychomotor pe r fo rmance i n Mathematics .
From t h e researcher's e x p e r i e n c e as a m a t h e m a t i c s , t e a c h e r
a n d from t e a c h e r s 1 r e p o r t s d u r i n g w o r k s h o p s a n d c o n f e r e n c e s ,
many s e c o n d a r y s c h o o l s t u d e n t s t e n d t o a v o i d l e s s o n s a n d
m o r e s o a s s e s s m e n t e x e r c i s e s i n M a t h e m a t i c s , Hence o n e
o f t h e t r u s t s o f t h i s work i s t o see w h e t h e r t h e use o f
p r o j e c t s c o u l d r e d u c e t h e a v o i d a n c e t e n d e n c y , F u r t h e r m o r e
t h e r e s e a r c h e r d i d n o t i n v e s t i g a t e t h e effects of use of .
p r o j e c t s o n s t u d e n t s ' c o g n i t i v e p e r f o r m a n c e b e c a u s e A l i
(1989) o b s e r v e d t h a t s t u d e n t s ' a c h i e v e m e n t i n M a t h e m a t i c s
( t h e i r c o g n i t i v e p e r f o r m a n c e i n M a t h e m a t i c s t e s t s ) do n o t
d e p e n d s o l e l y o n t h e amount of l e a r n i n g / m a s t e r y of t h e
u n i t s o f work on w h i c h t h e test i s b a s e d , T h e i r
p e r f o r m a n c e s may d e p e n d l a r g e l y on t r a i n i n g programmes
o n test t a k i n g a n d t e s t w i s e n e s s ,
1.6 HEhdHkCH y U E S T I O N j A N D HYPOTHESES
Below are t h e research q u e s t i o n s a n d h y p o t h e s e s
f o r m u l a t e d a n d u s e d f o r t h i s s t u d y ,
6 R e s e a r c h Q u e s t i o n s
1, Does a s i g n i f i c a n t m a j o r i t y o f t h e v a l i d a t o r s
( P l a t h e m a t i c s e d u c a t o r s a n d Measurement a n d E v a l u a t i o n
e x p e r t s ) a g r e e t h a t t h e d e v e l o p e d M a t h e m a t i c a l p r o j e c t s
are s u i t a b l e f o r a n d r e l e v a n t t o t h e t e a c h i n g 'and l e a r n i n g
o f JS5 M a t h e m a t i c s ?
( e v a l u a t
d o e s a s i g n i f i c a n t r n a j o r i t y o f t h e judge 's -
i o n e x p e r t s and M a t h e m a t i c s t e a c h e r s ) a g r e e t o
t h e c a t e g o r i z a t i o n o f t h e e x p e c t e d s t u d e n t s r e s p o n s e s ,
t o t h e d e v e l o p e d M a t h e m a t i c a l p r o j e c t s , i n t o c o g n i t i v e
a n d psychomo tor l e a r n i n g o u t c o m e s ?
3. To w h a t e x t e n t do teachers a g r e e among t h e m s e l v e s
i n terms o f t h e s c o r e s t h e y , by u s i n g t h e m a r k i n g s c h e m e s ,
a w a r d e d t o s t u d e n t s o c o g n i t i v e a n d p s y c h o m o t o r r e s p o n s e s
t o t h e d e v e l o p e d p r o j e c t s ?
4, To w h a t l e v e l d o t e a c h e r s , by making u s e of
t h e m a r k i n g s c h e m e s i n s c o r i n g s t u d e n t s 1 c o g n i t i v e a n d
p s y c h o n o t o r r e s p o n s e s t o t h e t h r e e p r o j e c t s , d i s c r i m i n a t e
among s t u d e n t s p e r f o r m a n c e s ?
5, d h a t a re t h e s i m p l e o r m u l t i p l e r e g r e s s i o n
e q u a t i o n s (o r l i n e s ) be tween o r among t h e means of t h e
s t u d e n t s ' c o g n i t i v e ( x ) , p s y c h o m o t o r (Y) a n d a f fec t ive
( A ) p e r f o r m a n c e scores as m o n i t o r e d by t h e p r o j e c t s ?
6, To w h a t e x t e n t does e x p o s u r e t o t h e u s e of
h a t h e r n a t 5 c a l p r o j e c t s a f f e c t s t u d e n t s affect ive
p e r f orrnance i n Ibiathematics?
2-6.2 R e s e a r c h Hypo theses
1, Nul l : The mean a f f e c t i v e pe r fo rmance score
i n Ma thema t i c s o f s t u d e n t s exposed to Mathema t i ca l
p r o j e c t s i s n o t s i g n i f i c a n t l y g r e a t e r t h a n t h e mean
a f f e c t i v e pe r fo rmance score i n Ma thema t i c s of s t u d e n t s
n o t exposed to t h e Ma thema t i ca l p r o j e c t s ( p L 0 . 0 5 ) .
2, Nu l l : The marker ( r a t e r ) r e l i a b i l i t y a n d
d i s c r i m i n a t o r y a b i l i t i e s o f t h e teachers, (making u s e
o f t h e mark ing schemes t o score t h e s t u d e n t s g c o g n i t i v e
and psychomotor r e s p o n s e s t o t h e ~ r o j e c t s ) , are n o t
s t a t i s t i c a l l y s i g n i f i c a n t . (p,LO,OS),
3 , Nul l : E a c h o f t h e s i m p l e o r m u l t i p l e
c o r r e l a t i o n and l i n e a r r e g r e s s i o n c o e f f i c i e n t s between
or among s t u d e n t s ' a f f e c t i v e , c o g n i t i v e and psychomotor
p e r f o r m a n c e s , as m o n i t o r e d by t h e p r o j e c t s i s n o t
s i g n i f i c a n t (pLO,OS) ,
CHAPTER TWO
LITERATURE REVIEW -.a-
2,O II~TROUUCTION
T h i s s t u d y aim^ J a t d e v e l o p i n g some p r o j e c t s
based on J u n i o r Seconda ry S c h o o l Ma thema t i c s C u r r i c u l u m
(JSSNC) , f o r t e a c h i n g and l e a r n i n g Mathema t i c s i n
N i g e r i a ' s J u n i o r Secondary S c h o o l s ( J S S ) and f o r
e v a l u a t i n g JSS s t u d e n t s 1 pe r fo rmance i n Mathemat ics .
I n t h i s c h a p t e r , therefore t h e r e s e a r c h e r r ev i ewed some
of t h e r e l e v a n t works on: t e a c h i n g and l e a r n i n g ,
t e a c h i n g and l e a r n i n g Mathematics, E v a l u a t i o n of
s t u d e n t s ' a c h i e v e m e n t s , C o n t i n u o u s Asses smen t (CAI i n
N i g e r i a S c h o o l s , e x p e c t e d l e a r n i n g ou tcomes o f
Ma thema t i c s e d u c a t i o n , i n s t r u m e n t s fo r e v a l u a t i n g t h e :-
t h e e x p e c t e d l e a r n i n g outcomes , p r o j e c t s and t h e
v a l i d a t i o n o f e v a l u a t i o n i n s t r u m e n t s . A l s o p r e s e n t e d
a t t h e end of t h i s c h a p t e r i s a br ief summary o f t h e
i s s u e s r a i s e d and d i s c u s s e d .
2.1 TEACHING AND LEARNING
E d u c a t i o n as s e e n by Dewey (1966, pp. 1-23) impl ies
t h e " r enewa l of l i f e by t r a n s m i s s i o n i t , i t i s a "forming
ac t i v i t y n i n v o l v i n g I1shaping a n d mounding1! i n d i v i d u a l s
i n ' t o s t a n d a r d f o r m s for s o c i a l l i v i n g . Okafo r (1984,
p. 16 ) d e f i n e s e d u c a t i o n as "a p r o c e s s of a c c u l t u r a t i o n
t h r o u g h which t h e i n d i v i d u a l i s h e l p e d to a t t a i n t h e
devclopmeri t o f h i s p o t e n t i a l i t i e s and t h e i r maximum
a c t i v a t i o n when n e c e s s x y , a c c o r d i n g t o r i g h t r e a s o n and
a c h i e v e t h e r e b y h i s p e r f e c t s e l f f u l f i l m e n t . " From t h e
f o r c g o i n g , therefore, e d u c a t i o n i m p l i e s a p r o c c s s which
i n v o l v e s p r e p a r i n g t h e i n d i v i d u a l f o r h i s / h e r p r e s e n t
o r f u t u r e roles as a n i n d i v i d u a l a n d as a mc1nbc.r of a
g i v e n s o c i e t y . I n o t h e r words , t h e main g o a l o f
e d u c a t i o n i s to s i g n i f i c a n t l y s h a p e or c h a n g e t h e
l e a r n e r ' s p a t t e r n s o f b e h a v i o u r so t h a t t h e y are u s e f u l
and a c c e p t a b l e to t h e s o c i e t y .
But e d u c a t i o n , however c o n c e i v e d a n d p r a c t i s e d , i s
e n t i r e l y d e p e n d e n t on t w o real i t ies, n a t u r e and n u r t u r e *
E d u c a t i o n o f any s o r t g o e s on b e c a u s e man i s educab le .
T h i s i s t o s a y t h a t man i s endowed by n a t u r e , w i t h t h e
f a c u l t i e s o r c a p a b i l i t i e s t o r e spond t o s h a p i n g , mould ing ,
c h a n g i n g a c t i v i t i e s or s e e k i n g s e l f - f u l f i l m e n t . A l s o i t
i s acknowledged by e d u c a t o r s , t h a t t h e e x t e n t to which
a n y i n d i v i d u a l % p a t t e r n s o f b e h a v i o u r c o u l d be moulded,
shaped or changed i s l i m i t e d by n a t u r e , A l though n a t u r e
sets t h e l i m i t t o which an i n d i v i d u a l c a n b e moulded,
m v y p ' v .
shaped or changed , ( and t h i s l i m i t i s r a r e ' y i f e v e r
a t t a i n e d ) , n u r t u r e or e n v i r o n m e n t sets t h e p a c e a n d
d e t e r m i n e s t h e q u a l i t i e ~ , t y p e s and d i r e c t i o n s of t h e
mould ing , s h a p i n g or c h a n g i n g t h e i n d i v i d u a l ' s bebav iou r .
It c a n t h e r e f o r e b e s a i d t h a t t h e p r o d u c t s of t h e above
a c t i v i t i e s i s i n f l u e n c e d g r e a t l y by env i ronmen t . Indeed
c e n t r a l to e d u c a t i o n , a s a human c o n c e r n o r t h a t w i t h o u t
wh ich t h e r e i s no e d u c a t i o n , are t h e t w i n s or t w o s i d e s
of t h e same c o i n t e a c h i n g and l e a r n i n g . It follows,
t h e r e f o r e , t h a t s i n c e e d u c a t i o n i s as o l d a s man and
s o c i e t y , t e a c h i n g a n d l e a r n i n g are no less so. L i k e
man and s o c i e t y , t h e s e p r o c e s s e s have been unde rgo ing
c h a n g e s i n c o n c e p t i o n and p r a c t i c e ,
From t h e v e r y e a r l y societies and e v e n u p t o r e c e n t
t r a d i t i o n a l societies, t e a c h i n g and l e a r n i n g i n v o l v e d
v e r y close one-to-one i n t e r a c t i o n between t h e t e a c h e r
and t h e l e a r n e r . The t e a c h e r exposed t h e l e a r n e r t o
e x p e r i e n c e t h e a c t u a l t a s k s a n d a c t i v i t i e s wh ich
c o n s t i t u t e d t h e o c c u p a t i o n t h e l e a r n e r was e x p e c t e d t o
t a k e to , The l e a r n e r watched and imitated h i s / h e r
t e a c h e r , and had t h e c h a n c e o r o p p o r t u n i t y t o a s s e s s
h i m s e l f / h e r s e l f by compar ing h i s / h e r p r o d u c t t o t h a t of
t h e t e a c h e r , I n t h e above p a t t e r n / a r r a n g e m e n t of t e a c h i n g
and l e a r n i n g , t r a n s f e r o f knowledge, t r a i n i -ig and
s k i l l s was n o t q u e s t i o n a b l e . A l s o t h e c o n t e n t s a n d
cr i ter ia on wh ich a s s e s s m e n t s were based , w e r e v e r y
much r e l e v a n t .
But t h e a b o v e real i t ies p r e s e n t i n t h e t e a c h i n g
and l e a r n i n g s i t u a t i o n s o f t r a d i t i o n a l societies are
r a r e l y p r e s e n t i n almost a l l t h e f o r m a l e d u c a t i o n set-
u p s o f nowadays, Teach ing and l e a r n i n g u n d e r t a k e n i n
most e d u c a t i o n a l i n s t i t u t i o n s of t h e s e d a y s l a c k t h a t
v e r y c l o s e one-to-one i n t e r a c t i o n between t h e t e a c h e r
and t h e t a u g h t . The t e a c h e r , t h e s e d a y s , deals w i t h
many l e a n e r s a t t h e same t i m e a l m o s t a lways . Should
t h i s be t h e s i t u a t i o n c o n s i d e r i n g what t e a c h i n g i s
c o n c e i v e d t o b e ? G e n e r a l l y t e a c h i n g , i n t h e words o f
Bur ton as q u o t e d by Chauhan (1979 , p. 41, "is
s t i m u l a t i o n , g u i d a n c e , d i r e c t i o n and encouragement o f
l e a r n i n g , " Zukowski (1980, p, 20) n o t e d :
o b v i o u s l y t e a c h i n g i s what t e a c h e r s do. They do many t h i n g s . The p r e s e n t i n f o r m a t i o n o r a l l y , t h e y g i v e i n s t r u c - t i o n s to p u p i l s t o d e a l w i t h , They estimate t h e p r o g r e s s o f p u p i l s . They c a r r y o u t a v a r i e t y of act ivi t ies t h a t c h a r a c t e r i z e t h e i n s t i t u t i o n w e c a l l t h e s c h o o l ,
But t h e way and manner d i f f e r e n t t e a c h e r s c a r r y o u t
t h o s e a c t i v i t i e s , t h a t c h a r a c t e r i z e t h e s c h o o l , d o
d i f f e r a t t h s least w i t h r e s p e c t t o g e n e r a t i o n s and
s c h o o l s o f t h o u g h t , I n g e n e r a l , t h e v a r i a t i o n s i n t h e
way and manner t e a c n e r s c a r r y o u t t h e a c t i v i t i e s
depend upon:
( i) t h e p r e v a l e n t p h i l o s o p h y o f t h e g e n e r a t i o n oP
s o c i e t y o f t h e t e a c h e r s a b o u t t h e n a t u r e of man;
(ii) t h e methods and p r i n c i p l e s o f t e a c h i n g wh ich t h e
t e a c h e r s were exposed to d u r i n g t h e i r s c h o o l d a y s
or t r a i n i n g p e r i o d s ; and
(iii) t h e p r e v a l e n t l e a r n i n g t h e o r i e s known t o t h e set
of t e a c h e r s ( B i g g e and Hunt, 1968). A l s o
v a r i a t i o n s i n t e a c h i n g s t r a t e g i e s may e x i s t
a l o n g s u b j e c t a r e a l i n e s b e c a u s e o f t h e rea l or
p e r c e i v e d d i f f e r e n c e s i n t h e n a t u r e and s t r u c t u r e s
o f s u c h s u b j e c t areas.
Indeed t h e p h i l o s o p h i e s a b o u t t h e n a t u r e of t h e
l e a r n e r and t h e i r a t t e n d a n t l e a r n i n g t h e o r i e s , t h a t
have i n f l u e n c e d t e a c h i n g and l e a r n i n g t h r o u g h t h e a g e s
a re many. However, t h e y c a n b e c a t e g o r i s e d i n t o a f e w
g r o u p s . The f i r s t c a t e g o r y and may be o n e of t h e
ear l ies t i s t h e b e l i e v e t h a t 'man', by n a t u r e , i s bad
' and a c t i v e i n h i s env i ronmen t . S e q u e l t o t h a t b e l i e f
i s t h e t h e o r y of m e n t a l d i s c i p l i h e . The t h e o r y h o l d s
t h a t t h c mind, l i k e t h e m u s c l e s of t h e body, c o u l d be
made s t r o n g e r and a c t i v e by m e n t a l exercise. Teach ing
and l e a r n i n g need n o t b e made i n t e r e s t i n g , r a t h e r some
c o n t e n t matter u n i n t e r e s t i n g and d i f f i c u l t were made
p a r t o f t h e l e a r n i n g e x p e r i e n c e s i n o r d e r t o d i s c i p l i n e
t h e mind. Next t o t h e above p h i l o s o p h y , and i n d e e d a n
a t t e m p t e d c o u n t e r t o i t , i s t h e b e l i e f t h a t 'man' i s
i n t r i n s i c a l l y good and a c t i v e i n h i s env i ronmen t ,
T h i s o u t l o o k on l i f e u n d e r l i e s r o m a n t i c n a t u r a l i s m and
n a t u r a l un fo ldmen t , T h i s p h i l o s o p h y and t h e o r i e s
f a v o u r e d making c o n t e n t , t e a c h i n g and l e a r n i n q , c h i l d /
l e a r n e r c e n t r e d , i n t e r e s t i n g and a c t i v i t y o r i e n t e d .
Unde r ly ing t h e P a v l o v i a n l e a r n i n g t h e o r y ( c l a s s i c a l
c o n d i t i o n i n g 1 m d e d u c a t i o n a l p r a c t i c e s a t t e n d a n t t o i t ,
i s t h e b e l i e f t h a t man i s by n a t u r e n e u t r a l and r e a c t i v e
t o h i s env i ronmen t . S k i n n e r ' s o p e r a n t c o n d i t i o n i n g ,
T h o r n d i k e ' s c o n n e c t i o n i s m o r t r i a l and error l e a r n i n g
and a host . o f o t h e r S t imulus-Response t h e o r i e s , are
b a s e d on the view t h a t 'man' i s n a t u r a l l y n e u t r a l and
a c t i v e i n h i s env i ronmen t . Teach ing and l e a r n i n g demand
a c t i v i t i e s , p r a c t i c e s and f e e d b a c k s v e r y r e l e v a n t t o t h e
t a s k s e x p e c t e d t o be a c c o m p l i s h e d b y t h e l e a r r ~ e r s ,
T h e Gestal t f i e l d t h e o r i s t s see 'man1 a l s o a s b e i n g
n e u t r a l b u t i r l t e r a c t i v e w i t h h i s e n v i r o n m m t . T h e s e
t h e o r i s t s b e l i e v e t h a t t e a c h i n g a n d l e a r n i n g are more
e f f e c t i v e i f t h e o b j e c t t o be l e a r n t i s t a k e n i n w h o l e
b e f o r e c o n s i d e r i n g t h e p a r t s , T e a c h i n g a n d l e a r n i n g
a c t i v i t i e s s h o u l d be made m e a n i n g f u l , i n s i g h t f u l . a n d
p r o b l e m s o l v i n g o r i e n t e d . T h e l a s t b u t n o t t h e leas t
g r o u p of l e a r n i n g t h e o r i e s t s t h a t h a s i n f l u e n c e d
t e a c h i n g a n d l e a r n i n g c o n s i s t s o f t h e c o g n i t i v e
d e v e l o p m e n t a l t h e o r i s t s . T h e s e t h e o r i s t s i n c l u d e
P i a g e t a n d B r u n n e r a n d t h e i r r e s p e c t i v e a s s o c i a t e s .
P i a g e t , h i s associates a n d N e o - P i a g e t i a n s stress t h a t
c o n t e n t a n d i n s t r u c t i o n s s h o u l d be made t o s u i t t h e
m a t u r i t y l e v e l / s t a g e o f c o g n i t i v e d e v e l o p m e n t of t h e
l e a r n e r s , B r u n n e r , h i s associates a n d f o l l o w e r s , r a t h e r
t h a n stress c o n t e n t matter , h o l d t h e v i e w t h a t t h e a l l
i m p o r t a n t t h i n g t o t e a c h i n g and l e a r n i n g i s t h e 'mode1
of p r e s e n t i n g t h e c o n t e n t to t h e l e a r n e r s .
The u s e o f p r o j e c t s , i n t e a c h i n g and l e a r n i n g
M a t h e m a t i c s and t h e e v a l u a t i o n o f c o n s e q u e n t o u t c o m e s ,
a s i s d o n e i n t h i s s t u d y , accommodates many o f t h e
e m p h a s i s of t h e p r e v a i l i n g l e a r n i n g t h e o r i e s , P r o j e c t s
u n d e r t a k e n by s t u d e n t s a re p r o t o t y p e s o r m o d e l s o f real
t a s k s . T h e s e p r o j e c t s s f f o r d s t u d e n t s t h e o p p o r t u n i t y
t o e x p e r i e n c e r ea l l i f e t a s k s or a s p e c t s of o c c u p a t i o n s
t h e s t u d e n t s may t a k e to a f t e r s c h o o ~ . u n d e r t a k i n g
p r o j e c t s as i s d o n e i n t h i s s t u d y i n t e g r a t e s t h e o r y
a n d p r a c t i c e b P r o j e c t s a l so p r o v i d e s t u d e n t s w i t h
e x e r c i s e s t o t r y o u t w h a t t h e y l e a r n t from c l n s s r o o m
d i s c u s s i o n s o r e x p o s i t o r y l e s s o n s , P r o d u c t s o f p r o j e c t s
are f i r s t a n d foremost c o n c e i v e d as w h o l e s or e n t i t i e s
a n d real probl .ems t o be s o l v e d . The s u c c e s s i n under -
t a k i n g a p r o j e c t p r o v i d e s i n s i g h t i n t o s u c h p r o b l e m s a n d
a l so s e r v e as r e i n f o r c e m e n t , Of c o u r s e , t h e p r o j e c t s
u n d e r t a k e n b y s t u d e n t s are g r a d e d i n terms o f t h e
s t u d e n t s ' ' m a t u r i t y 1 l e v e l s , A l s o u n d e r t a k i n g p r o j e c t s
i n M a t h e m a t i c s makes i t p o s s i b l e f o r t o p i c s t h a t would
o t h e r w i s e be p r e s e n t e d t h r o u g h t h e s y m b o l i c mode, t o be
p r e s e n t e d a l so t h r o u g h t h e e n a c t i v e a n d i k o n i c modes.
2 1 T e a c h i n q a n d L e a r n i n q M a t h e m a t i c s .--.,-.-..--..-
B e s i d e s t h e p h i l o s o p h i e s a n d t h e o r i e s t h a t
i n f l u e n c e t e a c h i n g and l e a r n i n g i n g e n e r z l , t h e r e are
some p e c u l i a r b u t p o p u l a r m e t h o d s or t e c h n i q u e s u s e d
b y t e a c h e r s of M a t h e m a t i c s , T h e s e m e t h o d s i n c l u d e
among o t h e r s :
( i) e x p o s i t o r y method;
(ii) t h e d i s c o v e r y or g u i d e d d i s c o v e r y method;
(iii) t h e problem s o l v i n g a p p r o a c h ;
( i v ) t h e d e d u c t i v e n e t h o a and
( v ) t h e i n d u c t i v e method.
Of a l l t h e a b o v e methods , t h e problem s o l v i n g method
i s of i n t e r e s t t o t h i s work. I n t h c words of C a r p e n t e r ,
C o r b i t t , Kepner , L i n d q u i s t and Reys (1980 , p. 338) :
i f i t w e r e n e c e s s a r y t o s i n g l e o u t oqc a r e a t h a t demands c r g e n t a t t e n t i o n , i t would be c l e a r l y problem s o l v i n g . A t a l l a g e L e v e l s and v i r t u a l l y e v e r y c o n t e n t a r e a , pe r fo rmance w a s e x t r e m e l y low on e x e r c i s e s r e q u i r i n g problem s o l v i n g o r a p p l i c a t i o n o f m a t h e m a t i c a l s k i l l s , I n g e n e r a l , r e s p o n d e n t s d e m o n s t r a t e d a l a c k o f t h e most basic problem s o l v i n g s k i l l s . . , . , drawing a p i c t u r e of a f i g u r e d e s c r i b e d i n a problem or c h e c k i n g t h e r e a s o n a b l e n e s s of r e s u l t s for example,
The above o b s e r v a t i o n p o i n t s t o the f a c t t h a t problem
s o l v i n g as a n a p p r o a c h t o t e a c h i n g Mathema t i c s is n o t
b e i n g used w e l l by t e a c h e r s o r h a s n o t y i e l d e d much
f r u i t s .
I n t h e Uni ted S t a t e s , t h e r e s u l t s o f r e c e n t
N a t i o n a l Asses smen t s o f E d u c a t i o n a l P r o g r e s s (NAEP)
i n Ma thema t i c s i n d i c a t e t h a t t h e a b i l i t y o f man t o add
o r compute w i t h members i s f a r f rom a l o s t a r t . However,
knowing how t o compute d o e s n o t e n s u r e t h a t a p e r s o n
w i l l know when t o cornputc, which o p e r a t i o n t o u s e i n a
p a r t i c u l a r s i t u a t i o n or how t o u s e a n answer once i t i s
o b t a i n e d ( C a r p e n t e r , Coburn, Reys a n d W i l l s o n , 1 9 7 5 ) ,
T h e above s t a t e m e n t s c a l l f o r a r e v i e w o f w h a t t e a c h e r s
o f Ma thema t i c s know and t a k e t o be problem s o l v i n g
approach , I n t h e view o f L e s h (19811, problem s o l v i n g
p r o c e s s t h a t e n a b l e s a n i n d i v i d u a l t o u s e m a t h e m a t i c a l
i d e a s i n real s i t u a t i o n s are n o t n e c e s s a r i l y t h e t y p e
t h a t have been d i s c u s s e d by P o l y a (19751, Landa ( 1 9 7 4 ) ,
Wicke lg ren (1974) and D a v i s (1973) o r o t h e r problem
s o l v i n g t h e o r i s t s / e d u c a t o r s who a r e p o p u l a r w i t h /
* Mathemat i c s e d u c a t o r s and t e a c h e r s . B e l l ( 2 9 7 9 ) i s of
t h e v iew t h a t r e s e a r c h a n d i n s t r u c t i o n on problem
s o l v i n g p r o c e s s have made l i t t l e p r o g r e s s i n a d d r e s s i n g
t h e i s s u e of w h a t i t is , beyond h a v i n g a n idea, t h a t
a l l o w s a n o r m a l l y i n t e l l i g e n t p e r s o n t o u s e t h e i d e a t o
d e a l w i t h Ma thema t i c s r e l a t e d problems i n every-day
s i t u a t i o n . I n my own t h i n k i n g , o n e of t h e ways to
s t a r t w i t h , i s c r e a t i n g o r d e s i g n i n g s u c h Mathemat ics
related p rob lems and p o i n t i n g o u t o r s t r e s s i n g t h e
q M a t h w n a t i c a l i d e a s ' o r too ls t h a t c o u l d be used t o
s o l v e such problems. P r o j e c t s as are c o n c e i v e d and
u n d e r t a k e n i s t h i s s t u d v t o w t h i s l i n e of t h i n k i n g ,
T r y i n g t h e p r o j e c t s o u t h e r e i s a w o r t h w h i l e endeavour
and t h a t i s p a r t of .ch is s t u d y ,
2,1,2 Teach inq and L e a r n i n q Mathema t i c s -.--- -.-- I n - .. N i q e r i a n - S e c o n d a u --a S c h o o l s -.---,
The t i t l e o f L a s s a ' s (1986) I n a u g u r a l A d d r e s s
" t h e s o r r y s t a t e o f Mathemat ics e d u c a t i o n i n N i g e r i a v t
seems to t e l l t h e w h o l e s t o r y a b o u t t e a c h i n g a n d l e a r n -
i n g Mathema t i c r i n N i g e r i a n Secondary S c h o o l s , T h i s
f o l l o w s b e c a u s e f i n d i n g s of many s t u d i e s p o i n t o u t
t h a t Ma thema t i c s i n o u r s e c o n d a r y s c h o o l s a r e p o o r l y
t a u g h t and l e a r n e d . K a l e j a i y e (1976) i n v e s t i g a t e d how
c e r t a i n t e a c h i n g p r o c e d u r e s and t e c h n i q u e s are used i n
P la themat ics c l a s s e s . H e found, among o t h e r t h i n g s ,
t h a t i n g e n e r a l :
(i) t e a c h i n g a i d s were e i t h e r n o t u sed or used
p o o r l y ;
(ii) t e a c h e r s d i d n o t u s e t h e d i s c o v e r y app roach
and d i f f e r e n t i a t e d t e a c h i n g much; and
( i i i ) w h i l e t e a c h e r s a s k e d many q u e s t i o n s d u r i n g
c l a s s e s , w r i t t e n e x e r c i s e s g i v e n t o s t u d e n t s were
i n a d e q u a t e or t o t a l l y a b s e n t ,
It? h i s s t u d y , I 1 D i f f i c u l t i a s F a c i n g Mathematics T e a c h e r s
i n Deve lop ing C o u n t r i e s - A C a s e S t u d y o f N i g e r i a t 1 , A l e
( 1 9 8 1 ) i d e n t i f i e d among o t h e r t h i n g s t h a t s t u d e n t s see
Mathemat i c s a s :
( i ) h a v i n g t o o mar. y incomprehensi l .w f o r m u l a e ;
( i i ) l a c k i n g problem s o l v i n g t e c h n i q u e s ;
(iii 1 i n v o l v i n g e x c e s s i v e c a l c u l a t i o n s and n o t h i n g b u t
c a l c u l a t i o n s which e n k i n d l e no i n t e r e s t ; and
( i v ) h a v i n g n o e x p e r i m e n t a l n a t u r e o r 1 a c k i n g real
e x p e r i e n t i a l s i t u i t i o n s and p r o d u c t s ,
T h e s e i n a d e q u a c i e s and d i f f i c u l t i e s i n ' t e ach ing and
l e a r n i n g Mathema t i c s a r e v e r y m uch r e f l e c t e d by s t u d e n t s ' . .
a c h i e v e m e n t i n t h e s u b j e c t o v e r t h e y e a r s . L a s s a
( 1 9 7 8 ) carried o u t a s u r v e y t o f i n d o u t s t u d e n t s 1 basic
r l 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 t h e s y l l a b u s , H e
a d m i n i s t e r e d a ma thema t i c s l o b j e c t i v e l tes t t o some
s t u d e n t s , The r e s u l t showed a n o v e r a l l a v e r a g e of 42%
fo r t h e s t a t e s and 32% as t h e mean score fo r t h e t h e n
t e n N o r t h e r n s t a t e s . T h i s pe r fo rmance i s poor. WAEC1s
r e p o r t ( 1 9 7 3 ) as q u o t e d i n L a s s a l s (1986) p r o f e s s o r i a l
i n a u g u r a l a d d r e s s s t a tes t h a t :
( i ) s t u d e n t s pe r fo rmance i n Ma thema t i c s w a s poo r ;
34
on t h e a v e r a g e work p r e s e n t e d t o e x a m i n e r s ,
t h e c o n c e n s u s o f o p i n i o n was t h a t m a j o r i t y o f
t h c s e t a k i n g Mathema t i c s w e r e i l l i t e ra t e s i n
t h e s i m p l e a r t s o f number;
t h e s t u d e n t s seem t o b e l i e v e t h a t t h e y c o u l d
p a s s by c h a n c e ; and
o n e wondered w h e t h e r somc of t h c s t u d e n t s had
even seen t h e i n s i d c o f a classroom.
I t h a s a l s o been o b s e r v e d t h a t N i g e r i a n Saconda ry
S c h o o l s t u d e n t s ' a t t i t u d e toward Mathema t i c s i s
g e n e r a l l y n e g a t i v e ( L a s s a , 1 9 7 8 and A l e , 1981). E z e i l o
( 1 9 8 8 ) showed t h a t c o m p a r a t i v e l y s e c o n d a r y s c h o o l
s t u d e n t s i n N i g e r i a per form v e r y p o o r l y i n Mathemat ics .
T h e s e o b s e r v e d poor p e r f o r m a n c e s i n Mathemat ics a r e
n o t c n c o n n e c t e d w i t h t h e i n e f f e c t i v e t e a c h i n g methods ,
t e c h n i q u e s and p r o c e d u r e s a s i d e n t i f i e d by A l e (19811,
I t i s o f t h i s r e s e a r c h e r ' s o p i n i o n t h a t p r o j e c t s i n
Ma thema t i c s i f a p p r o p r i a t e l y u sed a r e l i k e l y t o remove
t h e s h o r t f a l l s a n d d i f f i c u l t i e s a n d raise s t u d e n t s 1
l e v e l s -f pe r fo rmances i n t h e s u b j e c t , T h i s view I
d e r i v e s from t h e way p r o j e c t s i n Ma thema t i c s i s
c o n c e p t u a l i s e d i n t h i s s t u d y and as t r i e d o u t , t o g e t h e r
w i t h t h e v e r y n a t u r e o f p r o j e c t s i n Mathemat ics .
2 2 EVALUATION OF STUDENTS1 ACHIEVEMENT -----.-
I N MATHEMATICS
Remers e t a1 ( 1 9 6 0 , pp, 7-91 see e d u c a t i o n a l
e v a l u a t i o n as c o n c e r n i n g t h e q u e s t i o n , "what v a l u e M ;
a n d as s e e k i n g t h e a : . ~ s w e r s t o t h e t e a c h e r s 1 a n d l e a r n e r s v
q u e s t i o n "what p r o g r e s s am I making?", Ahmann e t a1
( 1 9 7 1 ) see e v a l u a t i o n as t h e s y s t e m a t i c p r o c e s s o f
d e t e r m i n i n g t h e e f f e c t i v e n e s s of e d u c a t i o n a l e n d e a v o u r s
i n t h e l i g h t of e v i d e n c e , A c c o r d i n g t o O l a t u n j i ( 1 9 7 4 ,
p o 18) e v a l u a t i o n "is t h e a p p r a i s a l o f a n o u t c o m e or
t h e a c h i e v e m e n r of p a r t i c u l a r ob j e c t i ~ e s . ~ ~ Ohuche a n d
A k e j u ( 1 9 7 7 , p. 18) d e f i n e e v a l u a t i o n a s !)a q u a l i t y
c o n t r o l s y s t e m a l l o w i n g for t h e d e t e r m i n a t i o n a t e a c h
s t e p i n t h e t e a c h i n g - l e a r n i n g p r o c e s s t h e e f f e c t i v e n e s s
o r o t h e r w i s e o f t h a t p r o c e s s a n d programming f o r move-
men t t o w a r d s ef f e c t i v e n e ~ s . ~ ~
From t h e d e f i n i t i o n s a b o v e , e d u c a t i o n a l e v a l u a t i o n ,
t h e r e f o r e , s h o u l d s e e k t o a n s w e r c o n c l u s i v e l y s u c h
q u e s t i o n s as: w h a t d i d t h e e d u c a t i o n a l p r o c e s s set o u t
t o do?; how f a r h a v e t h e o b j e c t i v e s been a c h i e v e d ? ;
w h a t s t e p s a n d or factors i n t h e p r o c e s s h a v e e i t h e r
e n h a n c e d o r h i n d e r e d t h e a c h i e v e m e n t of t h e o b j e c t i v e s ? ;
w h a t are t h e s t r e n g t h s a n d w e a k n e s s e s of t h e system'?;
a n d how c a n t h e w e a k n e s s be removed a n d t h e s t r e n g t h s
b e c o n s o l i d a t e d ? . Indeed a n y m e a n i n g f u l ant1 comprehen-
s i v e programme of e d u c a t i o n a l e v a l u a t i m u n d e r t a k e s
t h e f o l l o w i n g :
(i) c o l l e c t i n g d a t a and m o n i t o r i n g a c t i v i t i e s and
p r o g r e s s o f p .~ . ren t e d u c a t i o n a l sys t em;
(ii) a p p r a i s i n g t h e w o r t h of t h e e d u c a t i o n a l m a t e r i a l s
and a c t i v i t i e s and a l so judg ing t h e s u c c e s s o r
o t h e r w i s e o f t h e p r o d u c t s o f t h e sys t em;
(iii) d i a g n o s i n g t h e s t r e n g t l ~ s and weakness o f t h e
e n t i r e p a r e n t sys t em;
( i v ) c l a s s i f y i n g t h e p r o d u c t s o f t h e sys t em;
( v ) s e l e c t i n g and p l a c i n g / c h a n n e l i n g l e a r n e r s o r
p r o d u c t s i n t o f i e l d s w h e r e each g r o u p h a s a
c o m p a r a t i v e a d v a n t a g e o v e r o t h e r g r o u p s ; and
( v i ) m o n i t o r i n g , comparing and m a i n t a i n i n g s t a n d a r d s .
But t h e q u e s t i o n i s , how f a r i s e v a l u a t i o n o f s t u d e n t s f
a c h i e v e m e n t i n Mathemat ics i n N i g e r i a n s e c o n d a r y s c h o o l s
p e r f o r m i n g t h e above f u n c t i o n s or p a r t s t h e r e o f ?
E v a l u a t i o n o f s t u d e n t s a c h i e v e m e n t i n Ma thema t i c s
i n N i g e r i a n s e c o n d a r y schoois, f rom t h e on set h a s been
c a r r i e d o u t by Mathema t i c s t e a c h e r s , e x t e r n a l examina-
t i o n b o d i e s and i n s t i t u t i o n s o f h i g h e r l e a r n i n g .
Ma thema t i c s t e a c h e r s used o ra l and w r i t t e n tests t o
. assess t h e i r s t u d e n t s for t h e p u r p o s e s of t h e f o l l o w i n g :
( i ) t e a c h i n g a n d learning; a n d
( i i ) p r o m o t i o n o f l e a r n e r s f r o m o n e form t o t h e
n e x t h i g h e r fo rm,
B e f o r e t h e i n t r o d u c l . o n of c o n t i n u o u s assessment,
almost a l l t e a c h e r s of o u r s e c o n d a r y s c h o o l s u s e d o n l y
t h e r e s u l t s of o n e e n d - o f - s e s s i o n examina .k ion f o r t h e
p u r p o s e of p r o m o t i o n . T h e e x t e r n a l e x a m i n a t i o n b o d i e s
a l s o u s e d r e s u l t s f r o m o n e set o f e x a m i n a t i o n s t a k e n
a t t h e e n d of t h e l e a r n e r s 1 c o u r s e t o award certifica-
tes t o c a n d i d d r s s , I n s t i t u t i o n s of h i g h e r l e a r n i n g
a p p r a i s e t h e m a t h e m a t i c s a c h i e v e m e n t s o f a s e c t i o n of
s e c o n d a r y s c h o o l s t u d e n t s who t a k e t h e M a t h e m a t i c s
c o m p o n e n t of t h e i n s ti t u t i o n s l e n t r a n c e e x a m i n a t i o n s ,
Hence t h e programme of e v a l u a t i o n of a c h i e v e m e n t s i n
e a c h s u b j e c t o f f e r e d by o u r s e c o n d a r y s c h o o l s t u d e n t s ,
h a s o f t e n be d i s c r i b e d as o n e - s h o t - p a p e r a n d pen
e x a m i n a t i o n , B e s i d e s , from t h e r e s e a r c h e r s
e x p e r i e n c e s , m o s t i f n o t a l l t h e e x a m i n a t i o n s or tests
i n M a t h e m a t i c s a p p r a i s e o n l y t h e c o g n i t i v e l e a r n i n g
o u t c o m e s . Many wel l -mean ing p e r s o n s cried o u t a g a i n s t
t h e a b o v e e v a l u a t i o n s y s t e m a n d t h a t h a s c o n t r i b u t e d
toward t h e i n t r o d u c t i o n of C o n t i n u o u s A s s e s s m e n t (CAI,
B e s i d e s t h e c o n c e r n e x p r e s s e d a b o u t the. s h o r t c o m i n g s
o f t P e e v a l u a t i o n sys t em, a f e w s t u d i e s have been
c a r r i e d o u t on t h e e v a l t l a t i o n o f Mathemat ics ach ievemen t
i n N i g e r i a n Szcondarv Schoo l s , Most of t h e few s t u d i e s
i n v e s t i g a t e d t h e d i f f e r e n c e s i n a c h i e v e m e n t a l o n g t h e
l i n e s o f : (i) s e x o f s t u d e n t s , (ii) s c h o o l t y p e and
( i i i ) s c h o o l l o c a t i o n , Soms a l s o looked in i . 0 t h ?
a t t i t u d e o f s t u d e n t s toward Mathemat ics , It h a s been
o b s e r v e d t h a t t h e r e e x i s t s ( a t t h e s e c o n d a r y s c h o o l
l e v e l ) s i g n i f i c a n t d i f f e r e n c e between t h e a c h i e v e m e n t s
i n Ma thema t i c s af t h e male and fema'e s t u d e n t s , t o
t h e a d v a n t a g e o f t h e ma le s t u d e n t s (Ogunyemi a n d B e t t i e ,
1974 ; O b i o m a a n d Ohuche, l 98O) , But Obibokun ( 1 9 7 0 )
r e p o r t e d t h a t t h e r e i s no d i f f e r e n c e between male and
f e m a l e s t u d e n t s ' a c h i e v e m e n t s i n Ma thema t i c s , R e c e n t l y
d z e i l o ( 1 9 8 8 ) i n h i s key-note a d d r e s s d u r i n g t h e
a n n u a l c o n f e r e n c e o f Ma thema t i ca l A s s o c i a t i o n of
N i g e r i a (MAN) p r e s e n t e d ample s t a t i s t i c s , s h w i n g t h a t
male s t u d e n t s pe r fo rm better t h a n t h e f e m a l e s t u d e c t s
i n s e c o n d a r y s c h o o l ~ a t h e m a k i c s , Obioma and Ghuche
( 1 9 8 0 ) an$ 'bbioma ( 1 9 8 2 ) o b s e r v e d t h a t s t u d b n t s i n $, :.
u r b a n en< ,kd%mn ts per formed b e t t e r i n Ma thcma t i c s t h a n
t h e s t d c k n t s i n non-urban env i ronmen t s . Obioma (1982
n o t e d t h a t s t u d e n t s i n c o - e d u c a t i o n a l s c h o c l s performed
s i g n i f i c a n t l y b e t t e r i n Ma thema t i c s t h a n s t u d e n t s i~
uni - sex s c h o o l s . The s t u d i e s c i t e d above mon i to red and
compared achievement.:; i n Ma thema t i c s o f sornc c a t e g o r i e s
o f s t u d e n t s and suggL:sted r e a s o n s for t h e o b s c r v c d
d i f f e r e n c e s . But t h e s t u d i e s d i d n o t embody d e s i g n s
o r p r o c e d u r c s t o i d e n t i f y t h e ca .uses o f s u c h
d i f f e r e n c e s , A l s o mos t o f t h e s t u d i e s c o n c e r n e d t h c
a s s e s s m e n t o f t h e c o g n i t i v e achiev-men t,
However, L a s s a (1978) c o n d u c t e d a s t u d y , a s p e c t
o f which i n v e s k i g z t e d t h e t e n N o r t h e r n s tatcsl
s e c o n d a r y s c h o o l s t d d e n t s ' a t t i t u d e toward Mathemat ics .
H e o b s e r v e d t h a t a l t h o u g h on t h e a v e r a g e s t u d e n t s '
a t t i t u d e toward Mathema t i c s i s n e g a t i v e , t h e boys
t e n d e d more toward t h e p o s i t i v e t h a n t h e g i r l s . I n
t h e v i e w s o f P h i l i p (19701, Banks (1964-1, Lassa (1975
and 1 9 7 6 ) a s p r e s e n t e d by L a s s a ( 1 9 3 6 ) i n h i s i n a u g u r a l
a d d r e s s , t h e r e a s o n s for t h e i d e n t i f i e d d i f f e r e n c e s
a n d poor p e r f o r m a n c e i n c l u d e :
( i ) f s a r o f t h e s u b j e c t a s pas sed on t o p u p i l s o r
s t u d e n t s by t e a c h e r s ;
( i i ) teachers' a t t i t u d e and e f f e c t i v e n e s s ;
. ( i i i ) s e x roles e x p e c t a t i o n s ;
( i v ) s o c i a l and i n t e l l e c t u a l f u n c t i b r i s ; and
( v ) p a r e n t a l i n f l u e n c e ,
R e c e n t l y A l i (1988) n o t e d t h a t t h e a t t i t u d e s o f
l e a r n e r s toward Matk~c mat..i.cs p r e d i s p o s e t h e e f f e c t i v e -
n e s s of t e a c h i n g .
Of c o u r s e i t i s p o s s i b l e t o d e s i g n p r o j e c t s i n
Mathematics t h a t w i l l , i f w e l l u sed ,
( i improve t e a c h e r s e f f e c t i v e n e s s ;
(ii) b r i n g o u t t h e p o s i t i v e social and i n t e l l e c t u a l
f u n c t i o n s o f Ma thema t i c s ; and
(iii) i l l u s t r a t e many female roles and f u n c t i o n s ,
as w e l l a s v e r y many male roles and f u n c t i o n s
t h a t t h e i r pe r fo rmances depend on some knowledge
and s k i l l s a c q u i r e d from Mathemat ics ,
I t i s i m p o r t a n t t o p o i n t o u t h e r e t h a t t h e a p p r a i s a l
o f s t u d e n t s a a f f e c t i v e pe r fo rmance i n Ma thema t i c s i s n e c e ss-
a r y s i n c e a c c o r d i n g t o Sawyer (1969) Mathemat i c s i s an
a c t i v i t y and a c t i v i t i e s depend g r e a t l y on t h e a t t i t u d e
of t h o s e pe r fo rming a c t i v i t i e s toward t h o s e a c t i v l k i e s ,
A t t h i s p o i n t , i t i s i m p o r t a n t t o o b s e r v e t h a t a t t l t u d e
i s n o t t h e o n l y component o f t h e a f f e c t i v e b e h a v i o u r a l
o b j e c t i v e s which t h e t e a c h i n g and l e a r n i n g o f
M a t h e m a t i c s o u g h t t o a c h i e v e , The o t h e r componcn ts,
i n t e r e s t , v a l u e and a p p r e c i a t i o n , nced a l s o be a p p r a i s e d ,
But i t may be much a w a s t e o f t i m e an@ r e s o u r c e s i f
s e p a r a t e i n s t r u m e n t s a r - d e s i g n e d t o m e a s u r e e a c h
component and a1 s o mtr.asure e a c h component a t d i f f ? r c n t
times, I t may, t h e r : ~ _ t o r e , be w o r t h w h i l e t o d e v e l o p o n e
i n s t r u m e n t t o a p p r a i s e s t u d e n t s ' a f f e c t i v e p e r E o r ~ l , ~ x i c e s
i n Mathemat ics , and a p p r a i s e them s i m u l t a n c o u s l ; ~ a s i s
t h e case w i t h t h e c o g n i t i v e pe r fo rmances , I n t h e same
l i g h t , t h e psychomotor b e h a v i o u r a l ou tcomes of
Ma thema t i c s e d u c s t i o n s h o u l d be m o n i t o r e d and a l l t h e
componen t s of psychomotor b e h a v i o u i t h a t c o u l d be
m o n i t o r e d t h r o u g h Ma thema t i c s s h o u l d be a s s e s s e d
s i m u l t a n e o u s l y . But b o t h t h e Ma thema t i c s t e a c h e r s and
t h e e x t e r n a l e x a m i n a t i o n b o d i e s iA2vo lved i n a s s e s s i n g
s t u d e n t s t a c h i e v e m e n t i n Ma thema t i c s u p to t h e p r e s e n t
d a y , c o n c e r n themselves w i t h t h e c o y n i t i v e o b j e c t i v e s .
However, now and t h e n , q u e s t i o n s a r e set, by t h e
t e a c h e r s or e x t e r n a l e x a m i n e r s , on c o n s t r u c t i o n s or
g r a p h w o r k s , b u t t h e psychomotor s k i l l s e x h i b i t e d by
s t u d e n t s / c a n d i d a t e s , i n r e s p o n d i n g to such q u e s t i o n s ,
a r e r a r e l y i f e v e r s c o r e d , But w i t h t h e i n t r o d u c t i o n
of C o n t i n u o u s Asses smen t , n o t o n l y s h o u l d t h e e x h i b i t e d
psychomotor o u t c o m e s / s k i l l s be s c o r e d and r e c o r d e d ,
2-3 CONTINUOUS ASSESSMENT --A I N N I G E R I A N SECONDARY . .-.-.- SCHOGLS ...-
The Nj g e r i a n N a t i o n a l P o l i c y on E d u c a t i o n (1981 ,
p, 4 s t i p u l a t e s t h a t vq t .c luca t iona l a s s e s s m e n t and
e v a l u a t i o n w i l l be 2 ' b w a l i s e d by b a s i n g them i n v-hole
o r i n p a r t on C o n t i n ~ ~ o u s Assessmentsq (CAI. By t h e
document , CA w a s supposed to be started by 1 9 8 2 i n a l l
t h e s e c o n d a r y s c h o o l s i n N i g e r i a w i t h t h e t dke -o f f o f
t h e 3-3 s e c o n d a r y s c h o o l system. But o n l y t h e F e d e r a l
Government Seconda ry S c h o o l s (FGSS) and s e c o n d a r y
schools i n s e v w states s t a r t e d t o t r y o u t w h a t t h e
p o l i c y s t i p u l a t e s , D e s p i t e t h e c l a i m t h a t some
i n s e r v i c e t r a i n i n g and workshops on CA were o r g a n i s e d fo r
t e a c h e r s , t h e t h e n r e m a i n i n g t w e l v e states o f t h e f e d e r a -
t i o n i n d i c a t e d t h e y w e r e n o t y e t r e a d y . However, many
more states, as i t were, have j o i n e d i n o p e r a t i n g t h e
3-3 s e c o n d a r y s c h o o l sys t em,
C o n t i n u o u s Asses smen t h a s been d e f i n e d v a r i o u s l y
by p e o p l e b u t t h e F e d e r a l M i n i s t r y of E d u c a t i o n ' s
handbook on CA (1985) v i e w s i t as a method of f i n d i n g
o u t w h a t t h e p u p i l / l e a r n e r h a s g a i n e d from l e a r n i n g
a c t i v i t i e s i n t e r m s o f knowledge, t h i n k i n g , r e a s o n i n g ,
c h a r a c t e r deve lopment and i n d u s t r y . Indeed an a s s e s s m e n t
programme i s s a i d to be C A if it i s systeri lal ; ic ,
c o m p r e h e n s i v e , c u m m u l a t i v e a n d g u i d n r x e o r i e n t e d , The
p r o p e r i m p l e m e n t a t i o n o f C R demands t h a t :
( i) t h e r e m u s t bc a p l a n o r g u i d e l i n e w h i c h s p e c i f i e s ,
w h a t , idhen ~i,: , : l how to d o t h e t h i n g s t h a t s h o u l d
be d o n e ;
(ii) a t e a c h t i m e or p e r i o d , t h r o u g h o u t t h e
l e a r n e r 1 s s t a y i n s c h o o l , h i s / h e r p e r f o r m a n c e s
o r a c h i e v e m e n t s i n t h e affective, c o g n i t i v e and
p s y c h o m o t o r d o m a i n s of e d u c a t i o n a l o b j e c t i v e s ,
s h o u l d be m o n i t o r e d a n d r e c o r d e d ;
(iii) t h e r e p o r t i n g , c o n s i d e r a t i o n a n d j u d g e m e n t of
a l e a r n e r ' s p e r f o r m a n c e s or a c h i e v e m e n t s a t a n y
g i v e n time ( i n c l u d i n g t h e t i m e of h i s / h e r p a s s i n g
o u t ) s h o u l d t a k e a c c o u n t of a n d i n c l u d e ( n o t
n e c e s s a r i l y b y a s i m p l e a r i t h m c t i c a v e r a g i n g )
a l l t h e m o n i t o r e d a n d r e c o r d e d p e r f o r m a n c e s of
t h e l e a r n e r u p to t h e t i m e ; a n d
( i v ) t h o s e m o n i t o r e d a n d r e c o r d e d p e r f o r m a n c e s of t h e
l e a r n e r s h o u l d be u s e d , a t a n y p o i n t i n time, t o
g u i d e , d i r e c t a n d h e l p t h e l e a r n e r t o i m p r o v e h i s /
h e r p e r f o r m a n c e s a n d f i n a l l y to u s e t h e i n f e r e n c e s
d r a w n from t h e records of h i s / h e r p e r f o r m a n c e s to
l e a d t h e l e a r n e r t o a c c e p t a c a r e e r i n an a -ea where
h e / s h e h a s c o m p a r a t i v e a d v a n t a g e o v e r vtost o t h e r s ,
C o n t i n u o u s Assessment i s t h e r e f o r e p r e f e r r e d t o
t h e one-shot paper-n.14- pen e x a m i n a t i o n , T h i s i s ".he
c x e as p re sen t , ?d by Yoloye e t a1 (1980, pp, 4-51, when
t h a y s a i d t h a t CA:
e n s u r e s a c c o u n t i n g f o r pe r fo rmance t h r o u g h o u t
s c h o o l i n g ;
p romotes t h e r e a d i n e s s o f t e a c h e r s t o i n t r o d u c e
i n n o v a t i o n s ;
e n s u r e s t h a t a s s e s s m e n t i s an i n t e g r a l p?rk of
t e a c h i n g and l e a r n i n g ;
p r o v i d e s g u i d a n c e f o r m e a n i n g f u l l e a r n i n g and
c a r e e r c h o i c e ;
p r o v i d e s t h e d a t a f o r t h e a s s e s s m e n t of
i n s t r u c t i o n a l methods and m a t e r i a l s ; and
r e d u c e s t h e t e n s i o n c r e a t e d by e x a m i n a t i o n s
and a lso t h e o c c a s i o n s o f e x a m i n a t i o n m a l -
p r a c t i c e s ,
For t h e above merits of CA t o b e r e a l i z e d , t h e Handbook
o n CA s tates t h a t a v a r i e t y o f a s s e s s m e n t i n s t r u m e n t s
s h o u l d b e u s e d , T h e s e i n s t r u m e n t s i n c l u d e s ; c ~ ~ s ~ w o ~ I c ,
t a k e home e x e r c i s e s / ' a s s i g n m e n t s J , tests, p r o j e c t s ,
q u e s t i o n n a i r e s , ancedo ta l r e c o r d s , r a t i n g scales and
s o c i o g r a m s . I t i s f u r t h e r r e c o m m e n d ~ d t h a t t h e r e s h o u l d
be a conrmj-ttee t o m o n i t o r a n d su2erv :c .z~ t h e c o n d u c t of
CA a t tach of t h e f o l l o w i . n q l e v e l s ; t h e s c h o o l , t h e
l o c a l gove rnmen t cd c? Livn !:oard, t h e z o n a l e d u c a t j - o n
b o a r d and t h e ,qta LC - J u c a t i o n b o a r d . T h e s c recommenda-
t i o n s are weiL mde, b u t are t h e y b e i n g r i g h t l y and
a d e q u a t e l y imp lemen ted?
T h e r e are many factors , t h a t have b e e n i d e n t i f i e d
o r o b s e r v e d , wh ich h i n d e r t h e c o n d u c t o f CA i n s c h o o l s ,
I p a y e ( 1 9 8 2 ) n o t e d t h a t many t e a c h e r s g o t h r o l ~ g h t h e i r
N . C e E o , B,Ed, or e v e n t h e P,G-D,E, programmes w i t h o u t
g r a s p i n g enough, t h e b a s i c c o n c e p t s ( a n d s k i l l s ) i n
measurement and me thods of a p p l y i n g measurement
t e c h n i q u e s , R e s u l t s o f r e s e a r c h s t u d i e s show t h a t
t e a c h e ~ . s i n o u r s c h o o l s y s t e m s d o n o t p o s s e s s many of
t h e c o m p e t e n c i e s n e c e s s a r y for t h e deve lopmen t a n d
v a l i d a t i o n of a s s e s s m e n t i n s t r u m e n t s (Epunam, 1 9 8 3 ;
H a r b o r - I b e a j a , 1983; H a r b o r - I b e a j a and Nworgu, 1 9 8 3 ;
Iwuchukwu, 1984 ; Mkpa, 1985 ; and Nwagu, 1985)- A l s o
O k u n r o t i f a (1977 ) o b s e r v e d t h a t most of t h e a s s e s s m e n t
i n s t r u m e n t s i n u s e i n o u r s c h o o l s t o d a y are for t h e
measurement of c o g n i t i v e ou t comes of l e a r n i n g ,
It, t h e r e f ore, i m p l i e s t h a t t.eac h e r s s h o u l d be rnaclc; .to
know what- i n s t r u m e n t s t o u s e a n d c l s ~ ? c q u i r e t h e
c o m p e t c r ~ c i ~ c s f o r devcl .opi .ng the i n s t r u m e n t s , Z e s i d e s
t h e fac tors m i l i t a t - 2 1 ~ c ;:.(di:ls'i t h e c o n d u c t o f CA z s
n e n t i o n e d abovt::, Ycj:i.<,>(e e t a1 (1380) p o i n t o u t t h a t
t h e p r o b l e m area i n t h e u s e of CA r e s u l t s j.s the
c o m p a - & l l i ty of s t a n d a r d s . T h i s prob.!.e:r, a:. .i sos f j:on
d i f f e r e n c e s i n t h e : a s s e s s m e n t i n s t r u m e n t s , a x 1 ~ ~ l . s o
s c o r i n g a n d g r a d i n g of s t u d e n t s ' r e s p o n s e s . The
i n c o m p a r a b i l i t y of s t a n d a r d s i s w o r s e n e d b y the
d i f f e r e n c e s i n t h e l a t i t u d e s o r r a n g e s of scores
o b t a i n a b l e from v a r i o u s a s s e s s m e n t i n s t r u m e n t s
a d m i n i s t e r e d o n s t u d e n t s i n v a r i o u s s c h o o l s a n d states.
T h e d i f f e r e n c e s , referred t o a b o v e , become e v i d e n t
i f t h e s c o r i n g or g r a d i n g s y s t e m s , i n s c h o o l s o f Anambra
s t a t e g o v e r n m e n t are c o n t r a s t e d w i t h t h e s c o r i n g o r
g r a d i n g s y s t e m s i n most F e d x a l Government S e c o n d a r y
S c h o o l s or U n i t y S c h o o l s (FGSS). I n Anambra s t a t e
s e c o n d a r y s c h o o l s , t h e f o l l o w i n g modes of a w a r d i n g
m a r k s o b t a i n :
( i) t h r e e tests or c l a s s w o r k s or a p r o j e c t ( w h i c h , a s
t h i s r e s e a r c h e r h a s o b s e r v e d , d o e s n o t , i n most
cases, q u a l i f y a s a p r o j e c t ) a n d a m i d - t e r m tes t
are a d m i n i s t e r e d on s t u d e n t s d u r i n g t h e first
h a l f of e v e r y term, and a s t u d c n t t s r e s p o n s e t o
e a c h o f t h e a s s e s s m e n t i n s t r u m e n t i s marked o u t
of a maximum of t e n marks s u c h t h a t t h e f o u r
a s s e s s m e n t s : c o u n t for f o r t y p e r c e n t (40%)
o f t h e term's score);
(ii) t h r e e a s s e s s m e n t s made up of tests o r c l a s s w o r k s
o r / a n d a p r o j e c t a re a d m i n i s t e r e d on s t u d e n t s
d u r i n g t h e second-ha l f of e a c h t e r m a n d a
s t u d e n t ' s r e s p o n s e t o e a c h of t h e a s s e s s m e n t
i n s t r u m e n t i s marked o u t of a maximum of t e n
marks ( s u c h t h a t t h e t h r e e a s s e s s m e n t s a c c o u n t
f o r t h i r t y p e r c e n t (30%) of t h e termt s scores);
and
(iii) a n end-of-term e x a m i n a t i o n i s a d m i n i s t e r e d a t
t h e end of e a c h term on s t u d e n t s a n d a s t u d e n t ' s
r e s p o n s e t o t h e e x a m i n a t i o n i s s c o r e d o u t o f a
maximum of 30 marks ( t h i s a c c o u n t s f o r t h i r t y
p e r c e n t (30%) o f t h e term's s c o r e ) .
I n all, t h e s e scores amount t o a maximum of 100% p e r
term, Most of t h e FGSS or U n i t y S c h o o l s a d m i n i s t e r
a b o u t f i v e a s s e s s m e n t i n s t r u m e n t s on t h e i r s t u d e n t s
p e r term (made u p o f tests, a s s i g n m e n t s o r / a n d a p r o j e c t
and a n end-of-term e x a m i n a t i o n ) . Each of t h e s t u d e n t s 1
. r e s p o n s e s t o e a c h of t h e a s s e s s m e n t i n s t r u m e n t s i s
s c o r e d o u t o f a maximum of 100%. A t t h e end some
w e i g h t e d a v e r a g i n g i s done such t h a t each s t u d e n t ' s
s c o r e i n each s u b j e c t i s o u t of a maximum of 100% p e r
term. A 1 though i n 30th cases a s t u d e n t ' s score, p e r
s u b j e c t , p a r term, i s o u t o f a maximum of 100% t h e r e
e x i s t d i f f e r e n c e s ,
T o mark any m e a n i n g f u l Mathemat ics tes t o r
ctss ignments ( t h a t is/are n o t mu1 t i p l e c h o j ce test
items) o r a p r o j e c t o u t of a maximum of 1 0 marks i s
v e r y d i f f i c u l t or e f f e c t i v e l y v a l u e l e s s , To do & may
lead to: l e a v i n g o u t s c o r i n g some n e c e s s a r y s t e p s ,
i n t e r m e d i a t e correct a n s w e r s or some d e g r e e s of
a c c u r a c y ; and t h e a d m i n i s t r a t i o n of tests composed
o f o n e or two test i t e m s e a c h t i m e . A l s o t h e f a c t
t h a t d e c i s i o n s based on scores o u t o f a maximum of
1 0 marks d i f f e r from d e c i s i o n s based on scores o u t o f
a maximum o f 1 0 0 marks s h o u l d be n o t e d , F o r i n s t a n c e ,
7 3 s e v e n or t h r e e o u t of t e n marks (n or m) i s e q u i v a l e n t
70 30 t o s e v e n t y or t h i r t y o u t o f hundred marks (lTo o r - 100
a s f r a c t i o n b u t i n s t a t i s t i c a l t e r m s / d e c i s i o n s /
judgements (which a s s e s s m e n t r e s u l t s are u s e d f o r )
t h e t w o m e a s u r e s d i f f e r . The d i f f e r e n c e i s e v i d e n t
49
' if w e t a k e a bench mark of e a c h sys t em o f mzasure , f i v e
o u t o f t e n and f i f t y o u t o f hundred and c o n s i d e r
w h e t h e r s e v e n o u t o f tell o r s e v e n t y o u t o f hundred
d i f f e r s i g n i z i c a n t l y from t h e bench mark, A s i m p l e
c h i - s q u a r e test shows t h a t w h i l e seven o u t of t e n does
n o t d i f f e r s i g n i f i c a n t l y from f i v e o u t o f t e n , s e v e n t y
o u t o f hundred d i f f e r s s i g t l i f i c a n t l y f rom f i f t y o u t o f
a hundred.
Ano the r a r e a f u l l of d i f f i c u l t i e s , d i s p a r i t i e s and
errors, i s t h e s c o r i n g o f s t u d e n t s ' a c h i e v e m e n t s i n
t h e a f f e c t i v e and psychomotor domalns o f e d u c a t i o n a l
o b j e c t i v e s , I n t h e s e c o n d a r y s c h o o l s i n Anambra S t a t e
f o r i n s t a n c e , some of the a t t r i b u t e s o r s k i l l s , i n d i c a t e d
i n s t u d e n t s 1 a n n u a l r e p o r t fo rms , t o be s c o r e d as
a f f e c t i v e or psychomotor ou tcomes /ob j e c t i v e s , are o u t cf
p l a c e . Some o f s u c h o b j e c t i v e s / l e a r n i n g outcomes ,
i n d i c a t e d i n t h e i r r i g h t f u l domains , are too broad as
i t e m s to b e s c o r e d on a f i v e p o i n t s c a l e , F o r examples :
( i ) ''human r e l a t i o n s i n p r o j e c t e x e c u t i o n a n d
c o - o p e r a t i v e s i 1 r e q u i r e d t o b e r a t e d as psychomotor
ou tcomes are r a t h e r a f f e c t i v e l e a r n i n g outcomes ;
( ii) W e r s a t i l i ty/adaptability/resourcefulness/
a b i l i t y t o come up w i t h a l t e r n a t i v e suggestion^^^,
p l a c e d u n d e r a f f e c t i v e ou tcomes , a r e more of
c o g n i t i v e ou t comes ; and
( i i i ) tlhuman r e l a t i o n s i n p r o j e c t e x e c u t i o n and
c o - o p e r a t i v e n e s s I 1 or l t ve r sa t i l i t y / adap tab i l i t y /
r e s o u r c e f u l r ~ ~ ~ s s / a b i l i t y t o come u p w i t h a l ter-
n a t i v e s u g g e s t i o n s " , as d g r o u p o f l e a r n i n g
ou t comes i s c e r t a i n l y too broad, v a g u e and
complex t o be scored as an i t e m o n a f i v e p o i n t
r a t i n g s c a l e (Anambra S t a t e M i n i s t r y o f
E d u c a t i o n ; s t u d e n t s 1 t e r m l y and a n n u a l r e p o r t
forms 1.
To worsen m a t t e r e , i n most of t h e s c h o o l s o p e r a t i n g CA,
a s t u d e n t i s r a t e d on t h e s e a f f e c t i v e and psychomotor
l e a r n i n g ou t comes o n l y by h i s / h e r form m a s t e r / m i s t r e s s ,
T h e s e form masters/mistresses k e e p n o r e c o r d s whatso-
e v e r o f s t u d e n t s ' achievements/performances i n t h e
a f f e c t i v e and psychomotor l e a r n i n g ou t comes d u r i n g t h e
s e s s i o n , But t h e y rate t h e s e s t u d e n t s a n d award them
some p e r c e n t a g e scores on t h e s e l e a r n i n g ou t comes a t
t h e end of t h e y e a r / s e s s i o n . ( T h e s e o b s e r v a t i o n s a re
made by t h e r e s e a r c h e r w h i l e / d u r i n g s u p e r v i s i o n t o u r s
of s c h o o l s i n and d u r i n g CA c o m m u n i t i e s ~ m e e t i n g s of
Nsukka LoG0A, and z o n a l e d u c a t i o n l e v e l s ) , One t h e n
' w o n d e r s how t h e form masters and mistresses come by
t h e p e r c e n t a g e scores awarded t o s t u d e n t s as t h e i r
a f f e c t i v e , c o g n i t i v e and psychomotor pe r fo rmances
F i n a l l y , i t is s u r p r i s i n g t h a t a scale f o r r a t i n g
i n t e l l e c t u a l o r c o g n i t i v e pe r fo rmances o f s t u d e n t s
b e s i d e t h e scores awarded to them from t h e v a r i o u s
s u b j e c t s a s s e s s m e n t i n s t r u m e n t s . I n most s c h o o l s , as
o b s e r v e d from s t u d e n t s ' i n t e l l e c t u a l a c h i e v e m e n t
r a t i n g s , scores d o n o t i n a n y way reflect a r o u g h o r
m e a n i n g f u l a v e r a g e o f t h e s t u d e n t s ' scores i n t h e i r
v a r i o u s s u b j e c t areas,
To min imize o r overcome t h e s e numerous problems,
common or similar i n s t r u m e n t s f o r u s e i n a s s e s s i n g
s t u d e n t s i n o u r s c h o o l s s h o u l d be deve loped . Of
c o u r s e , most o f t h e t i m e s when i n s t r u m e n t s f o r
a s s e s s m e n t are men t ioned , p e o p l e 1 s minds r u n t o c l a s s -
work a s s i g n m e n t s , tests and e x a m i n a t i o n s , T h i s i s
t h e case, s i n c e t h e s e make u p a l m o s t a l l t h e a s s e s s m e n t
i n s t r u m e n t s u sed i n o u r s c h o o l s p r e s e n t l y , T h i s
s i t u a t i o n a p p l i e s more so t o Mathema t i c s e d u c a t i o n i n
o u r s c h o o l s , Almost a l l t h e Ma thema t i c s e x a m i n a t i o n s ,
tests, c o n t e s t s , a n d a s s i g n m e n t s w r i t t e n b y s e c o n d a r y
s c h o o l s t u d e n t s i n t h i s c o u n t r y t a k e t h e form o f
m u l t i p l e c h o i c e or e s s a y - t y p e tes t items, A l l 05 them
a p p r a i s e ( f o r a l l i n t e n t s and p u r p o s e s ) o n l y t h e
c o g n i t i v e a c h i e v e m e n t s of s t u d e n t s / c a n d i d a t e s , I t is ,
t h e r e f o r e , i m p o r t a n t t o d e v e l o p o t h e r i n s t r u m e n t s t h a t
c a n a p p r a i s e i n a d d i t i o n t o t h e c o g n i t i v e or s e p a r a t e l y
t h e a c h i e v e m e n t s o f t h e s t u d e n t s / c a n d i d a t e s i n t h e
o t h e r t w o domains as r e q u i r e d fo r CA, N e c e s s a r y for
t h e deve lopment of i n s t r u m e n t s t h a t c a n be used t o
a p p r a i s e ach ievemen t i n t h e v a r i o u s b e h a v i ~ u r a l
domains i s a p r o p e r g r a s p o f t h e p r o f i l e s t h a t make
u p t h e b e h a v i o u r a l domains .
2-4 EXPECTED LEARNING OUTCOMES OF MATHEMATICS EDUCATION -- The comprehens ive a s p e c t of CA i m p l i e s t h a t t h e
a c h i e v e m e n t s of s t u d e n t s i n t h e t h r e e domains
( a f f e c t i v e , c o g n i t i v e and psychomotor ) of e d u c a t i o n a l
o b j e c t i v e s s h o u l d b e a p p r a i s e d , r e c o r d e d and r e p o r t e d ,
The e v a l u a t i o n of ach ievemen t i n a n y of t h e g i v e n areas
demands t h a t t h e e v a l u a t o r s h o u l d be able t o s p e c i f y
and measu re t h e s k i l l s o r a t t r i b u t e s when s t u d e n t s /
c a n d i d a t e s e x h i b i t / m a n i f e s t such s k i l l s o r a t t r i b u t e s ,
Of c o u r s e much work h a s been d o n e i n c a t e g o r i z i n g human
b e h a v i o u s i n to some d i s t i n g u i s n a b l e components ,
' T y l e r (1986, p, 7 9 ) s a y s :
t h e a n a l y s i s o f human b e h a v i o u r i ? t o t h r e e c h i e f c a t e g o r i e s , c o g n i t i v e , a f f e c t i v e a n d psychomotor , i s a c o n s t r u c t employed t o d i f f e r e n t i a t e c e r t a i n hu, a n re? z t i o n r o u g h l y s i m i l a r t o t h e age-old d i s - t i n c t i o n s between t h i n k i n g , f ~ e l i n g and a c t i n g .
T y l e r ' s a s s e r t i o n i s t h e case s i n c e t h e r e i s h a r a l y
any human react!-on t h a t f a l l s c o m p l e t e l y i n t o o n l y o n e
o f t h e s e c a t e g o r i e s * But f o r t h e s a k e o f b r e v i t y ,
emphasy and c o n c e r n , e a c h human r e a c e i o n or p r e c i s e
b e h a v i o u r h a s been c a t e g o r i z e d a s b e i n g a f f e c t i v e ,
c o g n i t i v e or psychonto tor , depend ing on w h i c h aspc-c ts
s t a n d o u t s h a r p l y . S o l v i n g a Mathematics problem, f o r
example , i s u s u a l l y c lassif ied as c o g n i t i v e , even though
w r i t i n g numbers o r symbo l s ( and sometimes c o n s t r u c t i n g
and d rawing f i g u r e s or objects) i s i n v o l v e d and t h e
p r o c e s s i s accompanied by some f e e l i n g t o n e . ( T y l e r ,
1 9 8 6 ) .
The a b o v e p o s i t i o n of n o n - e x c l u s i v e n e s s and close-
r e l a t e d n e s s o f human b e h a v i o u r i n terms o f i t s b e i n g
a f f e c t i v e , c o g n i t i v e or psychomotor i s attested to/
c o r r o b o r a t e d by a number o f s t u d i e s . S i lve rman (1973)
u s e d 684 low-achiev ing 4 t h , 5 t h and 6 t h g r a d e s s t u d e n t s
' o f f i v e t i t t l e I N e w York p u b l i c s c h o o l s i n a n
e x p e r i m e n t a l s t u d y , Among h i s f i n d i n g s i s t h a t
a t t i t u d e toward Ma thema t i c s i s d i r e c t l y r e l a t e d t o
m a s t e r y of t h e u n i t of work i n Ma thema t i c s employed i n
t h e s t u d y , L a s s a ( 1 9 7 6 ) u sed 240 TTC s t u d e n t s i n
N o r t h e r n N i g e r i a t o i n v e s t i g a t e t h e r e l a t i o n s h i p
be tween a c h i e v e m e n t i n Ma thema t i c s and a t t i t u d e toward
Mathemat ics . H e f o u n d o u t t h a t fo r t h e to ta l s a m p l e
a c h i e v e m e n t is s i g n i f i c a n t l y r e l a t e d t o a t t i t u d e ,
A l s o Mukher jee and U m a r ( 1978) i n v o l v e d 1 0 0 s e c o n d a r y
s c h o o l Form f o u r b o y s i n a n e x p e r i m e n t . T h e i r f i n d i n g s ,
among o t h e r s , i n d i c a t e t h a t post-tests scores o f t h e
e x p e r i m e n t a l g r o u p i n a c h i e v e m e n t and a t t i t u d e a r e
p o s i t i v e l y and s i g n i f i c a n t l y c o r r e l a t e d , However, none
of t h e s t u d i e s c i t e d h e r e i n v e s t i g a t e d t h e r e l a t i o n s h i p
be tween psychomotor p e r f o r m a n c e and c o g n i t i v e or
a f f e c t i v e pe r fo rmance . A l s o none of t h e i n s t r u m a n t s
u s e d i n v o l v e d p r o j e c t s i n Ma thema t i c s , S t r u d e r (1971)
u s e d f o u r t h and s i x t h g r a d e s s t u d e n t s i n a correla-
t i o n a l s t u d y t o d e t e r m i n e w h e t h e r d i s c o v e r y me thods
( a s d i f f e r e n t from e x p o s i t o r y me thods ) i n Ma thema t i c s
i s c o r r e l a t e d t o c r e a t i v e t h i n k i n g and a t t i t u d e t oward
Ma thema t i c s . H e f ound among o t h e r t h i n g s t h a t none o f
' b o t h methods p roduced a s i g n i f i c a n t a t t i t u d i n a l c h a n g e
toward Mathemat ics . I t i s t h e r e f o r e n e c e s s a r y t o
e x p l o r e o t h e r a p p r o a c h e s of t e a c h i n g Ma thema t i c s t h a t
may r e s u l t i n s i g n i f i c a n t p o s i t i v e a t t i t u d i n a l c h a n g e
toward Ma thema t i c s md a l s o e x p o s e s t u d e n t s t o
p e r f o r m a n c e s i n t h e psychomotor domain. O f c o u r s e
s u c h an e x p l o r a t i o n r e q u i r e s t h e knowledge of w h a t
c o n s t i t u t e p e r f o r m a n c e s i n t h e v a r i o u s domains .
From l i t e r a t u r e , t h e a f f e c t i v e o b j c c t i v e s of
e d u c a t i o n i n c l u d e s u c h s u b c a t e g o r i e s a s : i n t e r e s t ,
a t t i t u d e , v a l u m g , and a p p r e c i a t i o n ( T y l e r 1986).
I n t e r e s t i s a t e n d m c y t o p r e f e r or e n g a g e i n a
p a r t i c u l a r t y p e of o b j e c t o r a c t i v i t y when t h e
G h o o s i i n d i v i d u a l h a s o p t i o n s . I n t e r e s t i m p l i e s i a n ovject
or a c t i v i t y when the i n d i v i d u a l h a s t h e t i m e or
o p p o r t u n i t y f a r v o l u n t a r y s e l e c t i o n , Deve lop ing
s t u d e n t i n t e r e s t i n Ma thema t i c s and m o n i t o r i n g same
s h o u l d be t h e c o n c e r n of S C ~ O O ~ S ~ T h i s i s t h e case
s i n c e i t i s o n l y t h r o u g h t h e u s e of some of a s t u d e n t ' s
free t i m e t o s t u d y M a t h e m a t i c s t h a t t h e s t u d e n t c a n
c o n s o l i d a t e what -ever h e / s h e h a s g a i n e d i n M a t h e m a t i c s
classes. A t t i t u d e i m p l i e s t h e f e e l i n g of f a v o u r a b l e -
n e s s toward some ' p s y c h o l o g i c a l o b j e c t ' . A t t i t u d e
* e n t a i l s t h e deZelopment of f e e l i n g o r t h e c o n t r o l of
f e e l i n g s i n c e r t a i n s i t u a t i o n s , T y l ~ r (1986) i s of t h e
v iew t h a t s u b j e c t t e a c h e r s s h o u l d s e e k t o open t h e
'minds1 and ' h e a r t s 1 o f t h e i r s t u d e n t s t o more v a r i e d
c o n s i d e r a t i o n s and t a a w i d e r r a n g e o f e x p e c t a t i o n s
and e x p e r i e n c e s t h a n t h e y h a v e . c o n s i d e r e d o r e n c o u n t e r e d
i n t h e p a s t , I n T y l e r ' s (1986 , p. 8 5 ) o p i n i o n "human
b e i n g s o f t e n d e v e l o p an e m o t i o n a l a t t a c h m e n t t o t h e i r
b e l i e f s y s t e m s , t h e i r h a b i t s , t h e i r p a t t e r n s of a t t e n -
t i o n and a c t i v i t y t h a t c l o s e o f f new p e r c e p t i o n s , new
i d e a s new a c t i v i t i e s , new s t a n d a r d s and new goals.If The
above o p i n i o n seems, to d e s c r i b e a p t l y t h e c o n d i t i o n
of most o f s e c o n d a r y s c h o o l s t u d e n t s ( o r r a t h e r m o s t
p e o p l e who come i n c o n t a c t w i t h Ma thema t i c s ) w i t h
r e s p e c t t o Mathema t i c s , Most s e c o n d a r y s c h o o l s t u d e n t s
feel and b e l i e v e t h a t Ma thema t i c s is t o u g h and meant
f o r t h e g i f t e d , The above f e e l i n g and belief p u t them
o f f from a t t e n d i n g t o Mathema t i c s a s t h e y ought .
T h e r e f o r e , t h e deve lopmen t o f g r e a t e r o p e n n e s s t o
Mathema t i c s o r t h e deve lopmen t o f o b j e c t i v e a n d f a v o u r -
a b l e r e s p o n s e s w i l l l e a d s e c o n d a r y s c h o o l s t u d e n t s t o
pe r fo rm c o m p a r a t i v e l y b e t t e r i n Ma thema t i c s t h a n t h e y
now do. S t u d e n t s f r e c e p t i o n of t h e p a r t i c u l a r matters
w i t h which Ma thema t i c s d e a l s , s h o u l d be a p r i o r i t y
o b j e c t i v e o f o u r s c h o o l s . A s t h i n g s s t a n d , t h e p r e s e n t
methods and t e c h n i q u e s o f t e a c h i n g and l e a r n i n g
Ma thema t i c s have f a i l e d us. Expos ing s t u d e n t s t o
p r o j e c t s as a s u p p l e m e n t to o t h e r t e c h n i q u e s may d o t h e
job,
Va lu ing a s an e d u c a t i o n a l o b j e c t i v e b e l o n g s t o * .
b o t h a f f e c t i v e and c o g n i t i v e domains. However, i f
p u r e l y a f f e c t i v e a s p e c t o f v a l u i n g i s b e i n g c o n s i d e r e d ,
t h e n i t i n v o l v e s m a t t e r s of d e e p e m o t i o n a l commitment, -.
To c h a r a c t e r i s e t h i s e m o t i o n a l d imens ion , v a l u e s may
t h e r e f o r e be s e e n a s t n o s e c h e r i s h e d objects, ac t iv i t ies
o r i d e a s which d r i v e or d i r e c t t h e i n t e r e s t s , a t t i t u d e s
and s a t i s f a c t i o n s 02 i n d i v i d u a l s c h e r i s h i n g t h e o b j e c t s ,
a c t i v i t i e s or i d e a s . Making s t u d e n t s v a l u e Mathemati ,cs
and a s s e s s i n g t h e i r rates and l e v e l s of v a l u i n g
l~i la tnernat ics , s h o u l d o f n e c e s s i t y be p a r t of t h e t e a c h i n g
a n d l e a r n i n g o f F l a t h e m a t i c s i n s c h o o l s , Here a g a i n ,
t h e r e i s t h e necd t o d e v e l o p and u s e new t e c h n i q u e s o f
t e a c h i n g a n d a s s e s s m e n t i n s t r u m e n t s s i n c e t h e o n e s i n
u s e have n o t produced s a t i s f a c t o r y r e s u l t s , T h i s i s
p a r t o f wha t m o t i v a t e d t h i s s t u d y ,
' the l a s t b u t t h e h i g h e s t l e v e l of t h e a f f e c t i v e '
l e a r n i n g ou tcomes i s a p p r e c i a t i o n . T y l e r (1986) i s
' o f t h e o p i n i o n t h a t t h e term a p p r e c i a t i o n i n t h e l i s t
o f e d u c a t i o n a l o b j e c t i v e s i s used mosr. i n t h e f i e l d s
o f l i t e r a t u r e , a r t a n d m u s i c ,
H e went f u r t h e r t o s u g g e s t t h a t when t h e term i s
used i n t h e f i e l d s o f s c i e n c e , social s c i e n c e and
Mathematics, i t s meaning seems l a r g e l y t o b e l o n g to
c o g n i t i o n , But I b e l i e v e t h a t s i n c e Ma thema t i c s d e a l s
w i t h :
( i ) p a t t e r n s o f i d e a s and r e a s o n i n g ;
( i i ) p a t t e r n s a n d g e n e r a t i o n of p l a n e s h a p e s and
s o l i d o b j e c t s ; and
(iii) p a t t e r n s and r u l e s o f r e l a t i o n s h i p s ( t h o u g h a l l
t h r e e , are d e v o i d o f t r a p p i n g s o f p a i n t and
p a g e a n t r y ) , Ma thema t i c s c a n no less be a p p r e c i a t e d
t h a n works o f ar t ,
T h i s ~ r g u e m e n t p o s s i b l y a p p l y t o t h e s c i e n c e s . H o w e v s ,
w h e t h e r l a r g e l y c o g n i t i v e or e n t i r e l y a f f e c t i v e ,
a p p r e c i a t i o n o f a n 'object ' l e a d s to:
d e r i v i n g s a t i s f a c t i o n i n t h e ' o b jectt
a p p r e c i a t e d ;
d e s i r i n g more o f t h e t h i n g a p p r e c i a t e d ;
d e s i r i n g to know more o f t h e t h i n g a p p r e c i a t e d ;
d e s i r i n g to e x p r e s s o n e - s e l f c r e a t i v e l y ;
i d e n t i f i c a t i o n o f o n e s e l f w i t h the o b j e c t
a p p r e c i a t e d ;
( v i ) d c a i r i n g t o e v a l u a t e t h e t h i n g a p p r e c i a t e d ; and
( v i i ) d e s i r i n g t o c l a r i f y o n e 1 s own t h i n k i n g w i t h
r e g a r d t o t h e l i f e p roblems r a i s e d by t h i n g
a p p r e c i a t e d ,
Fo l lowing from w h a t M a t h e r . a t i c s d e a l s w i t h and w h a t
a p p r e c i a t i o n e n t a i l s , i t becomes clear t h a t a v e n u e s
and s i t u a t i o n s abound t o make s e c o n d a r y s c h o o l s t u d e n t s
a p p r e c i a t e Mathemat ics , One o f s u c h a v e n u e s or
s i t u a t i o n s i s t h e u s e o f m e a n i n g f u l p r o j e c t s i n t e a c h i n g ,
l e a r n i n g and a s s e s s i n g a c h i e v e m e n t i n Ma thema t i c s and
t h i s v i s i o n mc t i v a t e d t h i s s t u d y .
It may be w o r t h w h i l e a t t h i s p o i n t , t o n o t e t h a t
Krea thwohl (1970) and h i s associates c a t e g o r i s e d t h e
a f f e c t i v e l e a r n i n g outcomes /ob jec t i v e s i n t o L taxonomy.
The taxonomy i n c l u d e s i n t h a t o r d e r :
(i r e c e i g i n g (ii) r e s p o n d i n g (iii) v a l u i n g
( i v ) o r g a n i s a t i o n , and ( v ) c h a r a c t e r i s a t i o n .
Krea thwohl (1970) and h i s a s s o c i a t e s w r i t e t h a t :
( a ) r e c e i v i n g i s m a n i f e s t e d i n :
(i) a w a r e n e s s , ( i i ) w i l l i n g n e s s t o r e c e i v e , and
( i i i ) c o n t r o l l e d or s e l e c t e d a t t e n t i o n ;
(b) r e s p o n d i n g i n c l u d i n g ;
(i) a c q u i e s c e n c e i n r e s p o n d i n g , ( i i ) w i l l i n g n e s s t o r e s p o n d , and
( i i i ) s a t i s f a c t i o n i n r e s p o n s e ;
v a l u i n g i s i n d i c a t e d by
( i) a c c e p t a n c e o f v a l u e , (ii) p r e f e r e n c e o f v a l u e , and
(iii) commitment t o v a l u e ;
o r g a n i s a t i o n is shown by
( i ) c o n c e p t u a l i z a t i o n of v a l u e , ar.d ( i i ) o r g a n i s a t i o n of a v a l u e sys t em; and
c h a r a c t e r i s a t i o n by v a l u e or v a l u e complex
c o m p r i s e s of:
( i ) g e n e r a l i z e d set, and (ii) c h a r a c t e r i s a t i o n ,
T h i s taxonomy, i t seems i s more s t r a n g e and complex
t h a n t h e c a t e g ~ r i s a t i o n n o t e d b e f o r e it. However,
i n t e r e s t , a t t i t u d e and v a l u i n g c o r r e s p o n d t o r e c e i v i n g ,
r e s p o n d i n g and v a l u i n g , when t h e t w o c l a s s i f i c a t i o n s
are c o n s i d e r e d s i d e by s i d e . But a p p r e c i a t i o n i n t h e
f i r s t c a t e g o r i s a t i o n c o u l d a p p r o x i m a t e l y be e q u a t e d
to o r g a n i s a t i o n and c h a r a c t e r i z a t i o n by a v a l u e or
v a l u e complex i n t h e second c l a s s i f i c a t i o n ,
The c o g n i t i v e l e a r n i n g outcomes h a v e been classi-
f i e d g e n e r a l l y i n t o s i x c a t e g o r i e s (Bloom et , a l ,
19711, These c a t e g o r i e s i n c l u d e :
( i ) knowledge, (ii) comprehens ion
(iii) a p p l i c a t i o n ( i v ) a n a l y s i s
( v ) s y n t h e s i s , and ( v i ) e v a l u a t i o n ,
I n p a r t i c u l a r c o g n i t i v e ou tcomes t h a t r e s u l t f rom
Mathema t i c s e d u c a t i o n a r e f o u r c a t e g o r i e s , namely:
(i ) c o m p u t a t i o n , (ii) comprehens ion ,
(iii) a p p l i c a t i o n , and ( i v ) a n a l y s i s ,
Computa t ion i n c l u d e s : knowledge o f s p e c i f i c f a c t s ,
knowledge o f t e r m i n o l o g i e s and a b i l i t y t o c a r r y o u t
a l g o r i t h m s .
Comprehension i s e x h i b i t e d i n o n e of t h e f o l l o w i n g ways:
(i knowledge o f c o n c e p t s ;
( i i ) knowledge o f r u l e s , p r i n c i p l e s a n d g e n e r a l i z a t i o n s ;
(iii) knowledge of Mathema t i ca l s t r u c t u r e ;
( i v ) a b i l i t y t o t r a n s f o r m problem e l e m e n t s from o n e mode t o a n o t h e r mode;
(v) a b i l i t y t o f o l l o w a l i n e of r e a s o n i n g ; and
( v i ) a b i l i t y t o r e a d and i n t e r p r e t e a problem.
A p p l i c a t i o n i s s u b d i v i d e d i n t o :
( i) a b i l i t y t o s o l v e r o u t i n e p rob lems ;
( i i ) a b i l i t y t o make c o m p a r i s o n s ;
(iii) a b i l i t y t o a n a l y s e d a t a , and
( i v ) a b i l i t y to r e c o g n i s e p a t t e r n s , i somorph i sms and symmet r i e s ,
A n a l y s i s i s d e m o n s t r a t e d by:
( i ) t h e a b i l i t y t o s o l v e non- rou t ine problems; (ii) a b i l i t y t o c o n s t r u c t p r o o f s ;
(iii) a b i l i t y t o d i s c o v e r r e a l t i o n s h i p s ; ( i v ) a b i l i t y t o criticise p r o o f s , and
( v ) a b i l i t y to f o r m u l a t e and v a l i d a t e g e n e r a l i s a - t i o n s ,
Indeed , t h a t c o g n i t i v e l e a r n i n g out.comes s h o u l d b e t h e
o b j e c t i v e s ( o r p a r t t h e r e o f ) o f s c h o ~ l i n g and i n
p a r t i c u l a r o b j e c t i v e s o f Mathemat ics e d u c a t i o n h a s n o t
been c o n t e s t - e d by ally one. O f c o u r s e s e c o n d a r y X h o o l s
i n t h i s c o u n t r y have e v e r s t r e s s e d t h e c o g n i t i v e
o b j e c t i v e s , Plost o f , i f n o t a l l , t h e Ma thema t i c s tests
a n d e x a m i n a t i o n s t a k e n by s t u d e n t s o f o u r school-s , u p
to t h e s e d a y s , a s s e s z t h e cognitive l e a r n i n g outcomes.
What i s r e q u i r e d i n t h i s area i s improvercent and
d i v e r s i f i c a t i o i ~ o f t h e i n s t r u m e n t s u s e d , s u c h t h a t t h e
p rob lems and e,.ercises set become v e r y r e l e v a n t t o
a c t i v i t i e s and p rob lems o f rea l l i f e ,
U n l i k e t h e c o g n i t i v e o b j e c t i v e s , t h e psychomotor
o b j e c t i v e s o f l e a r n i n g Mathema t i c s h a v e n o t been
s t r e s s e d . The psychomotor o b j e c t i v e s o r L e a r n i n g
outconles of s t u d e n t s i n t h e psychornotor area (as
a c q u i r e d from Mathema t i c s ) have n e v e r been f o r m a l l y
o r s y s t e m a t i c a l l y a p p r a i s e d , But s i n c e C,A,, a s
s t i p u l a t e d i n t h e N a t i o n a l P o l i c y on E d u c a t i o n ( 1 9 8 1 ) ,
r e q u i r e s t h a t s t u d e n t s ' a ch i evemen t i n t h e t h r e e domains
s h o u l d be assessed, i t becomes n e c e s s a r y t o examine
wha t c o n s t i t u t e t h e psychomotor ou tcomes o f Ma thema t i c s
e d u c a t i o n , Accord ing t o Onwuka (1981), t h e psychomotor
' s k i l l s c a n be c l a s s i f i e d i n t o s i x . T h e s e a r e : r e f l e x
movements, b a s i c fundamen ta l movements, p e r c e p t u a l
ab i l i t i es , p h y s i c a l a b i l i t i e s , s k i l l e d movenent and
n o n - d i s c u r s i v e communications, Each of t h e s e c a t e g o r i e s
i s f u r t h e r sub-diviut2d i n t o some s p e c i f i c s k i l l s a s
shown below: l i e f l e x movements as i t were, have no sub-
d i v i s i o n s r a t h e r s u c h movements as: k n e e j e r k , b l i n k i n g
of t h e e y e s , etc. a r e r e f l e x movements, Basic
f u n d a m e n t a l movements i n c l u d e s u c h s u b - c a t e g o r i e s as:
locomotor movements, non-locorno tor movementse and
m a n i p u l a t i v e movementsg, - p e r c e p t u a l ab i l i t i es are
s u b - c a t e g o r i s e d i n t o :
( i ) k i n e s t h e t i c d i s c r i m i n a t i o n , (ii) v i s u a l d i s c r i m i n a t i o n * ,
( i i i ) a u d i r o t y d i s c r i m i n a t i o n , ( i v ) tact i le d i s c r i m i n a t i o n , and
( v ) c o o r d i n a t e d p e r c e p t u a l a b i l i t i e s . *
P h y s i c a l a b i l i t i e s embody a s s u b g r o u p s of psychomotor
a b i l i t i e s :
(i) e ~ d u r a n c e (ii) s t r e n g t h
( i i i ) f l e x i b i l i t y w , and ( i v ) a g i l i t y .
S k i l l e d movement i s s u b - c l a s s i f i e d i n t o :
(i) s i m p l e a d a p t i v e s k i l l * , ( i i ) compound a d a p t i v e s k i l l , and
( i i i ) complex a d a p t i v e s k i l l .
R o n - d i s c u s s i v e communica t ion i n c l u d e s :
( i e x p r e s s i v e movarnen ts and (ii) i - n t e r p r e t a t i v e movements',
From t h i s researcher% e x p e r i e n c e s a s a Mathema t i c s
t e a c h e r , t e ach ing - l c . ~ r n i n c a c t i v i t i e s and meaningr 'ul
pe r fo rmances i n JSS Mathema t i c s c a l l f o r p a r t o r i.11
o f : b a s i c f u n d a m e n t a l movements, p e r c e p t u a l a b i l i t ics.
S k i 1 l e d movements and n o n - d i s c u s s i v e c o m m u i ~ i c s ~ t i o n s ( a s
i n d i c a t e d above by the s u b - c a t e g o r i e s s t a r t e d ( o r f o l l o w e d
by a s t e r i s k s ) , It is t h e r e f o r e i m p o r t a n t t h a t JSS
Mathemat i c s t e - x h i n g and l e a r n i n g s h o u l d i n c u l c a t e and
d e v e l o p t h e s e >sychornotor s k i l l s / a b i l i t i e s i n o u r JSS
s t u d e n t s , Having a g r e e d t h a t t h e s e s k i l l s / a b i l i t i e s
a r e t o be i n c u l c a t e d and d e v e l o p e d i n o u r JSS s t u d e n t s ,
i t becomes n e c e s s a r y t h a t t h e r a te and e x t e n t of
i n c u l c a t i n g a n d d e v e l o p i n g t h e s e s k i l l s and a b i l i t i e s
i n s t u d e n t s s h o u l d be m o n i t o r e d and n o t e d , T h i s ca l l s
for t h e u s e of r e l e v a n t a s s e s s m e n t i n s t r u m e n t s and
p r o j e c t s a s c o n c e i v e d i n t h i s s t u d y c o n s t i t u t e p a r t
o f t h e i n s t r u m e n t s ,
2.5 INSTRUIVIEiVTS FOR EVALUATING THE LEARNING -- -- --.--- -- OUTCOMES OF MUTHENATICS ED W-ATION
A s h a s been n o t e d e a r l i e r , t h e e v a l u a t i o n c a r r i e d
o u t i n o u r s e c o n d a r y s c h o o l s i s .mos t ly if n o t e n t i r e l y
c o n c e r n e d w i t h mon.i.l;.- r i n g F . ~ d a p p r a i s i n g s t u d e n tst
a c h i e v e m e n t s i n t h e c o g n i t i v e domain, I n t h e area of
Na then la t i c s , mos t o f t h e i n s t r u m e n t s employed are tests.
The o n l y a p p a r e n t v a r i a t i o n i s t h e u s e o f d i f f e r e n t
tes t forrrrats, T e a c h e r s i n o u r s c h o o l s , e s p e c i a l l y t h e
Ma thema t i c s t e a c h e r s , a r e , i n t h e l i g h t of t h e above ,
f a m i l i a r w i t h Lests b u t n o t w i t h t h e o t h e r e v a l u a t i o n
i n s t r u m e n t s , T h i s view i s c o n f i r m e d by t h e s t u d y of
Harbor-Ibeaja and Nworgu (19831, a b o u t t h e c o m p r e h e n s i v e
a s p e c t s of CA. They o b s e r v e d from t h e r e s p o n s 2 s of
s e c o n d a r y s c h o o l t e a c h e r s t o w h a t e v a l u a t i o n i n s t r u m e n t s
t h e y u s e and how o f t e n t h e y u s e t h e i n s t r u m e n t s , t h a t n t
t h e t e a c h e r s a r & a n i l i a c w i t h p r o j e c t s , socograms and
a n e c d o t a l r e c o r d s .
However, b e s i d e s tests, t h e r e are many i n s t r u m e n t s
u sed f o r t h e e v a l u a t i o n of personality and human
a b i l i t i e s , B road ly t h e s e i n s t r u m e n t s c a n be c l a s s i f i e d
i n t o : s e l f r e p o r t i n g i n s t r u m e n t s , i n s t r u m e n t s t h a t
m e a s u r e p e r s o n a l i t y by way of i n d i v i d u a l i m a g i n a t i o n s
and f a n t a s i e s , and i n s t r u m e n t s u , ~ e d to a p p r a i s e
i n d i v i d u a l s t h r o u ~ h t h e o p i n i o n o f oS. t ls rso Acco rd ing
t o T h o r n d i k e and Hagen (1969, p, 302) "one o b v i o u s
s o u r c e o f i n f o r m a t i o o a b o u t a p e r s o n i s t h a t p e r s c n
h i m s e l f . No a le el:;e has ~ n t i r n a t e and c o n t i n u o u s v iew
of Johnny a s Johnny ha..; of h i m s e l f , " R u t .rcco~-dir:1 t o
them, t t l e u s e o f s e l f - r e p o r t s h a s o b v i o u s l i m i t a t i o n s ,
An i n d i v i d u a l w i l l be r e a d y to t e l l a b o u t hirrlsclf i n
r e s p o n s e t o q u e s t i o n s , ( d e p e n d i n g upon) p r s - ~ i d e d , h e i s
w i l l i n g t o r e v e a l h i m s e l f , h e u n d e r s t a n d s t h e q u e s t i o n s
and t h e a t t r i b u t e s of h i m s e l f , h e i s e x p e c t e d t o
r e p o r t upon,
But t h e a b o v e l i m i t a t i o n s t o s e l f - r e p o r t i n g c a n
b e overcome. I f t h e i n f o r m a t i o n r e q u i r e d f rom a n
i n d i v j j u a l i s n o t t o be made p u b l i c o r h a s no a d v e r s e
i m p l i c a t i o n s t o self esteem, p r i d e , g a i n s and s e c u r i t y ,
t o h i d e t h e i n f o r m a t i o n i s n o t p r o b a b l e . The i s s u e o f
l a n g u a g e may b e removed t h r o u g h p r o p e r face v a l i d a t i o n
and t r i a l a d m i n i s t r a t i o n s , A l s o t h e u s e o f s e l f - r e p o r t s
become i n e v i t a b l e i f t h e i n f o r m a t i o n s o u g h t i s s u c h
t h a t o n l y Johnny knows how Johnny i s o r feels. I n
t h i s s t u d y , s t u d e n t s were r e q u i r e d t o r e p o r t t h e i r
a f f e c t i v e b e h a v i o u r s toward Ma thema t i c s i n g e n e r a l and
some Mathema t i c s p r o j e c t s i n p a r t i c u l a r . The above
r e p o r t s t h e s t u d e n t s g a v e by r e s p o n d i n g t o i t e m s o f
t w o i n s t r u m e n t s : Scale for A f f e c t i v e Pe r fo rmance i n
Ma thema t i c s (SAPIM) , and S c a l e f o r M o n i t o r i n g
A f f e c t i v e L e a r n i n g Uutcomes of Mathema t i ca l P r o j e c t s
SPWLOMP) . S i n c e t h e a f f e c t i v e pe r fo rmance of a
s t u d e n t i n Ma thema t i c s or a Mathema t i ca l p r o ject . ls
d i f f i c u l t t o summarize i n one o r t w o s t a t e m e n t s , a
number o f s t a t e m e n t s have t o be made to c o n s t i t u t e
e a c h o f t h e t w o i n s t r u m e n t s ( r a t . i n g scales). It t h e n
f o l l o w s t h a t t h e i n s t r u m e n t s w i l l be a L i k e r t scale
o r a T h u r s t o n scale, The t y p e of scale p r e f e r r e d i n
t h i s s t u d y i s a T h u r s t o n scale, s i n c e i t r e q u i r e s a
s t u d e n t t o i d e n t i f y and t i c k b e s i d e t h e s t a t e m e n t s t h a t
d e s c r i b e h i s / h e r b e h a v i o u r toward Mathema t i c s or a
Mathema t i ca l project, O f c o u r s e , t h e s t a t e m e n t s have
b e e n pre-weighted. The L i k e r t scale t y p e would have
r e q u i r e d a s t u d e n t t o d e c i d e w h e t h e r e a c h s t a t e m e n t
r e f l e c t s / d e s c r i b e s h i s / h e r b e h a v i o u r toward Mathema t i c s
a n d also to what e x t e n t t h e s t a t e m e n t does so w i t h
r e s p e c t t o some scale of p o i n t s ,
I n s t r u m e n t s t h a t t a p a p e r s o n ' s f a n t a s i e s o r
i m a g i n a t i o n s and f r o m s u c h d a t a e v a l u a t e t h e p e r s o n
i n c l u d e s u c h i n s t r u m e n t s as t h e t h e m a t i c a p p e r c e p t i o n
tests, i n k - b l o t tests among others, T h e s e sets of
tes t are not a p p r o p r i a t e fo r t h e a s s e s s m e n t o f s t u d e n t s
a c h i e v e m e n t s i n t h e a f f e c t i v e , c o g n i t i v e and psychomotor
domains . They a re d e s i g n e d p r i n c i p a l l y f o r t h e
a s s e s s m e n t o f t h e m a l a d j u s t e d o r abnormal b e h a v i o u r .
I n s t r u m e n t s t h a t a p p r a i s e i n d i v i d u a l s t h r o u g h t h e
o p i n i o n of o t h e r s are many. Such i n s t r u m e n t s i n c l u d e :
some r a t i n g scales ( i n v o l v i n g raters and ra tes ) ,
s o c i o g r a m s , a n e c d o t a l r e c o r d s a n d r e p o r t s , c h e c k l i s t s
a n d o t h e r o b s e r v a t i o n a l s c h e d u l e s o r t e c h n i q u e s by
which d a t a are c o l l e c t e d f o r t h e sake o f a s s e s s m e n t s ,
T h e s e i n s t r u m e n t s ( i f w e l l made and w e l l h a n d l e d )
c o u l d be u s e d ( i n c o m p a r a t i v e t e r m s w i t h i n a g r o u p ) t o
assess s t u d e n t s ' a f f e c t i v e , c o g n i t i v e and psychomotor
l e a r n i n g outcomes. S t u d e n t s , f o r i n s t a n c e , w i l l b e
a b l e t o l i s t , w i t h i n some t o l e r a b l e error l i m i t s ,
t h e f i v e b e s t ; s p r i n t e r s , wartists 's or t h e f i v e most;
p u n c t u a l , q u i e t , s o c i a b l e or a g g r e s s i v e s t u d e n t s
w i t h i n t h e i r class. By k e e p i n g r e c o r d s o f s i g n i f i c a n t '
e v e n t s o r r e a c t i o n s i n t h e l i f e o f a n i n d i v i d u a l , t h e
i n d i v i d u a l ' s a f f e c t i v e l e a r n i n g outcomes may be
a p p r a i s e d . I n t h e f i e l d of p l a y o r s p o r t s , a l i s t o f
" i n d i c a t o r s of f o u l p l a y s , m a l p r a c t i c e s o r a g g r e s s i o n ,
may b e t a k e n t o a ' p l a y g r o u n d a by a n a s s e s s o r t o wa tch
a g i v e n set o f p l a y e k s or a t h l e t e s f o r a number of
t i m e s . I f t h e a s s e s s o r t i c k s A g a i n s t t h e name of
e a c h p l a y e r / a t h l e t e whenever h e / s h e e x h i b i t s any of
such i n d i c a t o r s , t h e n t h e number o f t i c k e a g a i n s t
e a c h a t n l e t e l s / p l a y e r i s name g i v e s some measu re of t h e
a t h l e t e b s / p 1 a y e Y s s p i r i t o f s p o r t m a n s h i p o r l a c k of i t ,
W r i t t e n a s s i g n m e n t s , tests and p r o j e c t s may be
i n c l u d e d as i n s t r u m e n t s u sed t o a p p r a i s e i n d i v i d u a l s
t h r o u g h t h e o p i n i o n s o f edhhers , But t h e y d i f f e r from
some o f t h e o t h e r i n s t r u m e n t s i n t h e g r o u p i n a number
o f ways, I n w r i t t e n a s s i g n m e n t s , tests or p r o j e c t s ,
r e s p o n d e n t s a r e aware t h e y a r e b e i n g a s s e s s e d , so mos t
o f t'7e times t h e y t r y to p u t i n t h e i r b e s t . I n t h i s
s i t u a t i o n t h e y may b e r e g a r d e d as r e p o r t i n g t h e m s e l v e s .
A l s o when w r i t t e n s s s i g n m e n t s , tests or p r o j e c t s are
used to assess p e o p l e , there are d e p e n d a b l e r e c o r d s o f
t h e r e s p o n s e s of t h e p e o p l e t o t h e i n s t r u m e n t s .
T h e s e r e c o r d s c o u l d b e k e p t and i f need be, p r e s e n t e d
t o d i f f e r e n t assessors fo r t h e i r i n d e p e n d e n t judgements
o f t h e wor th of t h e r e s p o n s e s . However as h a s been
p o i n t e d o u t earlier, t e a c h e r s (of Mathema t i c s
e s p e c i a l l y ) a r e v e r y f a m i l i a r w i t h a s s i g n m e n t s and
tests b u t n o t w i t h p r o j e c t s . It may be wor thwhi le a t
t h i s p o i n t , t h e r e f o r e , t o rev iew w h a t p r o j e c t s (as i t
a p p l i e s t o t h i s work) are.
2,5,1 projects sf
O k u n r o t i f a (1977, p, 6 3 ) d e f i n e s p r o j e c t s as
ex tended p i e c e of work on g i v e n t o p i c s u s u a l l y p r e s e n t e d
i n loose l e a v e s of f o l d e r s w , The F e d e r a l M i n i s t r y o f
Educat ion (FME) (1985, p, 6) d e f i n e s a p r o j e c t as a
" l a r g e set e x e c c i s e from w h i c h t i m e c o n s t r a i n t s have
been l a r g e l y r z m ~ v e d . ~ ~ Onwuka (1981, p. 1 7 7 ) writes
t h a t a p r o j e c t " is a p u r p o s e f u l a c t i v i t y o r a meaning-
f u l w h o l e or u n i t of e x p e r i e n c e which proceedq i n a
s o c i a l environment . According to Onwuka (19811,
p r o j e c t s are c l a s s i f i e d i n t o f o u r t y p e s u s i n g t h e
f o l l o w i n g c r i t e r i a :
p r o j e c t s embodying some i d e a o r p lan i n e x t e r n a l
f orm ;
p r o j e c t s i n v o l v i n g t h e s o l u t i o n s o f some problem
or s t r a i g h t e n i n g o u t some i n t e l l e c t u a l
d i f f i c u l t y ;
p r o j e c t s aimed a t o b t a i n i n g some i t e m of
knowledge; and
( i v ) p r o j e c t s aimed a t p r o v i d i n g e n j o y a b l e
e x p e r i e n c e or a e s t h e t i c a p p r e c i a t i o n ,
I n w r i t i n y a b o u t p r o j e c t s Okeke (1982) s a y s t h a t
p r o j e c t s c o v e r a w i d e r a n g e o f work and i n v o l v e s u c h
a c t i v i t i e s as c o l l e c t i n g d a t a e n g a g i n g i n open-ended
( e x p e r i e n c e s ) e x p e r i m e n t s i n l a b o r a t o r i e s o r o u t d o o r ,
making c o m p l e t e o b j e c t s or p a r t s o f a l a r g e object ,
i d e n t i f y i n g and e n d e a v o u r i n g t o s o l v e problems i n
f i e l d s o f l e a r n i n g o r community l i v i n g and i n v e s t i g a -
t i n g u n d e r l y i n g c o n c e p t s and p r i n c i p l e s ,
The f i r s t a n d second d e f i n i t i o n s of p r o j e c t s g i v e n
above are, f o r t h e p u r p o s e of t h i s work, vague and
l i m i t e d i n e s s e n c e and scope . The t h i r d and f o u r t h
d e f i n i t i o n s a r e c o m p r e h e n s i v e s i n c e t h e y a p p l y t o mos t
s u b j e c t a r e a s o r f i e l d s o f l i f e . They are a p p a r e n t l y
more g e n e r a l t h a n w h a t Ma thema t i c s p r o j e c t s a t t h e
JSS are i n terms of s c o p e , As i t a p p l i e s t o t h i s
s t u d y , a p r o j e c t i n Ma thema t i c s i s made u p of t a s k s
e x p e c t e d t o be u n d e r t a k e n by a s t u d e n t o r g r o u p of
s t u d e n t s and accompl i shed i n n o t less t h a n two weeks
b u t i n more t h a n one y e a r , The u n d e r t a k i n g and
accompl i shmen t o f t h e t a s k s r e q u i r e a v a r i e t y o f
s k i l l s and knowledge a c q u i r e d from much more t h a n o n e
' t o p i c / u n i t o f work i n Mathematics. A p r o j e c t l e a d s
t o t h e p r o d u c t i o n of a m o d e l / p r o t o t y p e o f a commodity
o r t h e e x p o s i t i o n o f a M a t h e m a t i c a l l y r e l a t e d r e a l l i f e
problem. H Mathemat i c s p r o j e c t , w h i l e b e i n g t h e same
i n e s s e n c e w i t h t h e d e f i n i t i o n s of Onwuka ( 1 9 8 1 ) and
Okeke ( l 9 8 2 ) , i t seems t o i n c l u d e s i m u l t a n e o u s l y t w o
o r more t y p e s o f p r o j e c t s i d e n t i f i e d by Onwuka.
T h i s i s , however n o t c o n t r a d i c t o r y o r s u r p r i s i n g
s i n c e Onwuka (1981) o b s e r v e d , i n t e r e s t i n g l y , t h a t
t h e f o u r t y p e s o f p r o j e c t s i d e n t i f i e d c a n be pu r sued
c o n - c u r r e n t l y i n a p r o j e c t , Ano the r major t h r u s t
of t h e p r o j e c t s ( a s u n d e r s t u d i e d i n t h i s work ) c o n c e r n s
o n e o f t h e o b j e c t i v e s of p r o j e c t s t h a t r u n s a p p a r e n t l y
i n a l l t h e d e f i n i t i o n s o f p r o j e c t s g i v e n above. T h a t
o b j e c t i v e i s t h a t p r o j e c t s a r e o f t e n means t o problem
s o l v i n g .
Indeed p rob lems n o t e d / i n d i c a t e d above s h o u l d n o t
be s e e n a s t h e t y p e t h a t i s i n v o l v e d i n a s i n g l e
Ma thema t i c s test i tern or e x a m i n a t i o n q u e s t i o n , Such
p rob lems a s are c o n c e i v e d i n t h i s s t u d y h a v e similar
c h a r a c t e r i s t i c s w i t h s u c h r ea l l i f e p roblems b e i n g
r e f e r r e d to by Lesh. Lesh (1981, pp, 235-2621 i n
h i s s t u d y writes t h a t :
( i ) real problems seldom emerge as w e l l f o r m u l a t e d q u e s t i o n s ;
(ii) real problems o r i g i n a t e In t h e r e a l wor ld where i d e n t i f i c a - t i o n of r e l e v a n t c o n t e n t i s c r i t i ca l ;
(iii app l i ec ' prob:.ems seldom f a l l i n t o n e a t d i s c i p l i n a r y c a t e g o r i e s ;
( i v ) real problems o f t e n r e q u i r e s o l u t i o n s based on i n a d e q u a t e i n f o r m a t i o n ; and
( v ) r z a l problem s i t u a t i o n s sometimes c o n t a i n too much i n f o r m a t i o n o f t e n i n a d i s o r g a n i s e d form,
C e r t a i n l y , i t i s b e c a u s e of t h e s e c h a r a c t e r i s t i c s of
r ea l p rob lems wh ich p r o j e c t s set o u t t o s o l v e or
e x p o s e , t h a t make p e o p l e e x t o l t h e p r o j e c t t e c h n i q u e
t o t e a c h i n g and l e a r n i n g . As c o n t a i n e d i n a Handbook
on C o n t i n u o u s Assessment (FME, 1985, p. 61) t h e
p r o j e c t t e c h n i q u e u s u a l l y p r o v i d e s a d e e p e r i n s i g h t
i n t o s t u d e n t s 1 l e v e l o f f e e l i n g and t h e way t h e y
n o r m a l l y r e a c t t o s i t u a t i o n s . It went on t o s a y t h a t
t h e p r o j e c t t e c h n i q u e i s u s e f u l i n t h e c o g n i t i v e ,
a f f e c t i v e and psychomotor domains of l e a r n i n g , I f
w e l l h a n d l e d to p r o v i d e s u c c e s s f u l r e s u l t s , t h e
c o n f i d e n c e r e s u l t i n g from t h e s e n s e o f a c h i e v e m e n t
i s a c q u i r e d and s a t i s f i e d , The above r e a l i z a t i o n s are
s i g n i f i c a n t b u t how could t h e s e p r o j e c t s be made
a v a i l a b l e ? Could t h e i r u s e s b o t h a s t e a c h i n g and
l e a r n i n g tools and i n s t r u m e n t s f o r e v a l u a t i n g a c h i e v e -
ment i n Ma thema t i c s b e r e w a r d i n g a s p i c t u r e d above?
C e r t a i n l y a n s w e r s t o t h e s e q u e s t i o n s c o u l d b e p r o v i d e d
by t h e deve lopmen t , v a l i d a t i o n and u s e of some o f s u c h
p r o j e c t s i n Ma thema t i c s and e s p e c i a l l y a t t h e JSS.
2.6 VALIDATION OF INSTRUMENTS ---.- ,--
Whenever a n i n s t r u m e n t is t o b e u s e d , c e r t a i n
q u e s t i o n s ar ise c o n c e r n i n g w p s u c h an rLns t r u m e n t or
i n d e e d why t h a t s p e c i f i c i n s t r u m e n t s h o u l d be u s e d ,
E x p r e s s i n g t h e above view T h o r n d i k e a n d Hagen (1969,
p, 1 6 2 ) wrote:
Whenever a worke r i n psycho logy or e d u c a t i o n d e s i r e s t o measu re some q u a l i t y i n a g r o u p ( o f p e r s o n s ) or an i n d i v i d u a l , h e f a c e s t h e problem of c h o o s i n g t h e best i n s t r u m e n t f o r h i s pu rpose ,
Even i f t h e worke r d e c i d e s t o u s e a new i n s t r u m e n t or
h a s o n l y o n e i n s t r u m e n t , a n d none o t h e r , t o , h e s t i l l
s h o u l d w o r r y a b o u t how t h e i n s t r u m e n t does t h e job
h e d e s i r e s i t t o do. Accord ing t o T h o r n d i k e and
Hagen (1969), t h e w o r k e r may a lso be i n t e r e s t e d to
d e t e r m i n e n o t o n l y wh ich i s t h e best ( i n s t r u m e n t )
p r o c e d u r e among t h e o n e s a v a i l a b l e b u t how w e l l t h e
o n e h e / s h e h a s c h o s e n s a t i s f i e s h i s n e e d s by some
a b s o l u t e s t a n d a r d s . T h i s i s n e c e s s a r y s i n c e as
e x p r e s s e d by Lewis 119671, e v e r y s c o r e / m e a s u r e o o t a i n e d
by t h e u s e of some e v a l u a t i o n i n s t r u m e n t i s i n e v i t a b l y
i n f l u e n c e d by c h a n c e and s y s t e m a t i c errors. I n o the r
words , any o b t a i n e d score i s e i t h e r a n o v e r - e s t i z a t e
o r u n d e r - e s t i m a t e o f t h e t r u e t score. With t h e s e f n
mind, e v e r y r e s e a r c h e r s h o u l d endeavour t o u s e
i n s t r u m e n t s t.Qat g i v e scores which are m i n i m a l l y
i n f l u e n c e d by c h a n c e and s y s t e m a t i c errors. But when
t h e r e s e a r c h e r c a n n o t t e l l t h e i n s t r u m e n t among t h e
o n e s a v a i l a b l e or h a s o n l y o n e a v a i l a b l e , i t i s
i m p o r t a n t t h e r e s e a r c h e r e s t a b l i s h e s t h e e x t e n t score
o b t a i n e d by u s i n g t h e i n s t r u m e n t h e c h o o s e i s f r e e of
( i) c h a n c e errors ( r e l i a b i l i t y ) and
(ii) a l l errors; t h a t i s c h a n c e errors and
s y s t e m a t i c e r r o r s ( v a l i d i t y ) ,
Ohuche and Akeju (1977, p, 164) d e f i n e reliability
a s "+,he c o n s i s t e n c y , t r u s t w o r t h i n e s s or a c c u r a c y o f a
m a a s u r i n g i n s trumen t, " An i n s t r u m e n t wh ich
c o n s i s t e n t l y m e a s u r e s w h a t e v e r i t m e a s u r e s ( g i v i n g the
same or a p p r o x i m a t e l y t h e same s c o r e a l l t h e t i m e )
i s r e l i a b l e , T h e r e are b a s i c a l l y f o u r a p p r o a c h e s t o
e s t i m a t i n g , i n q u a n t i t a t i v e terms t h e r e l i a b i l i t y o f
a m e a s u r i n g i n s t r u m e n t , T h e s e a p p r o a c h e s are: test-
retest , e q u i v a l e n t form, s p l i t h a l f and r a t i o of
error ( c h a n c e ) v a r i m c e t o t o t a l v a r i a n c e a p p r o a c h e s .
Of c o u r s e , t h e r e is a p p a r e n t l y a f i f t h a p p r o a c h known
a s t h e Kuder-Richardson ( i n t e r n a l c o n s i s t e n c y 1
a p p r o a c h , T h i s f i f t h a p p r o a c h , however, e n t a i l s a
p r o c e d u r e which compu te s or g e t s a t t h e mean of a l l
p o s s i b l e s p l i t - h a l f r e l i a b i l i t y c o e f f i c i e n t s of a n
i n s t r u m e n t .
I n t h i s s t u d y , t h e s p l i t - h a l f and test-retest
a p p r o a c h e s h a v e been used t o a s c e r t a i n t h e r e l i a b i l i t y
c o e f f i c i e n t s o f SAPIM. The r a t i o of v a r i a n c e s a p p r o a c h
i s u s e d h e r e t o d e t e r m i n e t h e rater r e l i a b i l i t y
c o e f f i c i e n t o f t e a c h e r s a s m a r k e r s of s t u d e n t s f
r e s f i o n s e s t o t h e p r o j e c t s , The r a t i o of v a r i a n c e s
a p p r o a c h was worked o u t by E b e l (1951) when h e was
d e a l i n g w i t h r a t i n g s made by a g r o u p o f raters, T h i s
a p p r o a c h employs a two-way a n a l y s i s of v a r i ? n c e
( random model ) to factor o u t t h e error sum o f s q u a r e s
a n d t h e row sum of s q u a r e s ( SSe and S S r ) r e s p e c t i v e l y .
The mean error s q u a r e s and t h e mean row s q u a r e s are
'also o b t a i n e d ( t h a t i s MSe and MS,) The r e l i a b i l i t y
o f t h e r a t e r s ( r t t ) i s o b t a i n e d by d i v i d i n g t h e
d i f f e r e n c e be tween MSr and MSe by MSr, Thus t h e
r e l i a b i l i t y c o e f f i c i e n t ( r t t ) of t h e raters i s t h e
r a t i o o f t h e e r r o r f ree v h r i a n c e t o a l l v a r i a n c e
(Kbe l 1951, pp, 407-4241, r t t = MSr - MSe M~F-
Many s t u d i e s i n c l u d i n g Obioma (1985) i n h i s s tuUy
" t h e deve lopmen t and v a l i d a t i o n o f a d i a g n o s t i c
Ma thema t i c s test f o r JSS s t u d e n t s f s u s e t h e i n t e r n a l
c o n s i s t e n c y c o e f f i c i e n t s as t h e i n d i c e s of t h e
r e l i a b i l i t i e s of t h e i r i n s t r u m e n t s , The i n t e r n a l
c o n s i s t e n c y a p p r o a c h o r t h e Kudder-Richardson method
of e s t i m a t i n g r e l i a b i l i t y was n o t u sed i n t h i s s t u d y
b e c a u s e o f t h e n a t u r e o f p r o j e c t s and t h e s c o r i n g o f
t h e r e s p o n s e s . A l s o t h e r e s e a r c h e r i s n o t keen a t
compu t ing i n t e r n a l c o n s i s t e n c y c o e f f i c i e n t s s i n c e
e a c h p r o j e c t as a n e n t i t y and i s s c o r e d as such .
I f we estimate t h e e x t e n t t o wh ich a n i n s t r u m e n t
i s r e l i a b l e , t h a t i s free f rom c h a n c e e r r o r s , i t i s
more p e r t i n e n t t o a s c e r t a i n t h e e x t e n t t o wh ich t h e
i n s t r u m e n t i s free from s y s t e m a t i c errors, I n o t h e r
words , i t i s v e r y i m p o r t a n t t o d e t e r m i n e n o t o n l y how
re l iab le b u t a l s o how r e l e v a n t t h e i n s t r u m e n t is.
The above s t a n d i s s o s i n c e i n s t r u m e n t i s v a l i d o n l y
i f i t i s rel iable and r e l e v a n t , V a l i d i t y o r more so
t h e r e l e v a n c e of a:. i n s t r u m e n t is c o n s i d e r e d i n t e r m s
o f : ( i)
(ii)
( i i i )
t h e c o v e r a g e ( h o r i z o n t a l and v e r t i c a l ) of a
c o n t e n t on which t h e a c h i e v e m e n t s of p e o p l e
exposed t o t h e c o u r s e a r e b e i n g a s s e s s e d ; o r
t h e spec t ru rn / scope o f b e h a v i o u r p t t e r n s or
t h e facets of a n a t t r i b u t e / c o n s t r u c t a n
i n s t r u m e n t i s a p p r a i s i n g ; or
t h e n a t u r e o f t h e c r i t e r i o n which t h e i n s t r u m e n t
i n q ~ l e s t i o n i s u s e d t o e v a l u a t e .
With r e s p e c t to t h e s e c o n s i d e r a t i o n s , t h r e e co f i v e
t y p e s o f v a l i d i t y have been i g e n t i f i e d , K e r l i n g e r
(1973) c l a s s i f i e s v a l i d i t y a s f o l l o w s :
( i ) c o n t e n t v a l i d i t y , (ii 1 c r i t e r i o n - r e l a t e d v a l i d i t y and
(iii c o n s t r u c t v a l i d i t y ,
Horrock and Schoone r (1968) see v a l i d i t y as beincj of
f o u r k i n d s :
( i c o n t e n t v a l i d i t y , (ii c o n c u r r e n t v a l i d i t y ,
( i f i p r e d i c t i v e v a l i d i t y and ( i v ) c o n s t r u c t v a l i d i t y .
Indeed K e r l i n g e r l s e x p l a n a t i o n o f h i s c r i t e r i o n - r e l a t e d
v a l i d i t y e n c o r p o r a t e s Hor rock l s and Schooner1 s view
o f p r e d i c t i v e and c o n c u r r e n t v a l i d i t y . However, few
a u t h o r i t i e s l $ d f i f t!; t y p e o f v a l i d i t y known as f ice
v a l i d i t y , . o f t e n r e c e r r e d t o as q u a s i - t y p e of v a l i d i t y ,
R u s s e l ( 1 9 8 2 ) i n h i s work, " t h e c o n s t r u c t i o n ,
v a l i d a t i o n and u s e o f a s t a n d a r d i z e d tes t t o Yeasu re
Achievement i n High School Mathemat ics i n L i b e r i a v
c a r r i e d o u t some c o n t e n t v a l i d a t i o n o f his i n s t r u m e r t .
H e w e n t f u r t h e r t o d o c o n c u r r e n t v a l i d a t i o n of h i s
i n s t r u m e n t b.) c o r r e l a t i n g s t u d e n tst SMAT s c o r e s w i t h
t h e same s t u d e n t s 1 scores on t h r e e o t h e r tests o f h i s
c o u n t r y 1 s n a t i o n a l i m p o r t a n c e , Obioma (1985 ) and
Chacko ( 1 9 8 0 ) i n t h e i r r e s p e c t i v e works d i d some
c o n t e n t v a l i d a t i o n o f t h e i r r e s p e c t i v e i n s t r u m e n t s .
However, Obioma ( 1 9 8 5 ) i n d e v e l o p i n g and v a l i d a t i n g
h i s D i a g n o s t i c Ma thema t i c s T e s t f o r JSS s t u d e n t s
employed some f a c t o r i a l v a l i d a t i o n p r o c e d u r e s t o see
how r a c h i t e m i s p u r e l y or o t h e r w i s e l o a d e d i n t h e
f o u r a reas /cornponents o f JSS Mathematics .
I n t h i s s t u d y , t h e r e s e a r c h e r c a r r i e d o u t t h e
c o n t e n t v a l i d a t i o n o f SAPIM, SMALOMP and t h e t h r e e
p r o j e c t s u sed as e v a l u a t i o n i n s t -uments, C o n t e n t
80
v a l i d a t i o n of SAPIM and SMALOMP i s done by l i s t i n g t h e
a s p e c t s of a f f e c t i v e l e a r n i n g outcomes e x p e c t e d from
s e c o n d a r y school Mathzmakics e d u c a t i o n a n d t h e p r o j e c t s
r e s p e c t i v e l y . St:. .:einent were f o r m u l a t e d t o d e s c r i b e
or i n d i c a t e t h e p r e s e n c e o r o t h e r w i s e of such e x p e c t e d
l e a r n i n g outcomes i n a n i n d i v i d u a l who a g r e e s or
d i s a g r e e s w i t h t h e s t a t e m e n t s , The l i s t s of t h e
e x p e c t e d l e a r n i n g outcomes and t h e f o r m u l a t e d s t a t e m e n t s
were s u b m i t t e d t o e x p e r t s f o r v a l i d a t i o n and a l s o fcr
a s s i g n i n g w e l g h t s t o t h e s t a t e m e n t s . I n t h e c a s e of
t h e t h r e e p r o j e c t s , t h e c o n t e n t a r e a s / t o p i c s on which
each p r o j e c t i s set are i d e n t i f i e d . A l s o t h e knowledge
a n d s k i l l s n e c e s s a r y for e x e c u t i n g e a c h p r o j e c t a r e
i d e n t i f i e d and l i s t e d . The l i s t s were s u b m i t t e d to
e x p e r t s f o r v a l i d a t i o n , T h i s r e s e a r c h e r d i d n o t c a r r y
o u t c o n c u r r e n t c a l i d a t i o n of t h e p r o j e c t s b e c a u s e
t h e r e are no s t a n d a r d i z e d p r o j e c t s i n JSS Mathemat i c s
known t o t h e r e s e a r c h e r . A l s o f a c t o r i a l v a l i d a t i o n
i s n o t u sed i n t h i s s t u d y s i n c e t h e p r o j e c t s are meant
t o i ~ t e g r a t e t h e v a r i o u s s s p r c ts of Mathemat ics .
F u r t h e r m o r e , t h i s r e s e a r c h e r d o e s n o t see t h e need t o
d i f f e r e n t i a t e knowledge, r u l e s , c o n c e p t s and s k i l l s
a c q u i r e d from t h e v a r i o u s a s p e c t s of JSS ath he ma tics
'(Number a n d numnrst ion, A l g e b r a i c p r o c e s s e s . . , )
C e r t a i n l y a t t h e JSS l e v e l t h e s e s e c t i o n s of M a t h e m a t i c s
a re t i e d i o g e t h e r by t h e basic a r i t h m e t i c o p e r a t i o n s ,
2-7 SUMMARY O F IS: [IES D I ;CUSSED -- ---.a*-
T e a c h i n g a n d l e a r n i n g are human e n d e a v o u r s t h a t
are as old a s man a n d s o c i e t y , They ( a s man a n d
s o c i e t y ) h a v e b e e n u n d e r g o i n g c h a n g e s , I n t h e non-
f o r m a l s e t t i n g s , t e a c h i n g and l e a r n i n g i n v o l v e d one- to -
o n e i n t e r a c t i o n b e t w e e n t h e t e a c h e r a n d t h e l e a r n e r .
T h e l e a r n e r . i s e x p o s e d t o rea l t a s k s h e / s h e w i l l be
e x p e c t e d t o ~ n d e r t a k e i n l i f e . I n t h e formal s e t t i n g s ,
t h e r e emerged many p h i l o s o p h i c a l a n d p s y c h o l o g i c a l
t h e o r i e s t h a t d i r e c t l y or i n d i r e c t l y i n f l u e n c e t h e
p r o c e s s e s of t e a c h i n g a n d l e a r n i n g . However, n o
matter t h e t h e o r y , t h e one-to-one i n t e r a c t i o n b e t w e e n
t h e t e a c h e r a n d t h e l e a r n e r gave way to many-to-one
i n t e r a c t i o n s b e t w e e n t h e l e a r n e r s o n t h e o n e hand a n d
t h e t e a c h e r o n t h e o t h e r hand. I n t h e s e formal
s y s t e m s , s c h o o l s are e s t a b l i s h e d t o t e a c h fo r t r a n s f e r
of l e a r n i n g / t r a i n i n g . R a t h e r t h a n i n v o l v e l e a r n e r s
i n t h e rea l l i f e t a s k s , s u b j e c t areas or c o u r s e s are
c r e a t e d . d i t h t i m e t e a c h i n g a n d l e a r n i n g of t h e s e
' s u b j e c t s and c o u r s e s a c q u i r e d some p e c u l i a r c h a r a c t e r i s -
t ics, t e c h n i q u e s and methods.
I n Ma thema t i c s a number of methods o r a p p r o a c h e s
t o t e a c h i n g became p o p u l a r w i t h t h e t e a c h e r s b u t
t h e s e seem n o t t o have produced t h e e x p e c t e d r e s u l t s
i n Mathemat ics e d u c a t i o n , I n N i g e r i a n Seconda ry
S c h o o l s i n p a r t i c u l a r , j-t h a s been o b s e r v e d t h a t :
t e a c h e r s o f M a t h e m a t k s are i n s u f f i c i e n t and
i n e f f e c t i v e ; s t u d e n t s t c o g n i t i v e p e r f o r m a n c e s i n
Ma thema t i c s are l o w , and t h e i r a t t i t u d e s toward (o r
r a t h e r a f f ec c i v e p e r f o r m a n c e s i n ) Ma thema t i c s a re on
t h e a v e r a g e n e g a t i v e . The a b o v e o b s e r v a t i o n s have
l e d t o t h e d e s c r i p t i o n o f t h e t e a c h i n g and l e a r n i n g
of Mathema t i c s i n o u r ( s e c o n d a r y ) s c h o o l s as b e i n g i n
a s o r r y state. Not l e f t o u t o f t h e s o r r y s t a te i s
a l s o t h e e v a l u a t i m o f s t u d e n t s ' a c h i e v e m e n t i n
Mathemat ics , U p u n t i l now, e v a l u a t i o n of s t u d e n t s '
a c h i e v e m e n t i n Ma thema t i c s i s i n a d e q u a t e and e n t i r e l y
c o n c e r n e d w i t h t h e c o g n i t i v e ou tcomeso
T h e s e s i t u a t i o n s t r i g g e r e d off cal ls f o r c h a n g e s
and i n d e e d came c h a n g e s , One o f such c h a n g e s i s i n
t h e e v a l u a t i o n p o l i c y , C o n t i n u o u s Asses smen t (CA) h a s
been i n t r o d u c e d , A l s o t h e r e a r e some c u r r i c u l a r
r e f o r m s / c h a n g e s i n Mathematics . But t h e r e a r e l i t t l e
or no c h a n g e s i n t h e a p p r o a c h e s t o t e a c h i n g and
l e a r n i n g Yatherna t ics i n o u r s e c o n d a r y s c h o o l s , It
h a s a lso been n o t e d t h a t CA i s r u n n i n g i n t o s e r i o u s
p rob lems as r e g a r d s i t s 1,nplemen t a t i o n , T e a c h e r s of
o u r p r i m a r y and s e c o n d a r y s c h o o l s ( i n c l u d i n g t h e
Ma thema t i c s t e a c h e r s ) d o n o t p o s s e s s t h e n e c e s s a r y
c o m p e t e n c i e s and s k i l l s for t h e p r o p e r i m p l e m e n t a t i o n
of CA. Some of t h e p r o c e s s e s , t e c h n i q u e s and tools/
i n s t r u m e n t s b e i n g used for CA i n v a r i o u s q u a r t e r s
d i f f e r g r e a t l y and some o f t h e s e are e r r o n e o u s , A
r e v i e w of t h e sys t em i s t h e r e f o r e u r g e n t .
T h i s s t u d y c a l l s f o r t h e u s e o f p r o j e c t s i n
Ma thema t i c s a s a n app roach t o min imiz ing : t h t poor
t e a c h i n g and l e a r n i n g o f Mathemat ics ; and t h e
i n a d e q u a c i e s and errors i n v o l v e d i n t h e i m p l e m e n t a t i o n
of CH, T h i s c a l l s t a n d s on t h e p r e m i s e t h a t t h e u s e
o f p r o j e c t s i s i n n i n e w i t h t h e s t i p u l a t i o n s and
p r e s c r i p t i o n s of t h e con tempora ry l e a r n i n g and
, t r a n s f e r t h e o r i e s and methods o f t e a c h i n g Mathemat ics .
The u s e of p r o j e c t s i n t e a c h i n g , l e a r n i n g and e v a l u a -
t i o n of a c h i e v e m e n t s i n Mathemat ics i s s u p p o r t e d by
common s e n s e or r e a s o n and e x p e r i e n c e , It i s also
' r e levant t o o u r n a t i o n a l p o l i c i e s o f t h e p r e s e n t ,
S i n c e t h e merits and may b e t h e d e m e r i t s of t h e
u s e of p r o j e c t s as a n 3.pproach t o t h e t e a c h i n g and
l e a r n i n g ( i n c l u d i n g e v a l u a t i n g a c h i e v e m e n t ) of
s e c o n d a r y s c h o o l Ma thema t i c s i n N i g e r i a have n o t been
a s c e r t a i n e d e m p i r i c a l ] - y , t r y i n g o u t t h e u s e o f p r o j e c t s
i s c e r t a i n l y n e c e s s a r y - 111 t r y i n g some of s u c h
p r o j e c t s a s i n s t r u m e n t s of e d u c a t i o n a l p r a c t i c e
( e s p e c i a l l y f o r e v a l u a t i o n o f ach izvemen t ) , some a s p e c t s
o f r e l i a b i l i t y and v a l i d i t y o f t h e Ma thema t i c s p r o j e c t s
h a v e t o b e a s c e r t a i n e d . To t h i s e i ~ d , t h e v a r i o u s t y p e s
of r e l i a b i l i t y and v a l i d i t y of i n s t r u m e n t s of e d u c a t i o n
p r a c t i c e have been r e v i e w e d , and t h o s e t y p e s p e r t i n e n t
o r of i n t e r e s t t o t h e i n s t r u m e n t s used i n t h i s s t u d y
h a v e been i d e n t i f i e d . Indeed t h e r e v i e w of wha t t e a c h i n g and l e a r n i n g
h a v e been and w h a t t h e y are now, shows t h a t formal
e d u c a t i o n r a r e l y e x p o s e s l e a r n e r s a d e q u a t e l y to s o l v i n g
rea l l i f e problems and h e n c e l i m i t e d . t r a n s f e r s , I t
p o i n t e d o u t how non-formal s e t t i n g s d i d / d o e s t h a t and
t h e r e f o r e ca l ls for s u c h a p p r o a c h e s as t h e u s e of
p r o j e c t s ( a s c o n c e i v e d i n t h i s work) e s p e c i a l l y i n JSS
kathema tics of t h e N i g e r i a n e d u c a t i o n system. The
r e v i e w o f e v a l u a t i o n of s t u d e n t s a c h i e v e m e n t s and CA
h i g h l i g h t s : wha t e v a l u a t i o n o u g h t t o be; how i t s h o u l d
be c a r r i e d o u t ; w h a t h a v e been done or are b e i n g done
so f a r ; and t h e i n a d c q u a c i ~ s i n c o v e r a g e and a p p r o a c h e s
i n o u r p r e s e n t day e d u c a t i o n a l sys t em and more so i n
Ma thema t i c s e d u c a t i o n , The a s p e c t s o f r e v i e w a c c e n t u a -
t e d t h e need f o r t h e u s e o f l'/Iatiiem;itics p r o j e c t s
e s p e c i a l l y w i t h t h e i n t r o d u c t i o n c f CH and t h e emphas i s
o n p r e - v o c a t i o n a l e d u c a t i o n a t JSS, Examining w h a t
h a v e been w r i t t e n on e x p e c t e d l e a r n i n g outcomes o f
Ma thema t i c s e d u c a t i o n and t h e i n s t r u m e n t s f o r e v a l u a t i n g
s u c h outcomes e l u c i d a t e d : t h e knowledge and s k i l l s to
be e v a l u a t e d , how to e l i c i t such knowledge and s k i l l s ;
w h a t j n s t r u m e n t s t o u s e ; and t h e w e i g h t i n g , s c a l i n g or
norming o f such i n s t r u m e n t s . The above e x a m i n a t i o n w a s
i n e v i t a b l e for t h e deve lopmen t o f t h e p r o j e c t s u sed i n
t h i s work as tools for t e a c h i n g and l e a r n i n g and fo r
(CAI e v a l u a t i n g s t u d e n t s ' l e a r n i n g ou tcomes i n t h e
s u b j e c t , The s e c t i o n o f l i t e r a t u r e r e v i e w on v a l i d a t i n g
o f i n s t r u m e n t s p o i n t s o u t : w h a t a p p r o a c h e s t h a t c o u l d
b e used i n v a l i d a t i n g e v a l u a t i o n i n s t r u m e n t s which o f
s u c h a p p r o a c h e s have been used i n s imi lar s t u d i e s ; and
and which o f such s t i p u l a t e d or used a p p r o a c h e s s u i t
t h e v a l i d a t i o n o f t h e i n s t r u m e n t s u sed i n t h i s work.
CHAPTER THREE
RESEARCH NETHODOLOGY
3.0 INTRODUCTION
Under t h i s ck ~ p t e r , t h e r e s e a r c h d e s i g n and t h e
p r o c e d u r e s employed i n t h i s s t u d y are d i s c u s s e d .
The d i s c u s s i o n e n t a i l s :
i.
ii,
iii.
i v .
3.1
s t a t i n g and e x p l a i n i n g , though b r i e f l y , t h e
t y p e o f s t u d y c a r r i e d o u t ;
s p e c i f y i n g t h e p o p u l a t i o n , t h e s amples and
t h e s ampl ing t e c h n i q u e s i n v o l v e d ;
s t a t i r l g and d e s c r i b i n g t h e i n s t ~ w m e n t s u s e d i n
t h i s s ~ u d y , t h e methods of t h e i r deve lopment ,
and where a p p l i c a b l e t h e i r v a l i d a t i o n ; and
s t a t i n g and e x p l a i n i n g t h e methods and tools
u s e d to a n a l y s e t h e sets of d a t a t h a t were collected.
RESEARCH DESIGN
The d e s i g n of t h i s s t u d y i s two pronged. I n
t h e f i r s t d n s t a n c e , t h i s s t u d y i s a n t t i n s t r u m e n t a t i o n
r e s e a r c h , l 1 A s t u d y b e l o n g s to i n s t r u m e n t a t i o n
r e s e a r c h i f t h e s t u d y i n v o l v e s i n t r o d u c i n g new or
m o d i f i e d c o n t e n t p r o c e d u r e , t e c h n o l o g y or i n s t r u m e n t s
o f e d u c a t i o n a l p r a c t i c e ( ICEE, 19821, I n p a r t i c u l a r ,
t h i s work i n v o l v e d d e v e l o p i n g some model p r o j e c t s
b a s e d on JSSMC. T h r e e of t h e p r o j e c t s deve loped
are u s e d as tools f o r t e a c h i n g and l e a r n i n g Mathema t i c s
a t the JSS l e v e l a r d three others are u s e d as
i n s t r u m e n t s for t h e e v a l u a t i o n of s t u d e n t s @ a c h i e v e -
m e n t s i n t h e t h r e e domains of e d u c a t i o n a l o b j e c t i v e s ,
a t t h a t l e v e l ,
I n t h e second i n s t a n c e , t h i s s t u d y i s e x p e r i m e n t a l
i n n a t u r e , The p r e - t e s t , t r e a t m e n t and p o s t - t e s t w i t h
random or a l t e r n a t e r a n k s g r o u p i n g d e s i g n i s used i n
t h i s s tudy . 2he symbol ic p r e s e n t a t i o n below d e p i c t s
t h e d e s i g n .
Rg01X102 and ARg01X102 E x p e r i m e n t a l g r o u p s
Rg01X002 and ARg01Xo02 C o n t r o l g r o u p s
where ( i) Rg means random g r o u p i n g
(ii) ARg means a l t e r n a t e r a n k s g r o u p i n g
( i i i ) Ol means p r e t e s t u s i n g SAPIM
( i v ) 02 means p o s t t e s t u s i n g SAPIM
( v ) X1 i m p l i e s e x p o s u r e t o t h e u s e of p r c j e c t s
a s tools for t e a c h i n g and l e a r n i n g , and
f o r e v a l u a t i n g a c h i e v e m e n t
( v i ) Xo means n o e x p o s u r e t o the u s e of p r o j e c t s .
The a l t e r n a t e r a n k s g r o u p i s p r e f e r r e d for i t s
s t a t i s t i c a l a s w e l l a s i t s i n t s r p r e t q t i v e a d v a n t a g e s
( D a l t o n a1.d O v e r a l l 1 9 7 7 ) . It i s a l s o c o n v e n i e n t t o
u s e i n s c h o o l s ,
3-2 POPULATION OF T1-li7 STUDY -- -.--A-
T h i s s t u d y deve loped and v a l i d a t e d , a l b e i t on a
s m a l l s c a l e , p r o j e c t s for the t e a c h i n g and l e a r n i n g of
Matherna t ics i n t h e J u n i o r Seconda ry S c h o o l s ( JSS) i n
N i g e r i a , The p o p u l a t i o n o f t h i s s t u d y i s , t h e r e f o r e ,
t h e JSS s t u d e r t s o f t h e new 6-3-3-4 e d u c a t i o n sys t em
and t h e t e a c h e r s who t e a c h t h e s e s t u d e n t s Mathemat ics ,
T h e s e se t o f s t u d e n t s and t e a c h e r s b a s e t h e i r l e a r n i n g
a n d t e a c h i n g of Mathema t i c s on t h e F e d e r a l Mj n i s t r y
o f E d u c a t i o n , J u n i o r Seconda ry S c h o o l Ma thema t i c s
C u r r i c u l u m ( JSSMC). By t h e 1987 s c h o o l y e a r , s c h o o l s
i n 11 o u t of t h e '19 states i n t h e f e d e r a t i o n and a l l
t h e U n i t y / F e d e r a l Government Seconda ry S c h o o l s have
s t a r t e d t o implement t h e new 6-3-3-4 e d u c a t i o n a l s y s t e m
a s f a r as t h e s e c o n d a r y s c h o o l l e v e l ( t h o u g h t h e s e
s c h o o l r were a t d i f f e r e n t s t a g e s and, d e g r e e s o f
i m p l e m e n t a t i o n ) , So a l l t h e JSS s t u d e n t s i n t h e 11
states and t h e F e d e r a l Government S e c o n d a ~ . y S c h o o l s
c d n s t i t u t e d , as i t were, t h e t a r g e t p o p u l a t i o n of
t h i s s t u d y ,
3.3 SAMPLE ANI) SAMPLI.\IG TECHNIQUES
S i n c e t h e popu ' a t i o n ~f t h i s s t u d y i s c o n s t i t u t e d
o f t e a c h e r s and stuclet-i L s p i t s sample i s also made u p
of t h e s e t w o c a t e g o r i e s of p e r s o n s , The f i r s t
c a t e g o r y i s made u p of t w o hundred and f o r t y (240)
JSS s t u d e n t s drawn from f o u r s e c o n d a r y s c h o o l s s u c h
t h a t s i x t y (60) s t u d e n t s were s e l e c t e d from e a c h
s c h o o l , The f x r schools were p u r p o s i v e l y sampled,
T h i s i s to s a y t h a t the s c h o o l s were c h o s e n on t h e
bases t h a t :
( i ) t h e s c h o o l a u t h o r i t i e s a g r e e d t h a t t h e i r JSS
y e a r 111 s t u d e n t s be i n v o l v e d i n t h e s t u d y ;
( i i ) t h e t e a c h e r s of the c l a s s e s i n v o l v e d i n t h e
s t u d y , a g r e e d to unde rgo t r a i n i n g so as t o
t e a c h t h e i r s t u d e n t s a c c o r d i n g t o t h e p l a n of
t h e s t u d y ;
( i i i ) t h e s c h o o l s are i n p o s i t i o n s o r a t l o c a t i o n s
t h s t are e a s i l y a c c e s s i b l e to t h e r e s e a r c n e r
f o r e f f e c t i v e m o n i t o r i n g of t h e a c t i v i t i e s ; and
( i v ) e a c h s c h o o l had a t least f o u r classes of JSS
y e a r 111 s t u d e n t s .
Three s c h o o l s were chosen from I m o S t a t e and o n e
school from Anambra S t a t e , S t u d e n t s i .nvolved i n t h i y
s t u d y f rom e a c h o f t w o schools i n Imo were random1.y
a s s i g n e d t o t w o g r o u p s , The r e s u l t i n g g r o u p s i n e a c h
s c h o o l w e r e a l l randomly a s s i g n e d to e i t h e r treatment
or c o n t r o l , S t u d e n t s i n v o l v e d i n t h e s t u d y from each
of t h e r e m a i n i n g t w o s c h o o l s ( o n e i n I m o and t h o o t h e r
i n Anambra) were s o r t e d i n t o two a l t e r n a t e r a n k s g roups .
The sampled s t u d e n t s b e l o n g i n g t o t h e s e a1 t e r n a t e r a n k s
g r o u p s w e r e n o t e d and a l l o w e d to r ema in i n t h e i r i n t a c t
c l a s s e s . The two r e s u l t i n g a l t e r n a t e r a n k s g r o u p s i n
e a c h s c h o o l were t h e n randomly a s s i g n e d to t r e a t m e n t
and c o n t r o l . Below i s a t a b u l a r r e p r e s e n t a t i o n of t h e
sampl ing and g r o u p i n g d e s i g n s employed i n t h i s s t u d y ,
Random g r o u p s p r e t e s t , t r e a t m e n t s , post-test--- (i>
A 1 t e r n a t e r a n k s g r o u p s p r e t e s t , t r e a t m e n t s , p o s t - t e s t - - ( i i )
TABLE 3.1: T a b u l a r R e p r e s e n t a t ~ o n Of 'The Group inqs O f The Sample O f T h i s S t u d y
H. Expt. = Random and e x p e r i m e n t a l R , Cont. = Random and c o n t r o l
Ar, E x p t 0 = A l t e r n a t e r a n k s g r o u p e x p e r i m e n t a l A r , Cont . = A l t e r n a t e r a n k s g r o u p c o n t r o l
The second c a t e g o r y of p e r s o n s t h a t formed p a r t
of t h e sample of t h i s s t u d y i s t e n Mathema t i c s t e a c h e r s
who t e a c h JSS y e a r 111 s t u d e n t s o T h i s set of t e a c h e r s
i s made u p of t h e f o u r t e a c h e r s who t a u g h t t h e s t u d e n t s
i n v o l v e d i n t h e s t u d y and s i x o t h e r s randomly drawn
from s c h o o l s i n Nsukka local government area, T h e s e
t e n t e a c h e r s made u s e of t h e v a r i o u s mark ing schemes .
+I ; - 7r-'t3- -*
. for t h e t h r e e p r o j e c t s t o ( i n d i v i d u a l l y and v a r i o u s l y )
score t e n randomly s e l e c t e d s t u d e n t s r e s p o n s e s t o each
of t h e t h r e e p r o j e c t s u s e d for a s s e s s i n g t h e
experi rnen t a l g r o u p s ,
A n o t h e r c a t e g o r y of p e r s o n s ( t h o u g h n o t s t r i c t l y
p a r t of t h e sample o f t h i s s t u d y ) i s made u p o f ;en
E v a l u a t i o n e x p e r t s and Mathema t i c s e d u c a t o r s and
t w e n t y e x p e r i e n c e d s e c o n d a r y s c h o o l Ma thema t i c s
t e a c h e r s . T h i s set o f p e r s o n s , a s h a s been s t a t e d
earl ier i n t h i s w r i t e up, were i n v o l v e d i n v a l i d a t i n g
t h e i n s t r ~ r n e ~ t s .
3 - 4 INSTRUMENTS FOR DATA COLLECTION
B a s i c a l l y f o u r t y p e s of jr instruments w e r e u s e d to
collect d a t a f o r t h i s s t u d y . These i n s t r u m e n t s
i n c l u d e d :
( i ) a q u e s t i o n n a i r e - l i k e i n s t r u m e n t composed of
s c a l e d s ta te rnen ts c o n c e r n i n g Mathemat ics l e a r n t
i n s c h o o l s , s t u d e n t s t i c k e d (4) a g a i n s t t h e
s t a t e m e n t s which t h e y i n d i v i d u a l l y a g r e e d si t h
and l e f t o u t t h e s t a t e m e n t s t h e y i n d i v i d u a l l y
d i s a g r e e d w i t h , ( T h i s i n s t r u m e n t i s a scale for
measu r ing s t u d e n t s ' A f f e c t i v e Pe r fo rmances i n
Mathemat ics , I t S A ~ I M t t ) ;
a q u e s t i o n n a i r e - l i k e i n s t r u m e n t compr i s ing
s c a l e d s t a t e m e n t s c o n c e r n i n g some b e h a v i o u r a l
r e s p o n s e s of i n d i v i d u a l s toward a g i v e n
Mathematical p r o j e c t , s t u d e n t s t i c k e d QI/)
a g a i n s t such s t a t e m e n t s t h a t ag reed w i t h t h e i r
i n d i v i d u a l r e a c t i c ~ to a g i v e n p r o j e c t which
t h e y c a r r i e d o u t , his i n s t r u m e n t i s a scale
for Moni tor ing S t u d e n t s ' A f f e c t i v e Learn ing
Outcomes from Mathematical Projects llSMALOMPw ) ;
three Mathematical p r o j e c t s based on J u n i o r Secondary School Mathematics Cur r i cu lum (JSSMC) , t h e s t u d e n t s of t h e e x p e r i m e n t a l g r o u p s c a r r i e d
o u t t h e s e p r o j e c t s ; and
a set of marking schemes used by t e n t e a c h e r s t o score s t u d e n t s ' r e s p o n s e s t o t h e p r o j e c t s
i n (iii) above,
The scale f o r A f f e c t i v e Performance i n Mathematics
(SAPIM) i s made up of f i f t e e n (15) s c a l e d s t a t e m e n t s
(see Appendix I), The scaled s t a t e m e n t s have numer ica l
w e i g h t s a s s i g n e d t o them, The we igh t s r a n g e from one
(1) through seven (7). Weights, f i v e (5 ) , s i x (6), and
seven (7) a r e a s s i g n e d t o s t a t e m e n t s which e x p r e s s
modera te ly p o s i t i v e to v e r y p o s i t i v e a f f e c t i v e behav iour
toward Mathematics i n t h a t order, Weights three ( 3 ) ,
t w o (21, and o n e (1) are a s s i g n e d .to s t a t e m e n t s
e x p r e s s i n g modera te ly n e g a t i v e t o v e r y n e g a t i v e
b e h a v i o u r a l t e n d e n c i e s toward Mathemat ics , a l s o i n
t h a t o r d e r . The w e i g h t f o u r ( 4 ) i s a s s i g n e d to n e u ' t r a l
s t a t e m e n t s . S t u d e n t s were e x p e c t e d and i n s t r u c t e d to
t i c k ((/) a g a i n s t wh ich t h e y a s i n d i v i d u a l s v e r y much
o r e n t i r e l y a g r e e d w i t h . The s c o r e o f a s t u d e n t on
t h e scale, e x p r e s s e d a s a p e r c e n t i s c a l c u l a t e d in t h e
f o l l o w i n g ways:
(i) t h e w e i g h t s , c o r r e s p o n d i n g t o t h e items t i c k e d ,
are added;
(ii) t h e sum a s g o t f rom ( i ) i s d i v i d e d by t h e
number of items t i c k e d ; and
(iii) t h e q u o t i e n t o b t a i n e d i n (ii) i s d i v i d e d by
s e v e n and t h e n m u l t i p l i e d by hundred ,
Thus a s t u d e n t ' s score ( X ) on t h e s c a l e i s g i v e n by
X = Averaqe Weight , 100% 7 1
Mathemat i c s p r o j e c t s u sed as a s s e s s m e n t i n s t r u m e n t s
a r e t h r e e i n number (see Appendix 11). The p r o j e c t s
a r e t a s k s u n d e r t a k e n b y s t u d e n t s of t h e e x p e r i m e n t a l
g r o u p s , The t a s k s i n v o l v e d i n e a c h p r o j e c t arc
e x p r e s s e d as a series of sequenced and r e l a t e d a c t i v i t i e s
o r e x e r c i s e s l e a d i n g to t h e p r o d u c t i o n of a commodity
o r some p r e s e n t a t i o n s , S t u d e n t s are allowed a maximum
95
o f f o u r weeks t o a c c o m p l i s h e a c h p r o j e c t , However t h e
e x e c u t i o n of e a c h p r o j e c t i s e x p e c t e d to t a k e a n
a v e r a g e s t u d e n t a minimum o f t w o weeks, Each o f t h e
p r o j e c t s i s e x p e c t e d to and a c t u a l l y d i d e l i c i t from
t h e s t u d e n t s c o g n i t l v e , p..Bychomotor and a f f e c t i v e
r e s p o n s e s . ' T h e r e s p o n s e s t h a t be longed t o e a c h o f
t h e c o g n i t i v e and psychornotor domains were s c o r e d o u t
of a maximum of 100% u s i n g mark ing schemes drawn u p
f o r e a c h of t h e t h r e e p r o j e c t s ,
The mark ing schemes are g u i d e l i n e s f o r t h e t e a c h e r
( e v a l u a t o r ) tc score t h e r e s p o n s e s of t h e s t u d e n t s t o
t h e p r o j e c t s (see Appendix III), Each of t h e marking
s c h e m e s o u t l i n e s / s p e c i f i e s t h e s k i l l s o r knowledge
e x p e c t e d t o be e x h i b i t e d by s t u d e n t r e s p o n d e r ts and
t h e m a r k ( s ) a s s i g n e d t o e a c h e x h i b i t e d k n o w l e d g e / s k i l l ,
Each mark ing scheme s t i p u l a t e s a l s o t h e m a r k t s ) t o be
awarded to t h e v a r i o u s q u a l i t i e s , q u a n t i t i e s or l e v e l s
of knowledge or s k i l l s so e x h i b i t e d By t h e s t u d e n t s ,
The scale f o r M o n i t o r i n g A f f e c t i v e L e a r n i n g
Outcomes o f Mathemat ics P r o j e c t s (SWLOMP) i s made u p
o f e i g h t (8) s c a l e d s t a t e m e n t s (see Appendix I V ) , and
a n open ended q u e s t i o n ; how d o you f e e l a b o u t t h i s
p a r t i c u l a r p r o j e c t ? Aesponses t o t h i s U n s t r u c t u r e d / f ree
96 , . .
r e s p o n s e i t e m are compared and matched to t h e scaled
s t a t e m e n t s c l o s e l y a s s o c i a t e d o r v e r y s i m i l a r to them.
idlarks are a s s i g n e d t o s u c h r e s p o n s e s a c c o r d i n g l y .
T h e s e marks and the marks a s s i g n e d t o t h e s t a t e m e n t s
t i c k e d by a s t u d e n t are ac'.ded and t h e a v e r a g e computed,
T h e r e s u l t i n g a v e r a g e i s t h e n d i v i d e d by s e v e n and
m u l t i p l i e d by 100%. Thus a s t u d e n t ' s score on t h i s
scale i s g i v e n as i n t h e case o f SAPIM.
3.5 DEVELOPMENT A N D VALIDATION OF INSTRUMENTS
The d e v e l ~ p m e n t of SAPIM for JSS s t u d e n t s was d o n e
f o l l o w i n g a nvn~ber of s t e p s . F i r s t t h e b e h a v i o u r
p a t t e r n s t h a t c o n s t i t u t e a f f e c t i v e l e a r n i n g outcomes ,
a s c o n t a i n e d i n l i t e r a t u r e w e r e rev iewed, The r e v i e w
showed t h a t i n some view p o i n t s a f f e c t i v e b e h a v i o u r
p a t t e r n s are c h a r a c t e r i z e d by i n t e r e s t , m o t i v a t i o n ,
~ n x i e t y , s e l f c o n c e p t , v a l u i n g and a p p r e c i a t i o n . I n
some o t h e r view p o i n t s , a f f e c t i v e b e h a v i o u r p a t t e r n s
a r e m a n i f e s t e d i n terms of r e s p o n d i n g , r e c e i v i n g ,
v a l u i n g , o r g a n i z a t i o n , and c h a r a c t e r i z a t i o n by v a l u e
and complex. F u r t h e r more many a u t h o r i t i e s see t h e
a f f e c t i v e b e h a v i o u r p a t t e r n s o f human b e i n g s as a
taxonomy made u p o f ; i n t e r e s t , a t t i t u d e , v a l u i n g and
a p p r e c i a t i o n . Based on t h e above taxonomy t h e
r e s e a r c h e r c o n s t r u c t e d a t a b l e o f s p e c i f i c a t i o n showing
t h e number of i t e m s o f SAPIM t o be drawn f rom eeeh o f
t h e f o u r components o f t h e a f f e c t i v e b e h a v i o u r taxonomy,
B e a r i n g t h e s e i n mind, and c o n s i d e r i n g t h e items of
s i m i l a r i n s t r u m e n t s c o n t a i n e d i n l i t e r a t u r e , the
r e s e a r c h e r c o n s t r u c t e d s t a t e m e n t s which h e c o n s i d e r e d ,
when r e sponded t o by s t u d e n t s , would i n d i c a t e t h e i r
f e d i n g s , d i s p o s i t i o n s or r e a c t i o n toward Mathemat ics .
T h i r t y such (items) s t a t e m e n t s were made u p and
s u b m i t t e d to e ~ p e r t e d u c a t o r s f o r v e t t i n g / c r i t i q u i n g
(see Appendix i), The v e t t i n g / c r i t i q u i n g i m p l i e d t h a t t
t h e e x p e r t s j u d g e d / c o n s i d e r e d w h e t h e r t h e s t a t e m e n t s :
( i ) i n c l u d e d some t h a t are same o r o p p o s i t t i n
meaning;
( i i ) c o u l d a p p r a i s e s t u d e n t s ' a f f e c t i v e b e h a v i o u r
toward Mathema t i c s i f r e s p o n d e d t o by s t u d e n t s
and
(iii) c o u l d be r e sponded t o / u n d e r s t o o d by JSS s t u d c n t s ,
The twenty-two s t a t e m e n t s t h a t s u r v i v e d t h e
c r i t i q u i n g were t h e n s u b m i t t e d t o e d u c a t o r s , p o s t -
g r a d u a t e s t u d e n t s o f Measurement and E v a l u a t i o n and
some Mathema t i c s t e a c h e r s t o i n d i c a t e t h e d i r e c t i o n
and i n t e n s i t y of a f f e c t i v e b e h a v i o u r toward Mathemat ics ,
98
e a c h o f t h e s t a t e m e n t s e x p r e s s e s , To d o t h a t t h e y were
r e q u e s t e d t o a s s i g n w e i g h t s 5, 6 or 7 t o s t a t e m e n t
t h a t i n d i c a t e p o s i t i v e a f f e c t i v e b e h a v i o u r i n t h a t
o r d e r o f i n t e n s i t y . They were a l s o r e q u e s t e d t o a s s i g n
4 to s t a t e m e n t s t h a t a r e r . e u t r a 1 and 3, 2, o r Z t o
s t a t e m e n t s t h a t e x p r e s s m o d e r a t e l y n e g a t i v e or v e r y
n e g a t i v e a f f e c t i v e b e h a v i o u r toward Mathema t i c s i n
t h a t o r d e r , S t a t e m e n t s t h a t had n o n - s i g n i f i c a n t
d i s c r i p a n c i e s o v e r t h e w e i g h t s a s s i g n e d t o them by t h e
e d u c a t o r s and t e a c h e r s { s e e a p p e n d i x ) w e r e a c c e p t e d
and t h o s e f i f t z e n s t a t e m e n t s c o n s t i t u t e d t h e i n s t r u m e n t
SAPIM. The w e l g h t f i n a l l y a s s i g n e d t o e a c h item i s
t h e median of t h e w e i g h t s a s s i g n e d t o t h e i t e m by t h e
e d u c a t o r s , p o s t - g r a d u a t e s t u d e n t s and t h e t e a y h e r s ,
The r e s u l t i n g s c a l e was t h e n a d m i n i s t e r e d on a sample
of f o r t y class t h r e e s t u d e n t s of JSS l e v e l , The
r e l i a b i l i t y c o e f f i c i e n t of t h e i n s t r u m e n t ( t h r o u g h t h e
s p l i t h a l f a p p r o a c h ) w a s found to be 0-84, T h r e e weeks
la t ter , the scale w a s r e a d m i n i s t e r e d on t h e same
s t u d e n t s and t h e i n s t r u m e n t s ( t e s t - r e t e s t ) r e l i a b i l i t y
c o e f f i c , i . e n t was found t o be 0.92, Both c o e f f i c i e n t s
were computed t h r o u g h t h e Pea r son p r o d u c t moment
app roach ,
The deve lopment of t h e p r o j e c t s i n v o l v e d g o i n g
t h r o u g h t h e JSSMC and f o r m u l a t i n g p r o j e c t s t h a t c o u l d
be u n d e r t a k e n by JSS s t u d e n t s . The p r o j e c t s t h a t were
f o r m u l a t e d / c o n s t r u c t e d w e r e g rouped i n t o t h r e e (see
a p p e n d i x 1111, The f i rs t . j roup o f t e n p r o j e c t s , e a c h
h a s no d e t a i l s , They a re i n c l u d e d i n t h i s w r i t e u p t o
show t h a t many p r o j e c t s c o u l d be drawn from t h e JSSMC.
'The second g r o u p of t h r e e p r o j e c t s is f o r t e a c h i n g a n d
g i v i n g p r a c t i c e t o t h e e x p e r i m e n t a l g r o u p s , o f t h i s
s t u d y , on t h e e x e c u t i o n of s u c h p r o j e c t s , The l a s t
g r o u p o f p r o j e t - t s are t h r e e i n number, The t h r e e
p r o j e c t s are d e t a i l e d and have s p e c i f i c a t i o n s as to
w h a t are t o be done and t h e f i n a l p r o d u c t s t o e x p e c t .
T h e s e l a s t t h r e e p r o j e c t s were u s e d as a s s e s s r e n t
i n s t r i .men t s . (The d e t a i l s and s p e c i f i c a t i o n s , accompany-
i n g e a c h o f t h e t h r e e p r o j e c t s i n t h e l a s t g roup , p o i n t
t o t h e f a c t t h a t e a c h of t h e p r o j e c t s i n t h e first and
second g r o u p s c o u l d be m u l t i p l i e d by v a r y i n g t h e d e t a i l s
and s p e c i f i c a t i o n s t o accompany e a c h o f t hem) ,
To f u r t h e r d e v e l o p and v a l i d a t e t h e l a s t t h r e e
p r o j e c t s , a number of s t e p s were t a k e n , The t a s k s
i n v o l v e d i n u n d e r t a k i n g e a c h o f t h e p r o j e c t s and t h e
knowledge and s k i l l s called f o r w e r e listed, The l i s t
was s u b m i t t e d t o e x p e r i e n c e d Mathemat ics t e a c h e r s and
e d u c a t i o n a l e v a l u a t i o n e x p e r t s , T h e s e p e r s o n s c r i t i q u e d
t h e l i s t ~ u b m i t t e d t o t h e m i n terms o f t h e f o l l o w i n g
cr i ter ia :
c o u l d t h e t a s k s i n j v o l v e d , b e u n d e r t a k e n by JSS
s t u d e n t s who were w e l l exposed t o t h e JSSMC o r
a r e t h e t a s k s , what JSS s t u d e n t s c o u l d
a c c o m p l i s h ;
i s t h e l a n g u a g e used to e x p r e s s t h e t a s k s ,
d e t a i l s , s p e c i f i c a t i o n s o r i n s t r u c t i o n s
a p p r o p * , i a t e / u n d e r s t a n d a b l e f o r / t o JSS s t u d e n t s
y e a r 111;
a r e t h e knowledge and s k i l l s l i s t e d n e c e s s a r y
and s u f f i c i e n t f o r t h e s t u d e n t s to u n d e r t a k e
t h e t a s k s ;
c o u l d t h e s t u d e n t s e x h i b i t such knowledge and
s k i l l s ; and
c o u l d s u c h knowledge and s k i l l s , i f e x h i b i t e d
b e s c o r e d o b j e c t i v e l y ?
Based on t h e t a s k s , knowledge and s k i l l s a c c e p t e d
u n d e r t h e v a r i o u s c r i te r ia by a s t a t i s t i c a l l y
s i g n i f i c a n t m a j o r i t y of t h e v a l i d a t o r s ( b y t h e u s e o f
c h i - s q u a r e tes t ) , masking s c h e m e s were drawn up f o r
t h e t h r e e p r o j e c t s u s e d as e v a l u a t i o n i n s t r u m e n t s (see
a p p e n d i x V),) The mark ing schemes drawn u p were
s u b m i t t e d t o 15 e x p e r i e n c e d WAEC e x a m i n e r s i n
M a t h e m a t i c s t o c r i t i q u e ( see a p p e n d i x V I ) ,
F i n a l l y , t h e S c a l e for M o n i t o r i n g t h e A f f e c t i v e
L e a r n i n g ou tcomes of Ma t h e m a t i c s P r o j e c t s (SMALOMP)
was d e v e l o p e d f o l l o w i n g t h e same s t e p s t a k e n t o d e v e l o p
SHPIM, SMALOMP h a s e i g h t ( s t r u c t u r e d r e s p o n s e )
s t a t e m e n t s and a free r e s p o n s e i t e m . SMALOMP was t r i e d
o u t on f o r t y JSS y e a r 111 s t u d e n t s and i t s r e l i a b i l i t y
c o e f f i c i e n t ( t l r o u g h t h e s p l i t - h a l f a p p r o a c h ) was
f o u n d to be 0, i96 , SMHLOMP as it s t a n d s may be
c o n s i d e r e d s h o r t b u t i t s free r e s p o n s e i t e m g e n e r a t e d
f rom most s t u d e n t s an a v e r a g e o f f o u r s t a t e m e r , t s
i n d i c a t i n g some f e e l i n g t o n e , The i n s t r u m e n t i s
r e s p o n d e d t o a f t e r t h e e x e c u t i o n of each p r o j e c t to
m o n i t o r t h e s t u d e n t s ' f e e l i n g s toward t h e p r o j e c t ,
It i s t h e r e f o r e r e s p o n d e d t o f o r r e p e a t e d times, so
i f i t i s l o n g t h e s t u d e n t s m i g h t h a v e f e l t r e l u c t a n t t o
r e s p o n d i n g t o it. The i n s t r u m e n t , on t h e r e s e a r c h e r
c o n s i d e r a t i o n , would h a v e been l o n g e r b u t t h e v a l i d a t o r s
removed many o f t h e items i n c l u d e d i n t h e d r a f t , They
, s u g g e s t e d t h e i n s t r u m e n t s h o u l d c o n t a i n s u c h s i m p l e
items and needed n o t be long s i n c e i t i s f o r JSS
s t u d e n t s .
3-6 PROCEDURES FOR TREATMENTS AND POST-TESTS __
A t t h e b e g i n n i n g of t h e 1988 s c h o o l y e a r ,
a r r a n g e m e n t s were made such t h a t JSS y e a r I11 s t u d e n t s
f rom e a c h of s c h o o l s One and Two w e r e randomly a s s i g n e d
i n t o f o u r classes. I n e a c h o f t h e two s c h o o l s , t h i r t y
(30) JSS y e a r I11 s t u d e n t s were randomly s e l e c t e d from
e a c h o f t w o p a i r s o f classes t h a t made u p t h e f o u r
classes i n a s c h o o l , By a random p r o c e s s t h e two p a i r s
o f classes i n e a c h s c h o o l , o n e p a i r w a s a s s i g n e d t o
t r e a t m e n t , w h i l e t h e o t h e r p a i r w a s a s s i g n e d t o
c o n t r o l , Then SAPIM was a d m i n i s t e r e d on a l l t h e
s t u d e n t s i n a l l t h e f o u r classes i n e a c h o f s c h o o l s
1 and 2, However, o n l y t h e scores, o f t h e t h i r t y (30)
s t u d e n t s f rom e a c h of t h e two p a i r s o f c l a s s e s i n e a c h
s c h o o l , were recorded as t h e e x p e r i m e n t a l and c o n t r o l
g r o u p s ' p r e t e s t s c o r e s on SAPIM.
I n e a c h o f t h e r e m a i n i n g t w o s c h o o l s SAPIM w a s
a d m i n i s t e r e d on a l l t h e y e a r 111 s t u d e n t s i n t h e f o u r
classes, a t the b e g i n n i n g of t h e 1988 s c h o o l y e a r ,
Al though before t h e a d m i n i s t r a t i o n o f SAPIM, i n e a c h
s c h o o l , t h e JSS y e a r I11 s t u d e n t s had been grouped
i n t o f o u r c l a s s e s , t h e g r o u p i n g was n e i t h e r based on
r e l a t i v e pe r fo rmance i n n o r based on r e l a t i v e a f f e c t i o n
toward Mathemat ics , The cfrcruping w a s more o r less
random. N e v e r t h e l e s s , ba sed on SAPIM scores of t n e
s t u d e n t s , t w o a l t e r n a t e r a n k s g r o u p s o f t h i r t y ( 3 0 )
s t u d e n t s e a c h were i d e n t i f ied/selected i n e a c h of t h e
s c h o o l s 3 and 4. I n each s c h o o l t h e i d e n t i f i c a t i o n /
s e l e c t i o n w a s done i n s u c h a way t h a t e a c h a l t e r n a t e
r a n k s g r o u p o f 30 s t u d e n t s be longed t o a d i s t i n c t / a n d
e x c l u s i v e p a i r of i n t a c t classes, I n e a c h s c h o o l , one
of t h e t w o p a i r s of i n t a c t classes w a s a s s i g n e d t o
e x p e r i m e n t , w h i l e t h e o t h e r p a i r was a s s i g n e d to
c o n t r o l by a s i m p l e random p r o c e s s .
Each o f t h e random a s s i g n m e n t s done i n t h i s s t u d y ,
w a s by a s i m p l e b a l l o t i n g p r o c e s s , I t s h o u l d be n o t e d
t h a t a1 though random or a 1 t e r n a te r a n k s g r o u p i n g were
made, i n e a c h s c h o o l , t r e a t m e n t ( s 1 was/were g i v e n to
s t u d e n t s i n i n t a c t c l a s s e s . T h i s i s t h e c a s e g i n c e
i n e a c h s c h o o l , t h e e x p e r i m e n t a l g r o u p w a s l o c a t e d i n
t w o i n t a c t classes w h i l e t h e c o n t r o l g r o u p be longed
t o t h e o t h e r t w o i n t a c t c l a s s e s , The same Mathema t i c s
t e a c h e r t a u g h t e a c h p a i r of classes c o n t a i n i n g c h e
e x p e r i m e n t a l g r o u p a n d t h e c o r r e s p o n d i n g pair of
classes i n w h i c h members o f t h e c o n t r o l g r o u p s t a y e d .
T h e same u n i t s of work were d o n e i n a l l t h e classes.
B u t t h e t e a c h e r s g a v e i l l ~ s t r a t i o n s a n d p r a c t i c e o n
a c c o m p l i s h i n g t h e t a s k s t h a t c o n s t i t u t e some of he
p r o j e c t s i n g r o u p two o f t h e l i s t o f p r o j e c t s o n l y t o
t h e e x p e r i m e n t a l g r o u p s (see a p p e n d i x X I I ) , A l s o a t
t h e e n d of related u n i t s of work, A p r o j e c t w a s
a s s i g n e d t o t h e e x p e r i m e n t a l g r o u p t o a c c o m p l i s h /
u n d e r t a k e . SALOMP w a s a l so a d m i n i s t e r e d on t h e
e x p e r i m e n t a l g r o u p s e a c h t i m e t h e y f i n i s h e d e x e c u t i n g
a p r o j e c t ,
The u n i t s o f work u s e d f o r t h i s s t u d y i n c l u d e d :
( j ) d r a w i n g / c o n s t r u c t i n g o f n e t s o f r e g u l a r s o l i d
objects (a JSS y e a r 11 work w h i c h n o n e of t h e
t e a c h e r s a n d s t u d e n t s d i d t h e p r e v i o u s y e a r ;
(ii) s u r f a c e areas of c u b o i d s , c y l i n d e r s a n d c o n e s ;
(iii) v o l u m e s of c u b o i d s , c y l i n d e r s a n d c o n e s ;
( i v ) r o u g h estimates of s izes , q u a n t i t i e s ,
c a p a c i t i e s , etc, ;
( v ) b i n a r y c o u n t i n g s y s t e m , r e l a t e d t o punched
c a r d s (1 = y e s a n d 0 = n o ) ;
( v i ) c o d i n g i n f o r m a t i o n u s i n g b i n a r y numbers and
d e c o d i n g such i n f o r m a t i o n ;
( v i i ) i n t e r p r e t a t i o n of w o r d p rob lems i n t o e x p r e s s i o n s
and e q u a t i o n s and v i c e v e r s a ; and
( v i i i ) r e v i s i o n o f s t a t i s t j ~ s ,
U n i t s o f work ( v ) , ( v i ) , and ( v i i ) as p r e s e n t e d
above w e r e d o n e f o r t h r e e weekso So a t t h e end of t h e
f o u r t h week o f t h e f i r s t t e r m p r o j e c t o n e was a s s i g n e d
t o t h e e x p e r i m e n t a l g r o u p s , By t h e f i f t h week, u n i t s
o f work (i 1, (ii), (iii) and ( i v ) were t a k e n and
c o n t i n u e d to t 9 e e i g h t h week, By t h e end of week
e i g h t , p r ~ j e c t two was a s s i g n e d t o t h e e x p e r i m e n t a l
g r o u p s . Indeed by t h e end of week seven t h e members
of t h e e x p e r i m e n t a l g r o u p had a l l s u b m i t t e d t5eir
r e s p c n s e s t o p r o j e c t one. From t h e n i n e t h week t o t h e
e n d o f f irst t e r m , t h e t e a c h e r s a n d s t u d e n t s u n d e r t o o k
o t h e r u n i t s of work as t h e y f o u n d n e c e s s a r y and
s u i t a b l e . By t h e end o f t h e e l e v e n t h week, t h e
s t u d e n t s had s u b m i t t e d t h e i r r e s p o n s e s t o p r o j e c t t w o .
By t h e b e g i n n i n g o f t h e second week i n t h e second term,
u n i t of work ( v i i ) w a s s t a r t e d and was f i n i s h e d n o t
l a t e r t h a n t h e f o u r t h week i n a n y of t h e f o u r s c n o o l s .
So p r o j e c t t h r e e w a s a s s i g n e d t o t h e e x p e r i m e n t a l
g r o u p s (o r i n d e e d t o t h e classes i n which t h e
e x p e r i m e n t a l g r o u p s were) by t h e end of t h e f o u r t h week
o f seoond term, The s t u d e n t s ' r e s p o n s e s to p r o j e c t
three were s u b m i t t e d by s t u d e n t s by t h e end of week
seven . By t h a t t i m e SAPI r I was a d m i n i s t e r e d o n c e a g a i n
t o a l l t h e g r o u p s ,
The f o u r t e a c h e r s who d i d t h e t e a c h i n g and t h e
r e s e a r c h e r g raded t h e s t u d e n t s 1 r e s p o n s e s to e a c h of
t h e p r o j e c t , e a c h t i m e t h e y were s u b m i t t e d , A t t h e
end o f t h e e x p e r i m e n t , t h e r e s e a r c h e r randomly sampled
t h i r t y r e s p o n F e s t o t h e t h r e e p r o j e c t s ( t e n of e a c h
p r o j e c t s ) and s u b m i t t e d them to t h e t e n Mathema t i c s
t e a c h e r s ( sampled f o r t h a t p u r p o s e ) to g r a d e t h e
r e s p o n s e s . O f c o u r s e t h e t e n Mathemat ics t e a c h e r s made
u s e of t h e drawn u p mark ing schemes to g r a d e t h e
s t u d e n t s 1 works s u b m i t t e d t o them.
3 , 7 METHODS OF DATA ANALYSES
Data c o l l e c t e d a t t h e v a r i o u s s t a g e s of t h i s s t u d y
and by t h e u s e of v a r i o u s i n s t r u m e n t s are o r g a n i s e d
w i t h r e s p e c t to t h e p a r t i c u l a r r e s e a r c h q u e s t i o n or
h y p o t h e s i s for which t h e data were collected, Each
set of d a t a so o r g a n i s e d w a s a n a l y s e d w i t h t h e v i ew t o
a n s w e r i n g t h e r e l e v a n t r e s e a r c h q u e s t i o n or t e s t i n g t h e
r e l e v a n t h y p o t h e s i s .
~ f e s e a r c h q u e s t i o n o n e w a s a n s w e r e d by s t a t i n g t h e
number o f e x p e r t e d u c a t o r s a n d t e a c h e r s ( v a l i d a t o r s ) who
a g r e e d a n d who d i s a g r e e d t h a t each t a s k / p a r t of e a c h
p r o j e c t or e a c h e n t i r e p r o j e c t i s s u i t a b l e a n d r e l e v a n t .
The c h i - s q u a r e s t a t i s t i c was employed t o dec ide i f
t h o s e who a y r e e d t h a t e a c h t a s k / p a r t o f e a c h p r o j e c t i s
s u i t a b l e a n d r e l e v a n t , f o r r n e d / c o n s t i t u t e d a s i g n i f i c a n t
m a j o r i t y , Then t h e z-tes t o f d i f f e r e n c e b e t w e e n p r o p o r t i o n s ...
was u s e d t o d e c i d e w h e t h e r t h e p r o p o r t i o n o f v a l i d a t o r s
who a y r e e d t o t h e s u i t a b i l i t y a n d r e l e v a n c e of e a c h p r o j e c t
i s s i g n i f i c a n t l y g r e a t e r t h a n t h e p r o p o r t i o n t h a t d i s a g r e e .
R e s e a r c h q u e s t i o n two w a s a n s w e r e d by u s i n g t h e
c h i - s q u a r e s t a t i s t i c t o a s c e r t a i n i f a s i g n i f i c a n t m a j o r i t y
or n o t a c c e p t e d t h e c a t e g o r i z a t i o n of t h e s t u d e n t s '
r e s p o n s e s ( a s e l i c i t e d by t h e p r o j e c t s ) i n t o c o g n i t i v e
a n d p s y c h o m o t o r l e a r n i n g o u t c o m e s ,
R e s e a r c h q u e s t i o n t h r e e was a n s w e r e d by e m p l o y i n g
two-way a n a l y s e s of v a r i a n c e t o c o m p u t e a n d s tate t h e
r a t e r r e l i a b i l i t y c o e f f i c i e n t s o f t e a c h e r s i n s c o r i n g t h e
v a r i o u s s t u d e n t s ' r e s p o n s e s t o t h e p r o j e c t s , R e s e a r c h
q u e s t i o n f o u r w a s a n s w e r e d by e m p l o y i n g t h e same two-way
AhOVA t o g e t a t a n d s ta te t h e v a r i o u s levels a t w n i c h t h e
t e a c h e r s d i s c r i m i n a t e d among t h e s t u d e n t s t p e r f o r m a n c e s
a s i n d i c a t e d by t h e i n d i v i d u a l s t u d e n t s t sets of scores
a w a r d e d by t h e t e a c h e r s ,
~ n s w e r i n g research q u e s t i o n f i v e i n v o l v e d c o m p u t i n g
(by t h e u s e o f a c o m p u t e r ) a n d s t a t i n g t h e s i m p l e a n d
mu1 t i p l e l i n e a r r e g r e s s i o n c o e f f i c i e n t s among X, Y , a n d
z as d e f i n e d i n t h e q u e s t i o n .
R e s e a r c h q u e s t i o n s i x w a s i n d e e d a n s w e r e d by t e s t i n g
n u l l h y p o t h e s i s o n e , T h e n u l l h y p o t h e s i s o n e t h a t I t t h e
mean a f f e c t i v e p e r f o r m a n c e score o f s t u d e n t s e x p o s e d to
Matnematical p r o j e c t s i s n o t s i g n i f i c a n t l y g r e a t e r t h a n
t h e mean a f f e c t i v e p e r f o r m a n c e s c o r e o f s t u d e n t s n o t so
e x p o s e d " ( p L 0 . 0 5 ) w a s t e s t e d by t h e F - t e s t s ta t is t ics
. computed t n r o u g h ANCOVA.
N u l l h y p o t h e s i s ; t w o , w h i c h s ta tes t h a t t h e m a r k e r
( r a t e r ) r e l i a b i l i t y a n d t h e d i s c r i m i n a t o r y ab i l i t i e s of
t e a c h e r s , making u s e of t h e m a r k i n g schemes to score t h e
s t u d e n t s c o y n i t i v e a n d p s y c h o m o t o r r e s p o n s e s t o t h e
p r o j e c t s ) , a r e n o t s t a t i s t i c a l l y s i g n i f i c a n t ( p L 0 , 0 5 ) , '
w a s t e s t e d by u s i n g a two-way ANOVA F - t e s t s Non-
s i g n i f i c a n t d i f f e r e n c e s among s c o r e s a w a r d e d b y t e a c h e r s
t o i n d i v i d u a l s t u d e n t s as i n d i c a t e d by t h e n o n - s i g n i f i c a n c e
of PlSc/MSg shows r e l i a b i l i t y ( c o n s i s t e n c y a n d a g r e e m e n t )
i n s c o r i n g . Bu t s i f n i f i c a n t d i f f e r e n c e s among t h e mean
s c o r e s o f s t u d e n t s as i n d i c a t e d by t h e s i g n i f i c a n c e o f
t h e P - r a t i o I Y s ~ / M S E shows t h e d i s c r i m i n a t o r y a b i l i t y of
t h e t e a c h e r s ,
N u l l h y p o t h e s i s t h r e e : "each o f t h e s i m p l e or
m u l t i p l e c o r r e l a t i o n and l i n e a r r e g r e s s i o n c o e f f i c i e n t s
b e t w e e n or among s t u d e n t s ' a f f e c t i v e c o g n i t i v e and
p s y c h o m o t o r p e r f o r m a n c e scores, as m o n i t o r e d by t h e
p r o j e c t s , i s n o t s i g n i f i c a n t ( p 0 , 0 5 ) t t was t e s t e d by
l i n e a r r e g r e s s i o n a n a l y s e s t e c h n i q u e c a r r i e d o u t t h r o u g h
t h e s e r v i c e s o f a c o m p u t e r . T h e r e g r e s s i o n a n a l y s e s
t e c h n i q u e s employed h e r e are j u s t i f i e d s i n c e : "so l o n g
as t h e r e are N d i s t i n c t cases e a c h h a v i n g t w o ( o r m o r e )
n u m e r i c a l scores X, Y, ( a n d Z), t h e d e s c r i p t i v e statistics
c o r r e l a t i o n o f c o r r e l a t i o n a n d r e g r e s s i o n may be u s e d 1 '
(Hays , 1 9 7 3 , p, 636) . Hays c o n t i n u e d by w r i t i n g t h a t
I t i n s o d o i n g , w e describe t h e d a t a a s t h o u g h a l i n e a r
r u l e were t o be u s e d f o r p r e d i c t i o n a n d t h i s i s a
p e r f e c t l y a d e q u a t e way t o t a l k a b o u t t h e t e n d e n c y f o r
t h e s e n u m e r i c a l scores t o a s s o c i a t e (or g o t o g e t h e r )
i n a l i n e a r waynt (Hays 1 9 7 3 , p. 6361,
CHAPTER FOUH
DATA ANALYSES
I n this c h a p t e r , t h e a n a l y s e s o f d a t a and t h e
r e s u l t s of s u c h a n a l y s e s are p r e s e n t e d , Data o b t a i n e d
f r o m the a d m i n i s t r a t i o n s o f t h e v a r i o u s i n s t r u m e n t s o f
t h i s s t u a y on t h e s e v e r a l subjects or g r o u p s have been
s u m m d r i s d dnd p r e s e n t e d i n some o f t h e t a b l e s h c r c i n ,
Some ox ~ h e s e data have been a n a l y s e d on t h e b a s i s of
i n d i v i d u a l r e s e a r c h q u e s t i o n o r h y p o t h e s i s and t h e
r e s u l t s of t h e s e a n a l y s e s a r e a l s o s o p r e s e n t e d , On
t h e whole , t h e r e are s i x r e s e a r c h q u e s t i o n s and t h r e e
sets o r h y p o t h e s e s , so t h i s c h a p t e r h a s a b o u t e i g h t
s u b s e c t i o n s ,
4-1 n 1 W L Y S i S OF DATA RELATED TO RESEARCH 'QULLTION ONE
Research Q u e s t i o n 1: Does a s i g n i f i c a n t m a j o r i t y of t h e
v a l i d d t o r s a g r e e t h a t t h e d e v e l o p e d Ma thema t i ca l
p r o j e c t s are s u i t a b l e f o r and r e l e v a n t t o t h e t e a c h i n g
and l e a r n i n g of JSS Mathema t i c s?
T h i s q u e s t i o n i s a n s w e r e d b y s t a t i n g t h e number of
e x p e r t s a n d t e a c h e r s o u t o f 30 of them who a c c e p t e d /
a g r e e d t h a t :
( a ) e a c h o f t h e v a r i o u s t a s k s of e a c h p r o j e c t is:
( i) e x p r e s s e d i n a l a n g u a g e JSY s t u d e n t s c a n
f o l l o w ;
( i i 9 s u i t a b l e for a n d r e l e v a n t t o t h e t e a c h i n g a n d
l e a r n i n g o f JSS M a t h e m a t i c s ; and
( b ) e a c h o f t h e l is ted k n o w l e d g e o r s k i l l s e x p e c t e d
t o be u s e d by t h e s t u d e n t s t o u n d e r t a k e e a c h t a s k i s :
( i n e c e s s a r y
(ii) e x h i b i t a b l e b y t h e s t u d e n t s , a n d
(iii) o b j e c t i v e l y s c o r a b l e by t e a c h e r s (see
k p p e n d t x V I ) . P r o j e c t One h a s s e v e n t a s k s . T h e number of e x p e r t s
a n d t e a c h e r s who a g r e e d t h a t e a c h o f t h e s e v e n t a s k s i s
s u i t a S l e fo r t h e s t u d e n t s , and e x p r e s s e d i n a l a n g u a g e
t h e s t u d e n t s c a n f o l l o w , r a n g e s from a minimum of 23 t o
a maximum of 3 0 o u t of 30, T h e r e are 15 k n o w l e d g e a n d
s k i l l s e x p e c t e d t o be u s e d b y s t u a e n t s t o u n d e r t a k e a n d
a c c o m p l i s h p r o j e c t s o n e , Of a l l t h e 1 5 , a maximum o f 30
b u t a minimum of 2 3 o u t of t h e 30 e x p e r t s a n d t e a c h e r s
a g r e e d t h a t e a c h of k n o w l e d g e a n d s k i l l s i s n e c e s s a r y ,
e x h i b i t a b l e a n d scorable i f e x h i b i t e d ,
P r o j e c t t w o h a s f i v e t a s k s and 1 9 knowledge and
s k i l l s i n v o l v e d , Of t h e f i v e t a s k s , two attracted 22
a g r e e m e n t s o u t o f t h e e x p e c t e d 30 a g r e e m e n t s t o t h e
s u i t a b i l i t y and l a n g u a ~ e a p p r o p r i a t e n e s s o f t h e t w o t a s k s .
F o r t h e r e m a i n i n g t h r e e t a s k s , more t h a n 22 e x p e r t s a n d
t e a c h e r s o u t of 30 of them a g r e e d t h a t t h e t a s k s are
s u i t a b l e and e x p r e s s e d i n a l a n g u a g e t h e s t u d e n t s c a n
u n d e r s t a n d , N ine t een c u t of 30 v a l i d a t o r s a g r e e d t h a t
o n e s k i l l ( 2 d ( i ) c a n be o b j e c t i v e l y s c o r e d , F o r t h e
s k i l l / k n o w l e d g e l a b e l l e d 2 ( e ) ( i ) 2Cl o u t o f 30, 1 9 o u t
of 29 and 1 9 o u t o f 27 v a l i d a t o r s a g r e e d t h a t e a c h s k i l l /
knowledge i s n e c e s s a r y , e x h i b i t a b l e and o b j e c t i v e l y
s c o r a b l e r e s p e c t i v e l y . Fo r t h e rest o f t h e s k i l l s /
knowledge, a minimum o f 22 v a l i d a t o r s o u t o f 30 a g r e e d
t h a t e a c h o n e i s n e c e s s a r y , e x h i b i t a b l e a n d o b j e c t i v e l y
s c o r a b l e o n c e e x h i b i t e d ,
Projects t h r e e h a s f i v e t a s k s and 1 6 knowledge and
s k i l l s i n v o l v e d . Twenty-one o f e x p e r t s and t e a c h e r s
( v a l i d a t o r s ) o u t o f 30 of them a g r e e d t h a t o n e of t h e
t a s k s , l a b e l l e d 3e i s w i t h i n t h e r e a c h o f t h e s t u d e n t s ,
A minimum of 23 and a maximum o f 30 o u t of t h e 30
v a l i d a t o r s a g r e e d t o t h e s u i t a b i l i t y of t h e t a s k s , t h e
a p p r o p r i a t e n e s s o f l a n g u a g e e x p r e s s i o n , t h e n e c e s s i t y
113
o f t h e s ~ i l l s , t h e p o s s i b i l i t y o f t h e s t u d e n t s t o
e x h i b i t t h e s k i l l s and t h e s c o r a b i l i t y of t h e s k i l l s
o n c e e x h i b i t e d ,
Employing t h e c h i - s q u a r e s t a t i s t i c s and a s s u m i n g a
15 t o 1 5 a g r e e t o d i s a g r e e d i s t r i b u t i o n , i t i s f o u n d t h a t :
2 1 o u t o f 30 c o n s t i t u t e s a s i g n i f i c a n t m a j o r i t y ; also 19
o u t o f 27 i s a s i g n i f i c a n t m a j o r i t y ( p L 0 , 0 5 ) , T h e s e
r e s u l t s i n d i c a t e t h a t t h e s i g n i f i c a n t m a j o r i t y of t h e
e x p e r t e d u c a t o r s a n d Piathematics teachers a g r e e t h a t t h e
p r o j e c t s are s u i t a b l e f o r a n d r e l e v a n t to t h e t e a c h i n g a n d
l e a r n i n g o f J;s M a t h e m a t i c s . They a l so a g r e e t h a t t h e
t a s k s a re e x p r e s s e d i n a l a n g u a g e JSS s t u d e n t s c a n f o l l o w
a n d t h a t t h e e x p e c t e d s k i l l s to be e x h i b i t e d by s t u d e n t s
a re o b j e c t i v e l y scorable. (See A p p e n d i x V I , T a b l e 1 for
t h e d i s t r i b u t i o n o f t h e v a l i d a t o r s r e s p o n s e s / r e a c t i o n s ,
to t h e t a s k s , knowledge a n d s k i l l s ) ,
F i n a l l y z - tes t s ta t i s t ics f o r t h e d i f f e r e n c e s
b e t w e e n t h e p r o p o r t i o n s o f v a l i d a t o r s who a g r e e d a n d t h o s e
who d i s a g r e e d t h a t e a c h p r o j e c t i s s u i t a b l e a n d r e l e v a n t '
are h i g h l y s i g n i f i c a n t , T h e tests i n d i c a t e t h a t t h e
p r o p o r t i o n o f v a l i d a t o r s who a g r e e d i s s i g n i f i c a n t l y
g r e a t e r t h a n t h e p r o p o r t i o n w h o d i s a g r e e d (p/O.O5)
S e e 'l'able 2, A p p e n d i x V I , p a g e 2198.
4-2 A I W L Y ~ & S HIUU KEAULTS CONCLHNlIJG HLSLAKCH jJULs'l'lCIW TLJO
M e s e a r c h Q u e s t i o n 2; Does a s i g n i f i c a n t m a j o r i t y of t h e
j u d g e s a y r e e to t h e c a t e g o r i z a t i o n of t h e e x p e c t e d
s t u d e n t s r e s p o n s e s to t h e d e v e l o p e d p r o j e c t s i n t o
c o g n i t i v e a n a p s y c h o m o t o r l e a r n i n g o u t c o m e s ?
The 1 5 i ~ l a t h d r n a t i c s e d u c a t o r s a n d t e a c h e r s a n d
e v a l u a t i o n e x p e r t s who j u d g e d w h e t h e r t h e c a t e g o r i z a t i o n
o f t h e e x p e c t e d l e a r n i n g o u t c o m e s i n t o c o g n i t i v e and
psychor r \o to r d s p e c t s i s i n order , e i t h e r were o n e t i m e
e x p e r i e n c e d ~ A L C e x a m i n e r s or are still so p r e s e n t l y .
T h e table below s h o w s a summary of t h e i r r e s p o n s e s ,
1 2 1 IJurnber O u t Of 'The F i f t e e n J u d q e s riho Aqreed O r u i s a g r e e d T h a t 'i'he C a t e q o r i z a t i o n s A r e I n O r d e r ,
Worn T a b l e 4-2.1, t h e j u d g e s a g r e e d t h a t t h e
~ g r e e
U i s - a y r e e
c a t e g o r i z a t i o n s o i t h e e x p e c t e d l e a r n i n y o u t c o m e s i n t o
c o g n i t i v e and p s y c h o m o t o r c o m p o n e n t s are i n order.
P r o j e c t One
C ~ y n i - t i v e
1 3 -
2
t Projec t Two
Psycho- rno to r
1 2
3
P r o j e c t . T h r e e
Cogn i - t i v e
14
1
Cogni - t i v e
14
1
Psycho- rno t o r
23
2
psycho- motor
14
1
r i ssuming e q u a l d i s t r i b u t i o n t o a g r e e a n d d i s a g r e e
r e s p o n s e s a n d u s i n g t h e c h i - s q u a r e statistics, t h e
number of j u d g e s who a g r e e d t o t h e c a t e g o r i z a t i o n s
i n e a c h case c o n s t i t u t e s a s i g n i f i c a n t m a j o r i t y
( p L O . 0 5 ) .
463 A l W L Y S C S AND KESULTS v J I T H RESPECT T O --- .-- - ..- --- -.-- IIEAEAKCH Q U E S T I O N THREE -.-. --. -.-
Research - Q u e s t i o n 3: To wha t e x t e n t do t e a c h e r s , by
u s i n g t h e in-rking s c h e m e s i n s c o r i n g s t u d e n t s 1 c o g n i t i v e
and psychomotor r e s p o n s e s t o t h e p r o j e c t s , a g r e e among
t h e m s e l v e s w i t h r e g a r d t o scores t h e y awarded i n e a c h case?
The e x t e n t o f ag reemen t i s judged or r e l a t e d by t h e
m a g n i t u d e of t h e r a t e r e l i a b i l i t y c o e f f i c i e n t of t h e
m a r k e r s i n e a c h case,
TABLE 4,3,1: Marker R e l i a b i l i t y C o e f f i c i e n t s 3f The Ma thema t ics T e a c h e s
I P r o j e c t
One I._____...
C o g n i t i v e 0 0 9 9
Psychomotor 0 ,97 0.99 d
From T a b l e 4 ,3 ,1 , ( r a t e r ) marker r e l i a b i l i t y
c o e f f i c i e n t s r a n g i n g f rom a minimum o f 0.97 to a maximum
of 0.99 are c e r t a i n l y v e r y h i g h , T h e s e v a l u e s i n d i c a t e
t h a t i n s c o r i n g t h e s t u d e n t s c o g n i t i v e and psychomotor
r e s p o n s e s t o t h e p r o j e c t s , u s i n g t h e mark ing schemes,
t h e m a r k e r s or , t h e t e a c h e r s v a r i e d m i n i m a l l y or
i n s i g n i f i c a n t l y among themse lves . I t t h e r e f o r e f o l l o w s
t h a t t h e t e a c h e r s a g r e e d among t h e m s e l v e s to a v e r y l a r g e
e x t e n t i n s c o r i n g t h e s t u d e n t s * r e s p o n s e s t o t h e p r o j e c t s .
( F o r t h e r a w scores see Appendix X I , T a b l e s 1-6).
4.4 - RESULTS OF AldALYSLS d1TH RESPECT TO *----
RESEARCH QUESTION FOUR - -
R e s e a r c h --.- Q u e s t i o n Four : To wha t e x t e n t d o t e a c h e r s , by
u s i n g t h e mark ing schemes i n s c o r i n g s t u d e n t s 1 c o g n i t i v e
and psychomotor r e s p o n s e s t o t h e t h r e e p r o j e c t s ,
d i s c r i m i n a t e among t h e s t u d e n t s t p e r f o r m a n c e s ?
The above q u e s t i o n may b e p u t i n a n o t h e r way, t h u s , d o
t h e mean scores awarded by t h e m a r k e r s t o t h e i n d i v i d u a l
s t u d e n t s d i f f e r s t a t i s t i c a l l y among t h e m s e l v e s ?
TABLE 4,2.l: 'I'wo Way ANOVH S t a t i s t i c s Based On S c o r e s Awarded -.--- To The Handom&Selected -.. Responses By The Ten Marke r s
I ' C o g n i t i v e One Psychomotor
C o g n i t i v e Two Psychomotor
T h r e e C o g n i t i v e
Psychomo tor
S i g n i f i c a n t I t
S i g n i f i c a n t I
T a b l e 4.4.1 shows t h a t f o r e a c h c a t e g o r y of
r e s p o n s e s t o e a c h p r o j e c t , t h e means o f t h e t e n sets of
scores ( e a c h set h a s t e n scores) as awarded t o t h e
r e s p o n s e s by t h e t e n m a r k e r s (see Appendix X I , T a b l e 1)
are s i g n i f i c a n t l y d i . 3 e r r n t beyond t h e 95% c o n f i d e n c e
i n t e r v a l . 'This i m p l i e s t h a t a t l e a s t i n 95% o f s u c h
o c c a s i o n s o f s c o r i n g s t u d e n t s r e s p o n s e s t o p r o j e c t s ,
t h e t e a c h e r s w i l l d i sc r : -mina te among t h e p e r f o r m a n c e s
o f s t u d e n t s , The r e s u l t s show, t h e r e f o r e , t h a t by
u s i n g t h e mark ing schemes i n s c o r i n g s t u d e n t s t
c o g n i t i v e a n d psyahomotor r e s p o n s e s to t h e p r o j e c t s ,
t h e t e a c h e r s t o a v e r y l a r g e e x t e n t d i s c r i m i n a t e d among
t h e s t u d e n t s v perfo:mances,
8.5 RESULTS OF ANALYSES dITH RESPECT TO P.ESEARCH QUESTION -. FIVE
R e s e a r c 5 Q u e s t i o n 5: What are t h e s i m p l e and m u l t i p l e - - r e g r e s s i o n e q u a t i o n s among t h e means of s t u d e n t s f
c o g n i t i v e , psychomotor a n d a f f ec t i v e pe r fo rmances a s
m o n i t o r e d by t h e t h r e e p r o j e c t s ?
X, Y, and Z r e p r e s e n t t h e means of s t u d e n t s f c o g n i t i v e ,
psychomotor and a f f e c t i v e pe r fo rmance scores i n t h e
t h r e e p r o j e c t s r e s p e c t i v e l y , The r e g r e s s i o n c o e f f i c i e n t s
w e r e computed by t h e u s e of a compu te r , Below are soma
tables showing some o f t h e r e s u l t s a s s o c i a t e d wi th
comput ing r e g r e s s i o n c o e f f i c i e n t s ,
TABLE 4.5 -1: Means, Skandar-d D e v i a t i o ~ And Number o f C a s e s O f The
I Variable Mean Y Z 7 1 D e v i a t i o n --....-. I-.--
1 1
TABLE 4 S , 2 : C o r r e l a t i o n Coef fLc>.€!! Between P a i r s O f The - - ----.- variables X, Y and - - -- Z
C o n s i d e r i n g t h e number of cases i n v o l v e d t h e c o r r e l a t i o n
c o e f f i c i e n t s shown i n T a b l e 4,5,1 are a p p r e c i a b l y h i g h
o r v e r y h i g h ,
TABLL 4.5.3: S i m p l e L i n e a r i iegr-e-ssion -- -- -.-- - -- ..-d
C o e f f i c i e n t s Between ------ - -- - 4..-
Variables And T h e i r .-..--.-- -- .-- - C o r r e s p o n d i n q -- -- C o n s t a n .--. - .--- ts,
From T a b l e 4.5,3 t h e r e g r e s s i o n e q u a t i o n s are:
(v) 2 = 0.85X + 9.49 a n d
( v i ) Z = 0.85Y + 8.12
T h e da ta above show t h a t a l l t h e s i m p l e l i n e a r
r e g r e s s i o n c o e f f i c i e n t s a re p o s i t i v e .
TABLE 4-5.4: M u l t i p l e -.--. L i n e a r C o e f f i c i e n t s A m 9 ~ a r i a b l > s X and Z -----L-.-- - . - . -
120
From T a b l e 4-5.4, t h e m u l t i p l e l i n e a r r e g r e s s i o n
e q u a t i o n s are t h e r e f o r e :
(1) X = 1,482 - 0,46Y - 0.84
( 2 ) Y = 1,652 - 0,66X + 1.96 and
( 3 ) Z = 0.54X + 0,42Y + 2.08,
The l i n e a r e q u a t i o n 1, e x p r e s s i n g X i n terms of Z
and Y , shows t h a t t h e c o e f f i c i e n t o f Z i s p o s i t i v e
w h i l e t h a t o f Y i s n e g a t i v e , A t t h e same t i m e , t h e
m a g n i t u d e of t h e c o e f f i c i e n t of Z i s a b o u t t h r e e t i m e s
t h e m a g n i t u d e o f t h e c o e f f i c i e n t of Y o I n l i k e manner ,
t h e l i n e a r e q u a t i o n 2, e x p r e s s i o n Y i n terms of Z and X,
i n d i c a t e s t h a t t h e c o e f f i c i e n t of Z i s p o s i t i v e x h i l e
t h a t of X i s n t g a t i v e . However, t h e magn i tude of t h e
c o e f f i c i e n t o f Z i s a b o u t t w o and a h a l f times t h e
m a g n i t u d e of t h e c o e f f i c i e n t of X, I n e q u a t i o n 3,
e x p r e s s i n g Z i n terms o f X and Y , t h e c o e f f i c i e n t s o f
b o t h X and Y are p o s i t i v e , But t h e r a t i o o f t h e
c o e f f i c i e n t s o f X and Y i s a p p r o x i m a t e l y 5:4, That i s
t o s a y t h a t f o u r times t h e c o e f f i c i e n t of X i s
a p p r o x i m a t e l y e q u a l t o f i v e times t h e c o e f f i c i e n t of Y,
4.6' IIESULTS OF IU~I~LYSLS Aid0 'L'GSTS ;JITI-l ItLSPLCT TO HYPOTHESlS ONE
Nu1 1 Hypo t h e s i s 1 ; 'I'he rnean a f f e c t i v e pe r fo rmance s c o r e
i n P la themat ics o f s t u d e n t s exposed t o t h e u s e of
P la therna t ica l p r o j e c t s i s n o t s i q n i f i c a n t l y g r e a t e r t h a n
t h e mean a f f e c t i v e pe r fo rmance score i n Ma thema t i c s o f . s t u d e n t s n o t exposed to t h e Ma thema t i ca l p r o j e c t s
CpL0.05) .
A l t e r n a t i v e H y p o t h e s i s 1: The mean a f f e c t i v e
pe r fo rmance score i n P la themat ics of s t u d e n t s exposed
t o t h e u s e o f Ma thema t i ca l p r o j e c t s i s s i g n i f i c a n t l y
g r e a t e r t h a n t h e mean a f f e c t i v e pe r fo rmance score i n
P ia themat ics o f s t u d e n t s n o t exposed t o the Mathemat i ca l
p r o j e c t s ( p G 0 . 0 5 ) .
& l o w are tables showing r e s u l t s o f a n a l y s e s
o f d a t a and tests r e l a t e d t o H y p o t h e s i s One.
TABLE 4,6.1: P r e - t e s t And Post-test Mean -- S c o r e s And S tandard D e v i a t i o n s ,On SAPIM Accordinq To Subqroups And Groups -
score I ! --.__I
.-L" , A 1 t e r n a t e Ranks ~ a n d o m i s e d
Groups A l l Groups Groups -
P r e - test
Experi- Scores menta l
P r e - 57.80 59.77 test
Post- I test
Exper'- o n t r o l cOntrO1f men ta l
62.031 58.95 59-95 -7- ---
Table 4-6-1 shows t h e fo l lowing :
1, With r e s p e c t to t h e P r e t e s t , t h e mean score of t h e
10032 12.61 4 11058 11.57 1lm1O 1 -
e x p e r i m e n t a l g roups of t h e a l t e r n a t e r a n k s t g roups i s
n 60 60 60
67-05 59.28 68.93
g r e a t e r than t h e mean s c o r e of t h e c o n t r o l g r o u p s of t h e
60 120 --I-
120
61.22 67.99 60.25
same a l t e r n a t e r a n k s t g roups by 0.32, T h e i r s t a n d a r d
test 9.85 11.61 11.17 1 11.56 1 1O074 11.55
TY ~e o f Over A l l o r Grand No. o f
test Mean S tandard D e v i a t i o n Cases
d e v i a t i o n s d i f f e r by 0,11.
With r e s p e c t t o t h e same p r e t e s t u s i n g SAPIPA, t h e
mean score o f t h e c o n t r o l g r o u p s o f t h e r andomised
g r o u p i n g s i s g r e a t e r t h a n t h e mean s c o r e o f t h e e x p e r i -
m e n t a l g r o u p s o f t h e same randomised g r o u p i n g s by 2.26,
The s t a n d a r d d e v i a t i o n s of t h e i r scores d i f f e r by 1.03.
T k z mean score f o r a l l t h e e x p e r i m e n t a l g r o u p s of t h e
s t u d y i s less t h a n t h e mean score for t h e c o n t r o l g r o u p s
b y 1.00, But t h e s t a n d a r d d e v i a t i o n of t h e s c o r e s of
t h e e x p e r i m e n t a l g r o u p s i s g r e a t e r t h a n t h a t of t h e
c o n t r o l g r o u p s by 0.47. A s r e g a r d s t h e p o s t - t e s t
scores
(i>
( i i )
( i i i )
( i v )
o f t h e sampled s t u d e n t s on SAPIM:
t h e mean score of t h e c o n t r o l g r o u p s of t h e
a l t e r n a t e r a n k s g r o u p i n g s i s 7,77 less t h a n t h e
mean score of t h e c o r r e s p o n d i n g e x p c r i m c n t a l
g r o u p s ;
t h e mean score of t h e c o n t r o l g r o u p s of t h e
randomised g r o u p s i s 7.71 less t h a n t h e mean
scorc of t h e c o r r e s p o n d i n g e x p e r i m e n t a l g r o u p s ;
t h e mean score o f a l l t h e c o n t r o l g r o u p s of
t h i s s t u d y i s less t h a n t h e mean s c o r e of t h e
e x p e r i m e n t a l g r o u p s by 7.74; and
t h e s t a n d a r d d e v i a t i o n s of the scores of t h e
c o n t r o l g r o u p s are c o m p a r a t i v e l y a b i t g r e a t e r
i n each case t h a n t h a t o f t h e i r c o r r e s p o n d i n g
e x p e r i m e n t a l g r o u p s ,
T a b l e 4,6.1 i n d i c a t e s f u r t h e r m o r e t h a t : t h e T c a n s
and s t a n d a r d d e v i a t i o n s o f t h e scores of t h e c o n t r o l
g r o u p s i n t h e p r e t e s t d i f f e r v e r y s l i g h t l y f rom such
statistics o f t h e same g r o u p s i n t h e p o s t - t e s t ; w h i l e
t h c rncan scores o f c;rc e x p e r i m e n t a l g r o u p s i n t h e p r e -
t e s t a r e a l l a p p r e c i a b l y less t h a n t h e mean scores of
t h e same g r o u p s i n t h e p o s t - t e s t , On t h e o t h e r hand, t h e
s t a n d a r d d e v i a t i o n s o f t h e scores of t h e e x p e r i m e n t a l
g r o u p s i n t h e p r e - t e s t are a l l s l i g h t l y g r e a t e r t h a n
t h o s e of t h e same g r o u p s i n t h e p o s t - t e s t , w h i l e t h e r e
i s n o such t r e n d a p p a r e n t w i t h t h e s t a n d a r d d c v l a t i o n s
of t h e scores of t h e c o n t r o l g r o u p s as r e g a r d s t h e
p r e t e s t and t h e p o s t - t e s t .
F i n a l l y , T a b l e 4.6.1 s h o w s t h a t f o r t h e p r e t e s t :
(i) t h e mean scores of t h e t r e a t e d g r o u p s and t h e
c o n t r o l g r o u p s o f t h e a l t e r n a t e r a n k s n g r o u p i n g s
d i f f e r by 0-32, w h i l e t h e s t a n d a r d d e v i a t i o n s o f
t h e scores of t h e same cjroups d i f f e r by 0.11; and
( i i ) as r e g a r d s t h e randomised g r o u p i n g s , t h e means
and t h e s t a n d a r d d e v i a t i o n s o f t h e scores o f t h e
t r e a t e d g r o u p s and t h e c o n t r o l g r o u p s d i f f e r by
-2.26 and 1-03 r e s p e c t i v e l y .
' I , . , ' . . . . .
TABLE 4,6.2: AIJCOVA TABLE - A l l - - - Experimental Groups Versus A l l Cont ro l Groups.
By I: Group of s t u d e n t s (Experimental o r c o n t r o l
Wi th P. (Cova r i a t e ) S tuden t s P r e t e s t s c o r e s on SAPIM
Signi f i- cance o f
F --- -
CovarLate raw r e g r e s s i o n c o e f f i c i e n t ' i s 0-80,
F- ra t io
a ..-
Degrees j ou rces of Sum of of Var i a t i on Freedom - -
From Table 4.6.2, t h e main e f f e c t s (which i s t h e
Mean Square
d i f f e r e n c e due t o t he t rea tment ) 1s s i g n i f i c a n t
( p L 0.00). That means t h a t i t i s s i g n i f i c a n t beyond
t h e set - l e v e l ( p L 0-05) f o r t e s t i n g t h e n u l l
Main E f f e c t s 23321.19 1 2
hypothes i s , This r e s u l t shows t h a t t h e ad jus t ed mean
score of t h e exper imental groups i s s i g n i f i c a n t l y g r e a t e r
than t h e ad jus ted mean score of t h e c o n t r o l groups
( p L 0.05). T h e n u l l hypothes i s is , t h e r e f o r e , n o t
uphcld, r a t h e r t h e a l t e r n a t i v e hypothes i s i s upheld on
t h e evidence of the d a t a ,
j-- 1 4591.48 i 1 4591.48 / P 119129.041 1 19529.04 485.34 0.00 i
-- -4 / Sxplained , 23321.19 2 0.00
Residual 1 9536.35 1 232 40.24
To ta l 137.48 1 -
289.79 11660.59 0.00
TABLE 4.6.3: - ANCOVH T a b l e For A l t e r n a t e Ranks Groupinqs
Q: S t u d e n t s ' P o s t - t e s t s c o r e s on SAPIM by .... I: S t u d e n t s 1 Group (Exper imen ta l and C o n t r o l ) w i t h -
P: f e o v a r i a t e ) S t u d e n t s 1 P r e t e s t s c o r e s on SAPIM
C o v a r i a t e R a w R e g r e s s i o n C o e f f i c i e n t i s 0.87,
S i g n i f i- c a n c e of
T a b l e 4,6,3 shous t h a t t h e main e f f e c t s ( t h a t i s
. - 4
F- r a t i o
S o u r c e s o f
172.57
281.38
62-74 1- :!- 1 172,57
I
1
f Main ~ f f a c t s ! 11594,71 i 2 1 5797.35
e x p o s u r e t o u s e o f p r o j e c t s i n Mathematics) i s s i g n i f i c a n t
Degrees Sum of ! Mean
L I
P -- f Expla ined
1 R e s i d u a l
( p L 0.00i . I n o t h e r words, s i n c e f o r t h e a l t e r n a t e
V a r i a t i o n 1 squares /Freedom Of I Squa re
r a n k s t g r o u p i n g s , t h e p o s t - t e s t mean score o f t h e
1
2 0 4 0 0 B 2 9452.63 1 1
l l S 9 4 . 7 l i 2
3930.46 i 11'1
e x p e r i m e n t a l g r o u p s i s g r e a t e r t h a n t h e post-test mean
94 5 2
2O*O. 62
- 5797.35
33-59
1 T o t a l i 15525.16 ; 119 -4.
s c o r e of t h e c o n t r o l g r o u p s (see T a b l e 4.6.1), t hen t h e
130.46
r e s u l t s o f T a b l e 4,6,3 i n d i c a t e among o t h e r t h i n g s that
t h e a d j u s t e d mean =core of t h e e x p e r i m e n t a l g r o u p s i s
s i g n i f i c a n t l y g r e a t e r t h a n t h a t o f t h e c o n t r o l g r o u p s
beyond t h e 0.05 l e v e l , Hence t h e n u l l h y p o t h e s i s i s n o t
uphe ld , the a l t e r n a t i v e h y p o t h e s i s i s upheld i n s t e a d . I
TABLE 4-6.4: AiVCOVA Table f o r Randomised Groupinqs
Q: S t u d e n t s t P o s t - t e s t s c o r e s on SAPIM, by
I : Group of s t u d e n t s (Exper imenta l o r c o n t r o l 1, with
P: (Cova r i a t e ) S t u d e n t s P r e t e s t s c o r e s on SAPIM
G r c f of V a r i a t i o n
C o v a r i a t e raw r e g r e s s i o n c o e f f i c i e n t i s 0,87.
C o r r e l a t i o n c o e f f i c i e n t between Q and P, ( rQp) 2 0.82,r Qp = 0.68,
Sum of F-
Squares Freedom
Table 4.6.4 shows t h a t based on t h e d a t a c o l l e c t e d
from t h e randomised groupings , t h e main effects is
' 1 Main E f f e c t s 11495.02 1
s i g n i f i c a n t a t 0,GO. Thi s means t h a t s t u d e n t s ' exposure
I 1
I
i F
Explained
t o Mathematics p r o j e c t s h a s a s i g n i f i c a n t e f f e c t s on
s t u d e n t s 1 a f f e c t i v e performance i n Mathematics beyond
t h e set l e v e l of 0.05, The n u l l hypothes i s , on t h e
b a s i s ~f t h i s r e s u l t , i s n o t upheld. Evidence pe rmi t s
ho ld ing on t o t h e a l t e r n a t i v e hypo thes i s t h a t t h e
a d j u s t e d mean g a i n s of t h e exper imenta l groups i s
! 2
2487.02 1 1
9827.95 1
11493.02 2 5746.51 122,28
Res idua l 11? 47-00
TcL81 1 I 1 119 142.79
128
s i g n i f i c a n t l y g r e a t e r t h a n t h e a d j u s t e d mean score of
t h e c o n t r o l g r o u p s ( p L O . 0 5 ) .
The basic a s s u m p t i o n s u n d e r l y i n g t h e u s e o f ANCOVA
are rnet i n t h e c a s e s above . T h e c o v a r i a t e scores, P,
c o r r e l a t e h i g h l y w i t h t h e p o s t - t e s t scores 4, as shown
by t h e c o r r e l a t i o n c o e f i i c i e n t o f 0.82 a n d t h e , r a w
r e g r e s s i o n c o e f f i c i e n t s o f 0.80, 0.87, a n d 0.78. A l s o
t h e s e raw r e g r e s s i o n c o e f f i c i e n t s d o n o t v a r y s i g n i f i c a n t l y
( p 0 .05) .
F u r t h e r m o r e T a b l e s 4.6.3 a n d 4.6.4 show t h a t :
t i ) the v a r i a t i o n c o n t r i b u t e d by t h e main e f f e c t
f o r t h e r a n d o m i s e d g r o u p i n g s i s less t h a n t h e
v a r i a t i o n c o n t r i b u t e d by t h e same s o u r c e for
t h e a1 t e r n a t e r a n k s g r o u p i n g s ;
(ii) t h e v a r i a t i o n c o n t r i b u t e d by t h e c o v a r i a t e f o r
t h e r a n d o m i s e d g r o u p i n g s i s g r e a t e r t h a n t h a t ,
f o r t h e a l t e r n a t e r a n k s ' g r o u p i n g s ; a n d
(iii) t h e error v a r i a n c e ( r e s i d u a l ) f o r t h e r a n d o m i s e d
g r o u p i n g s i s g r e a t e r t h a n t h e error v a r i a n c e f o r
t h e a l t e r n a t e r a n k s g r o u p i n g s .
4.7 RESULTS OF ANALYSIS AND TESTS KlLWITED TO HYPOTHESIS TWO
fdu l l H y p o t h e s i s - 2: The m a r k e r ( ra te r ) r e l i a b i l i t y a n d
d i s c r i m i n a t o r y a b i l i t i e s o f t h e t e a c h e r s , (making u s e of
t h e m a r k i n g s c h e m e s to score t h e s t u d e n t s ' c o g n i t i v e a n d
p s y c h o m o t o r r e s p o n s e s t o t h e p r o j e c t s ) , are n o t s t a t i s t i c a l l y
s i g n i f i c a n t ( p L O . 0 5 ) .
A l t e r n a t i v e Hypo thes i s 2: T h e marker ( r a t e r ) r e l i a b i l i t y
anu d i s c r i m i n a t o r y a b i l i t y of t h e t e a c h e r s , (making u s e
o f t h e inarkiny schemes t o s c o r e t h e s t u d e n t s ' c o g n i t i v e
and psychomotor r e s p o n s e s t o t h e p r o j e c t s ) , are
s t a t i s t i c a l l y s i g n i f i c a n t ( p L 0 . 0 5 ) .
Melow is a t a b l e showing v a r i o u s two-way ANOVA
s ta t i s t ics computed from s c o r e s awarded t o t h e t e n
randomly s e l e c t e d r e s p o n s e s by t h e ten t e a c h e r s .
TABLE 4.7.1: Two-day ANOVA S t a t i s t i c s , Based On % o r e s Awarded To T h e Handomlv S e l e c t e d Responses Uy The Ten Markers , Compris inq ; MSH, MSc, MSE and F - r a t i o and t h e S t a t i s t i - .- cal S i q n i f i c a n c e Decis ion .
. p r o j e c t
One
Two
T h r e e
Response Cateqory
t i v e
Psycho- rno Lor
t i v e
motor
Cogni- t i v e
Psycho- motor
ElsR ( i )
1133.69
1111.64
1510.22
2277.45
2170.48
1408.48 1
MSC ( i i I i M S E
7.177
4.74
2.10
2.379
2.262
2.28
2.25
3.59
1.24
2.73
1.78
1 - 7 6
' D e c i s i o n ~ L 0 . 0 5
ig)S&e ( i ) sigo (ii) n. s i g . (i) s i g . (ii) n. s i g .
( i ) s i g . ( i i ) s i g .
(i) s i g . ( i i ) sige ( i) s i g . ( i i ) n. s i g .
F-rat 'io ( i )
504.09
309.30
1120.88
835.45
1220.75
802.10
(ii)
3.191
1.319
1.69
0.873
1.272
1.29
From t h e t ab le 4.7.1 o f p a g e 129 , a l l . t h e F - r a t i o s
M S ~ / M S ~ are h i g h l y s i g n i f i c a n t , t h e y are a l l s i g n i f i c a n t
beyond t h e set p r o b a b i l i t y l e v e l o f 0.05. T h e s e s i g n i f i -
c a n t r a t i o s i n d i c a t e t h a t t h e mean c o g n i t i v e a n d psycho-
motor scores of t h e t e n r e s p o n s e s are s t a t i s t i c a l l y
d i f f e r e n t f o r e a c h of t h e t h r e e p r o j e c t s , So t h e m a r k e r s i
( t e a c h e r s ) n o t e d the d i f f e r e n c e s among the r e s p o n s e s a n d
s c o r e d t h e m a c c o r d i n g l y . T h i s r e s u l t shows t h a t t h e raters
d i s c r i m i n d t o r y a b i l i t y i s s t a t i s t i c a l l y s i g n i f i c a n t .
From t ab le 4,7,1 a l so , f i v e o u t o f s i x F - r a t i o s .- -
M S c / M S E are i n s i g n i f i c a n t a t t h e 0.05 l e v e l , T h a t i s to
s a y t h a t i n s c o r i n g t h e p s y c h o m o t o r r e s p o n s e s to t h e t h r e e
p r o j e c t s a n d t h e c o g n i t i v e r e s p o n s e s t o p r o j e c t s two a n d
t h r e e , t h e mean s c o r e s awarded by t h e t ' e a c h e r s t o e a c h
a s p e c t o f e a c h p r o j e c t d o n o t d i f f e r s i g n i f i c a n t l y a t
t h e 0.05 l e v e l , The t e a c h e r raters, i n t h e f i v e cases
were i n a g r e e m e n t , They c o n s i s t e n t l y awarded scores t h a t
a re n o t s i g n i f i c a n t l y d i f f e r e n t t o each i n d i v i d u a l s t u d e n t ' s
p s y c h o m o t o r r e s p o n s e s t o e a c h o f t h e t h r e e p r o j e c t s . The
teachers a l s o ( c o n s i s t e n t l y ) awarded n o n - s i g n i f i c a n t l y
d i f f e r e n t scores to e a c h i n d i v i d u a l s t u d e n t ' s c o g n i t i v e
r e s p o n s e s to e a c h o f t h e two p r o j e c t s o u t of t h e t h r e e ,
OYhe n u l l h y p o t h e s i s t h a t t h e t e a c h e r s ' r e l i a b i l i t y a n d
d i s c r i m i n a t o r y a b i l i t i e s i s n o t s t a t i s t i c a l l y s i g n i f i c a n t
i s n o t u p h e l d . The a l t e r n a t i v e h y p o t h e s i s i s u p h e l d i n s t e a d ,
4.8 RESULTS OF ANALYSLL -- AND TELTS RLLiiTED TO HYPOTHESXS THREE
i i u l l h y p o t h e s i s : Each of t h e s i m p l e o r m u l t i p l e
c o r r e l a t i o n and l i n e a r r e g r e s s i o n c o e f f i c i e n t s between o r
among s t u d e n t s ' a f f e c t i v e , c o g n i t i v e and psychomotor
per formance s c o r e s a s moni tored by t h e Mathematical p r o j e c t s
i s n o t s i g n i f i c a n t ( p C 0 . 0 5 ) .
i i l t e r n a t i v e - i y p o t h e s i s 3 : Each of t h e s i m p l e o r m u l t i p l e
c o r r e l a t i o n and l i n e a r r e g r e s s i o n c o e f f i c i e n t s between o r
among s t u d e n t s ' a f f e c t i v e , c o g n i t i v e and psychomotor
per formance s c o r e s a s moni tored by t h e Mathematical p r o j e c t s
i s s i g n i f i c a n t ( p LO.05).
Below i s a t a b l e s u m m a r i z i n g / i n d i c a t i n g t h e s t a t i s t i c s
t o be t e s t e d f o r s i g n i f i c a n c e or o the rwise .
'IiiBLE 4.8: S imple o r ib1ul t ip le C o r r e l a t i o n - And L i n e a r Heqress ion C o e f f i c i e n t s Between O r Amonq X , Y and Z and T h e i r Correspond- inq C o n s t a n t s .
I *
~ u l t i p l e L i n e a r R e q r e s s i o n
COeffi- cients
-0.46
1.48 -0.66
1.65 0.54
0.42
5 imple L i n e a r Heqress ion
Oepen-lLnde- d e n t
X
Y
Z
c o n s t a n t
-0.84
1.96
2.08
' o r r e l a t i o n E o e f f i c i e n t
C o e f f i - ci en t
0.79
1.03 0.75
0.97 0.85
0 -85
pendent
Y
L X
A X
Y
Simple
0.77
C o n s t a n t
1 1 1 7
-2.51 17.61
3.62 9.49
8.12 .
Mul- ti le
0.96 0.94 1 0.77 ' 0.91 0.94
0.96
0.98 0 -91
n l t h o u g l l i n a n s w e r i n g r e s e a r c h q u e s t i o n f i v e ( p a g e s
l l i ' - l2O a l l t n e s i m p l e and mu1 t i p l e c o r r e l a t i o n
c o e i f i c i r n i s a r e , by t h e i r numera l v a l u e s , c o n s i d e r e d t o
be h i g h o r v e r y h i g h , t h e r e i s s t i l l t h e need t o test
w h e t h e r t l ~ e y a r e s t d t i s t i c a l l y s i g n i f i c c l n t ( p < O.C5).
T h i s test is n e c e s s a r y s i n c e c o r r e l a t i o n c o e f f i c i e n t s arc
n o t a b s o l u t e by t h e i r v a l u e s . T h e i r s i g n i f i c a n c e s o r o t h e s -
w i s e depcnd a lso on t h e numbers o f scores ( s a m p l e sizes)
i n v o l v e d .
If tests ;re n e c e s s a r y to d e t e r m i n e w h e t h e r cjiven
c o r r e l a t i o n c o e f f i c i e n t s are s t a t i s t i c a l l y s i g n i f i c a n t o r
o t h e r w i s e , tests a r e more s o i m p o r t a n t t o a s c e r t a i n t h e
s i y n i i i c t i n c e of g i v e n s i m p l e and m u l t i p l e r e g r e s s i o n
c o e f f i c i e n t s f o r a number o f r e a s o n s . S i g n i f i c a n c e s o f
r e g r e s s i o n c o e f f i c i e n t s l i k e c o r r e l a t i o n c o e f f i c i e n t s
a r e n o t c o n s i u e r e d by t h e i r n u m e r i c a l v a l u e s a l o n e , t h e
s a m p l e s i z e s i n v o l v e d a r e v e r y i m p o r t a n t . I n t h e c a s e
o f m u l t i p l e r e g r e s s i o n ( f o r t h e p u r p o s e o f p r e d i c t i o n ) ,
t h e s i g n i f i c a n t l e v e l a t wh ich e a c h i n d e p e n d e n t v a r i a b l e
i s c o n t r i b u t i n g to p r e d i c t i n g t h e d e p e n d e n t v a r i a b l e and
a l s o t h e c o m p a r a t i v e c o n t r i b u t i n g l e v e l s are a p p r o p r i a t e l y
d e t e r r ; . ~ i n e d by tests a s are i n t h e page t h a t follovs,
TABLE 4,8,1: - R e s u l t s Of t-tests And - -*- l... . t l .
Between Var i ab l e s
Var i ab l e s ~ o r G l a - Regre- t ton t-test s s i o n
Depen- Indepen- Coeff i - f o r r. Coef f i - d e n t d e n t c i e n t n (r)
X Y 0-77 13-07 0.79 0-06 170,7
X Z 1 0.94 29-66 1 - 0 3 0-03 878.4
X 0.77 113.07 1 0.75 0.06 170-7
; 0.04 f 535.6
878.4
0.85 : 535.6 -- ..------A i T h e r e s u l L s presented i n Table 4-8.1 show t h a t a l l
t h e s imple c o r r e l a t i o n and l i n e a r r e g r e s s i o n c o e f f i c i e n t s
between p a i r s of X, Y and Z are s i g n i f i c a n t l y s ta t i s t ica l
( d i f f e r e n t from z e r o ) , T h i s i s t h e c a s e s i n c e a l l t h e
c a l c u l a t e d t-test and F - t e s t s t a t i s t i c s a r e g r e a t e r
than t h e t a b l e v a l u e s of t and F - /F(0.05, 1.118) and
t (0 ,05 , 118) w h i c h a r e 3.92 and 1.98 r e s p e c t i v e l ~
TABLE 4.8.2: R e s u l t s Of F - t e s t s On Mu1 t i p l e L i n e a r R e q r e s s i . - And C o r r e l a t i o n C o e f f i c i e n t s Amonq X, Y and 2.
F - t e s t I f o r B
-
T h e r e s u ? t s p r e s e n t e d i n T a b l e 4.8.2 i n d i c a t e t h a t
( V a r i a b l e s !
Indepen-
a l l t h e m u l t i p l e c o r r e l a t i o n and l i n e a r r e g r e s s i o n
c o e f f i c i e n t s a r e s t a t i s t i c a l l y s i g n i f i c a n t ( p L 0,05).
C o r r e l a - F-test t i o n ' f o r !Regress-
The s t a t e m e n t above i s t h e c a s e s i n c e a l l t h e computed
C ~ e f f i- c i e n t
F - r a t i o s ( t e s t s t a t i s t i c s ) f o r m u l t i p l e r and B
l ion M u l t i ~ l e coeffi- ] Error r
-. --- 654.09 -0.46
1 - 4 8 0 0 0 7 - -.----. --a-
Mu1 ti-
r e s p e c t i v e l y are g r e a t e r than t h e t a b u l a t e d v a l u e o f
X I Y I 1
i Z
F(O.05, 2, 1 1 7 ) which i s 3-07,
p l e r
0.96
Y i
The r e s u l t s i n bo th T a b l e s 4.8.1 and 4,8,2
Y
i n d i c a t e t h a t t h e s i m p l e and mu1 t i p l e c o r r e l a t i o n
c o e f f i c i e n t s are a l l s t a t i s t i c a l l y s i g n i f i c a n t ( p - /0.05),
81
1.65 1 :::: x -0,66 0,09 .-.#
The n u l l h y p o t h e s i s 3 i s n o t upheld, i n s t e a d i n t h e
Z ! X
Y
0-98 0.54
0.42 0 -03 fl -
l i g h t o f t h e d a t a a v a i l a b l e t h e a l t e r n a t i v e
h y p o t h e s i s 3 i s upheld . T h i s i s t o s a y , t h e r e f o r e ,
t h a t s t u d e n t s 1 c o g n i t i v e , psychomotor and a f f e c t i v e
pe r fo rmance scores as o b t a i n e d th rough t h e t h r e e
p r o j e c t s are s i g n i f i c a n t l y c o r r e l a t e d .
~ l s o from b o t h T a b l e s 4.8-1 and 4.8,2, i t i s
e v i d e n t t h a t a l l t h e s i m p l e and m u l t i p l e l i n e a r
r e g r e s s i o n c o e f f i c i e n t s are s i g n i f i c a n t l y d i f f e r e n t
from z e r o ( p L O,O5), T h e r e f o r e t h e n u l l h y p o t h e s i s
f o u r i s ( re j c c t e d ) n o t uphe ld . I n s t e a d t h e a l t e r n a -
t i v e h y p o t h e s i s 4, which states t h a t e a c h of tlle
s i m p l e and m u l t i p l e l i n e a r r e g r e s s i o n c o e f f i c i e n t s
among t h e v a r i a b l e s X, Y and Z i s s t a t i s t i c a l l y
s i g n i f i c a n t ( p L 0,051 i s uphe ld ,
CHAPTER F I V E
INTI~RPRETATION AND OISCUSSION OF RESULTS
P r e s e n t e d u n d e r t h i s c h a p t e r a r e t h e f o l l o w i n g :
I n t e r p r e t a t i o n of r e s u l t s of t h i s s t u d y ;
E d u c a t i o n a l I m p l i c a t i o n s o f t h e f i n d i n g s of t h i s s t u d y ;
D i s c u s s i o n o f t h e l i m i t a t i o n s of t h i s s t u d y ;
Recommendations based o n t h e f i n d i n g s of t h i s s t u d y ;
S u g g e s t i o n s f o r f u r t h e r s t u d i e s ; and
C o n c l u s i o n and Summary o f t h i s s t udy .
LNTERPHLTATION OF RESULTS -- - The i n t e r p r e t a t i o n o f r e s u l t s o f t h i s s t u d y i s
b a s e d on c o n s i d e r i n g t h s e x t e n t t h e v a r i o u s oP j e c t i v e s
of t h i s s t u d y h a v e been a c h i e v e d , T h e s e c o n s i d e r a t i o n s
are u n d e r t a k e n w i t h r e s p e c t t o t h e sub -head ings l i s t e d
below:
( i ) t h e s u i t a b i l i g y and r e l e v a n c e o f t h e p r o j e c t s t o t e a c h i n g and l e a r n i n g JSS Mathema t i c s ;
(ii) t h e v a l i d i t y and u s u a b i l i t y of SAPIM and SMAL.0MP as a s s e s s m e n t i n s t r u m e n t s ;
(iii) t h e a c c e p t a b i l i t y and s u i t a b i l i t y o f t h e mark ing schemes f o r s c o r i n g s t u d e n t s ' r e s p o n s e s t o t h e p r o j e c t s ;
t h e r e l i a b i l i t y o f t e a c h e r s i n s c o r i n g s t u d e n t s u r e s p o n s e s t o t h e p r o j e c t s ;
t h e e f f e c t o f t h e u s e o f p r o j e c t s on s t u d e n t s t a f f e c t i v e b e h a v i o u r t o w a r d M a t h e m a t i c s ;
t h e e x t e n t o f c o r r e l a t i o n s among t h e s t u d e n t s t c o g n i t i v e , p s y c h o m o t o r a n d a f f e c t i v e p e r f o r m a n c e scores a s m o n i t o r e d by p r o j e c t s ;
s i g n i f i c a n c e of t h e inter l i n e a r r e g r e s s i o n c o e f f i c i e n t s among t h e s t u d e n t s ' p e r f o r m a n c e s i n t h e v a r i o u s d o m a i n s ; and
t h e r e a c t i o n s of t h e s c h o o l s t o t h e u s e of p r o j e c t s ,
R e l e v a n c e a n d b u i t a b i l i a - -..-. - -~ - - -.-.-a o f t h e P r o j e c t s -- t o a n d for7reachig a n d L e a r n i n g J S S M a t h e m a t i c s -----.-.--- -- *- ---- --.-- - - --
The p r o j e c t s d e v e l o p e d a n d u s e d i n t h i s s t u d y are
r e l e v a n t t o t e a - h i n g a n d l e a r n i n g M a t h e m a t i c s i n o u r
T h i s p o s i t i o n i s based o n t h e f o l l o w i n g facts:
t h e p r o j e c t s are drawn f r o m s e c t i o n s ( y e a r I1
a n d y e a r 111) o f t h e J S S M a t h e m a t i c s c u r r i c u l u m ;
t h e p r o j e c t s embody g r o u p s o f re la ted t a s k s ,
a c t i v i t i e s and e x p e r i e n c e s s u g g e s t e d by t h e J S S
M a t h e m a t i c s c u r r i c u l u m d e s i g n e r s as r e l e v a n t to
a n d n e c e s s a r y f o r t h e t e a c h i n g a n d l e a r n i n g J S S
Ma thema t ics ;
a l s o a h i g h l y s i g n i f i c a n t m a j o r i t y o f e x p e r t
e d u c a t o r s a n d M a t h e m a t i c s t e a c h e r s , a s shown b y
t h e r e s u l t s p r e s e n t e d u n d e r s e c t i o n 4 - 1 o f t h e
p r e v i o u s c h a p t e r , a c c e p t e d t h a t e a c h s e c t i o n /
a s p e c t of e a c h of t h e p r o j e c t s i s r e l e v a n t t o
t h e t e a c h i n g a n d l e a r n i n g o f J S S M a t h e m a t i c s ,
The p r o j e c t s a r e deemed s u i t a b l e t o t h e p e o p l e
i n v o l v e d i n t e a c h i n g a n d l e a r n i n g of JsS M a t h e m a t i c s
b e c a u s e a h i g h l y s i g n i f i c a n t m a j o r i t y of Measurement a n d
valuation e x p e r t s a n d M a t h e m a t i c s t e a c h e r s who
c r i t i q u e d a n d v a l i d a t , : d t h e p r o j e c t s a p p r o v e d of them a s
b e i n g s o . Above a l l , t h e p r o j e c t s h a v e , i n t h e c o u r s e
of t h i s s t u d y , b e e n u s e d f o r t e a c h i n g and l e a r n i n g J L S
P l a t h e m a t i c s a n d a l s o f o r e v a l u a t i n g JSS s t u d e n t s t
a c h i e v e m e n t s i n t h e same s u b j e c t area, T h e s t u d e n t s ,
who w e r e e x p o s e d t o t h e u s e o f t h e p r o j e c t s a s t e a c h i n g
a n d l e a r n i n g i n s t r u m e n t s , r e s p o n d e d p o s i t i v e l y t o t h e
p r o j e c t s , The s t u d e n t s a l so u n d e r t o o k and a c c o m p l i s h e d ,
( t h o u g h t o v a r y i n g b u t r ea l i s t i c e x t e n t s ) , t h e t h r e e
p r o j e c t s a d m i n i s t e r e d / a s s i g n e d t o them a s e v a l u a t i o n
i n s t r u m e n t s , The a b o v e o b s e r v a t i o n s v i n d i c a t e che
e x p e r t e d u c a t o r s ' a c c e p t a n c e of t h e p r o j e c t s a s s u i t a b l e
i n s t r u m e n t s f o r t e a c h i n g a n d l e a r n i n g JSS M a t h e m a t i c s ,
Based o n t h e a b o v e facts a n d r e a l i s a t i o n s , some p r o j e c t s
a c c e p t a b l e t o e x p e r t e d u c a t o r s a n d M a t h e m a t i c s t e a c h e r s
i n o u r s e c o n d a r y s c h o o l s ( a s r e l e v a n t t o a n d s u i t a b l e
for t e a c h i n g a n d l e a r n i n g JSS M a t h e m a t i c s ) h a v e b e e n
d e v e l o p e d ,
5 ,1 ,2 V a l i d i t y -...---- a n d U s u a b i l i t y of SAPIM-and SMALOMP -&,
as A s s e s s m e n t I n s t r u m e n t s -.------.--.*---------..--.-
D u r i n g t h e i r p r o c e s s e s of d e v e l o p m e n t , t h e scale
f o r A f f e c t i v e P e r f o r m a n c e I n M a t h e m a t i c s (SAPIM) a n d
scale f o r M o n i t o r i n g f i f f e c t i v e L e a r n i n g Outcomes o f
M a t h e m a t i c s P r o j e c t s ( sMHLOP?) w e r e c r i t i q u e d a n d
v a l i d a t e d b y some e x p e r t e d u c a t o r s . T h e s t a t e m e n t s t h a t
c o n s t i t u t e d t h e i n s t r u m e n t s were a s s i g n e d w e i g h t s by
some e x p e r t e d u c a t o r s a n d p o s t g r a d u a t e s t u d e n t s of
e d u c a t i o n a l e v a l u a t i o n , Based o n t h e a b o v e two
a c t i v i t i e s SAPIM a n d SMALOMP h a v e r e a s o n a b l e d e g r e e s
o f face a n d c o n t e n t v a l i d i t y .
T h e r e s e a r c h e r c a r r i e d o u t t w o t r i a l a d m i n i s t r a t i o n s
of AAPIM o n f o r t y JSS y e a r 111 s t u d e n t s a n d s c o r e d
t h e i r r e s p o n s e s . Based o n t h e i r s c o r e s , t h e r e l i a b i l i t y
c o e f f i c i e n t s of 0.84 ( b y t h e s p l i t - h k l f a p p r o a c h ) a n d
0 0 9 2 ( b ) t h e test-retest a p p r o a c h ) w e r e o b t a i n e d .
' T h e r e c o e f f i c i e n t s i n d i c a t e t h a t SAPIM h a s a h i g h d e g r e e
o f r e l i a b i l i t y . A l s o b a s e d o n t h e p r e - t r e a t m e n t a n d
p o s t - t r e a t m e n t scores of two h u n d r e d a n d f o r t y s t u d e n t s
o n SAPIM, t h e r e l i a b i l i t y c o e f f i c i e n t of SAPIM i s f o u n d
t o be 0.82. F u r t h e r m o r e , t h e c o v a r i a t e raw r e g r e s s i o n
c o e f f i c i e n t s of 0 ,800 , 0.87 a n d 0 - 7 8 w e r e o b t a i n e d from
SHPIM s c o r e s o f g r o u p s o f s t u d e n t s by u s i n g iiNCOVAo
(See T a b l e s 4,6.2, 4 . 6 , 3 and 4,6.4 r e s p e c t i v e l y ) .
T h e s e r e s u l t s c o n f i r m t h a t SHPIM i s h i g h l y re l iable ,
The a d m i n i s t r a t i o n o f SAPIM on s t u d e n t s a t t h e
d i f f e r e n t t i m e s and t h e r e s p o n d i n g o f t h e s t u d e n t s t o
t h e i n s t r u m e n t w e r e p roblem f r e e . A l s o t h e s c o r i n g
o f s t u d e n t s r e s p o n s e s to t h e i n s t r u m e n t was, a t e a c h
o f t h e times n o t d i f f i c u l t . These r e a l i s a t i o n s
i n d i c a t e t h a t SAPIM c a n b e used e a s i l y .
The t r i a l a d m i n i s t r a t i o n o f t h e s c a l e f o r
M o n i t o r i n g t h e A f f e c t i v e L e a r n i n g Outc,omes o f
Ma thema t i c s P r r j e c t s (SMALOMP) on ( 5 0 ) f i f t y J S S y e a r
111 s t u d e n t s and t h e s c o r i n g o f t h e r e s p o n s e s o f t h e
s t u d e n t s t o i t were u n d e r t a k e n i n t h i s s tudy . By u s i n g
t h e scores o f t h e f i f t y s t u d e n t s , t h e r e l i a b i l i t y
c o e f f i c i e n t o f sMALOMP w a s found to be 0.80 ( t h r o u g h
t h e s p l i t - h a l f a p p r o a c h ) . T h i s c o e f f i c i e n t i n d i c a t e s
t h a t SPALOMP i s s i g n i f i c a n t l y r e l i a b l e t o a h i g h d e g r e e ,
The a d m i n i s t r a t i o n o f SMALOMP on s t u d e n t s and t h e
r e s p o n d i n g o f t h e s t u d e n t s t o t h e i n s t r u m e n t were, i n
e a c h of t h e t h r e e times d u r i n g t h e main s t u d y , h i t c h
f r e e and easy . The s c o r i n g o f t h e s t u d e n t s 1 r e s p o n s e s
t o SMALOMP d i d n o t t a k e much t i m e . I t c a n t h e r e f o r e
1 4 1
b e s a i d t h a t SMALOMP i s u s u a b l e ,
5 ,1 ,3 A c c e p t a b i l i t y and S~itab$>~-ty-~-,o,f,--t>~e-,M-a~rkin~ Schemes f o r [email protected]_s.e-~~,~~ .t.h.f!.- . ~ . r ~ j g _ t ~ --------- .- ---
The r e s u l t s p r e s e n t e d u n d e r s e c t i o n 4.2 o f t h e
p r e v i o u s c h a p t e r , s h c v t h a t t h e j u d g e s , who c r i t i q u e d
t h e mark ing schemes , a c c e p t e d t h e c a t e g o r i s a t i o n of t h e
e x p e c t e d l e a r n i n g ou tcomes , o f e a c h o f t h e p r o j e c t s ,
i n t o c o t n i t i v e and psychomotor s k i l l s , T h i s accep t . ance
a g r e e s , ( i s i n 1 i n e ) w i t h t h e v i e w s t h a t p r o j e c t s a s
e v a l u a t i o n i n s t r u m e n t s c a n be used t o a p p r a i s e
p e r f o r m a n c e i n a n y o r a l l of t h e t h r e e doma ins of
e d u c a t i o n a l o b j ~ c t i v e s ( a s p r e s e n t e d i n s e c t i o n 2,5.1)
of t h e literature r e v i e w ,
B e s i d e s a c c e p t i n g t h e c a t e g o r i s a t i o n o f t h e e x p e c t e d
l e a r n i n g ou t comes , t h e j l ldges app roved of t h e marks
a l l o c a t i o n t o t h e v a r i o u s knowledge and s k i l l s e x p e c t e d
t o be u s e d by t h e s t u d e n t s i n r e s p o n d i n g / e x e c u t i n g t h e
p r o j e c t s , Moreover , t h e t e a c h e r m a r k e r s a g r e e m e n t i n
s c o r i n g some s t u d e n t s ' r e s p o n s e s t o t h e p r o j e c t s ( a s
p r e s e n t e d u n d e r s e c t i o n 4.3 of c h a p t e r f o u r ) was o n l y
p o s s i b l e b e c a u s e t h e t e a c h e r s c o u l d f o l l o w / u s e t h e
mark ing schemes, From t h e f o r e g o i n g s , i t h a s been
p o s s i b l e d r a w i n g u p mark ing schemes a c c e p t a b l e t o e x p e r t
e d u c a t o r s and Mathema t i c s t e a c h e r s
f o r s c o r i n g s t u d e n t s ' r e s p o n s e s t o
142
and a l s o s u i t a b l e
t h e p r o j e c t s .
5.1,4 - A b i l i 3 o f T e a c h e r s ,------.--. i n S c o r i n g - -. Student-s-!. . -...- ..- C o s n i t i v e and PSYC homoto r Re,spPoon,s_ses- 3-0 t h e ~ ~ r & ~ ~ t ~ ~ - '
Ten randomly sampled s t u d e n t s r e s p o n s e s t o e a c h
o f t h e t h r e e p r o j e c t s were s u b m i t t e d t o tei-1 JYS
Mathema t i c s t e a c h e r s f o r s c o r i n g , making u s e of t h e
mark ing schemes. By us:'.ng t h e s c o r e s awarded to t h e
r e s ? o n s e s by t h e t e a c h e r marke r s , i t was found t h a t
t h e m a r k e r s ' r a t e r r e l i a b i l i t y c o e f f i c i e n t , i n s c o r i n g
e a c h of t h e c o g n i t i v e and psychomotor r e s p o n s e s t o
e a c h o f t h e t h r e e p r o j e c t s , i s v e r y h igh . (see s e c t i o n
4.3, T a b l e 4.3.1 of c h p t e r f o u r ) , The s i x rater
r e l i a b i l i t y c o e f f i c i e n t s were s i g n i f i c a n t ( p L 0.05).
The above r e s u l t s mean t h a t t h e t e a c h e r s a g r e e d among
t h e m s e l v e s i n s c o r i n g e a c h o f t h e r e s p o n s e s . The
r e s u l t s a l s o i m p l y t h a t i n more t h a n 95% of s u c h
s i t u a t i o n s o r e x e r c i s e s t h e t e a c h e r s w i l l award a p p r o x i -
m a t e l y t h e same scores t o r e s p o n s e s s u b m i t t e d t o them
f o r marking.
The a b i l i t y o f m a r k e r s i s n o t o n l y d e t e r m i n e d by
t h e i r rater r e l i a b i l i t y / a g r e e m e n t among t h e m s e l v e s ,
Marke r s c o u l d a g r e e among t h e m s e l v e s i f t h e y award t h e
same score t o e v e r y r e s p o n s e t h e r e b y f a i l i n g t o
d i s t i n 5 u i s h / d i s c r i m i n a t e among t h e d i f f e r e n t l e v e l s o f
p e r f o r m a n c e embodied i n t h e r e s p o n s e s , So t h e a b i l i t y
o f m a r k e r s i s a l s o d e t e r m i n e d by t h e e x t e n t t h e y
d i s c r i m i n a t e d among t h e p e r f o r m a n c e s o f ratccs w i t h
r e s p e c t t o t h e scores awarded to ratees, R e s u l t
p r e s e n t e d i n table 4 -4 .1 o f c h a p t e r f o u r i n d i c a t e t h a t
t h e mean scores awarded t o t h e ten r e s p o n s e s t o e a c h
p r o j e c t i n e a c h o f t h e t w o c a t e g o r i s a t i o n s a r e
s i g n i f i c a n t l y d i f f e r e n t ( p - / 0.05), S i n c e t h e t e a c h e r s -.
marked t h e r e s p o n s e s i n d e p e n d e n t l y , c o l l a b o r a t i o n i s
r u l e d o u t , and a b i l i t y t o d i s c r i m i n a t e among r e s p o n s e s
by t h e teacher^ i s v i n d i c a t e d ,
5.1.5 p f f e c t s ---.- of Use of P r o j e c t s --- on -,- .-- S t u d e n t s ' --- ...- ...-. - A f f e c t i v e Behaviour Toward Ma thema t i c s - -----a .-..-- --.-
The r e s u l t s p r e s e n t e d u n d e r s e c t i o n 4.6 and
s p e c i f i c a l l y i n T a b l e s 4,6.3, 4,6 ,4 , 4,6.2 and 4.6,1 o f
c h a p t e r f o u r mean a l o t , Whether by a l t e r n a t e r a n k s
g r o u p i n g o r r andomised g r o u p i n g o r t h e t y p e s of g r o u p i n g s
combined , t h e e x p e r i m e n t a l g r o u p 1 s mean score i n
a f f e c t i v e b e h a v i o u r toward Ma thema t i c s i s s i g n i f i c a n t l y
g r e a t e r t h a n t h e mean score i n a f f e c t i v e b e h a v i o u r
toward Ma thema t i c s o f t h e c o n t r o l g r o u p i n t h e same
a f f e c t i v e b e h a v i o u r , T h i s means t h a t JSS s t u d e n t s
exposed t o t h e u s e o f p r o j e c t s ( a s i n s t r u m e n t s f o r
t e a c h i n g and l e a r n i n g and e v a l u a t i o n o f a c h i e v e m e n t )
i n Mathemat ics g a i n / i m p r o v e more i n t h e i r a f f e c t i v e
b e h a v i o u r toward Mathemat ics t h a n t h e y would if n o t
s o exposed , Exposure o f s t u d e n t s t o Ma thema t i ca l
p r o j e c t s r e d u c e s t h e s t u d e n t s t v a r i a b i l i t y i n a f f e c t i v e
b e h a v i o u r toward Mathemat ics , T h i s i s shown by t h e
fact t h a t t h e e x p e r i m e n t a l g r o u p s ' s t a n d a r d d e v i a t i o n
scores i n t h e p o s t - t e s t are less t h a n t h e g r o u p s '
s t a n d a r d d e v i a t i o n scores i n t h e p r e - t e s t on SAPIM,
The t r e n d abovd i s p a r t l y e x p l a i n e d by t h e f ac t t h a t
s t u d e n t s of t h e e x p e i r m e n t a l g r o u p s who had h i g h scores
on SAPIM d u r i n g t h e p r e t e s t g a i n e d l i t t l e o r n o t h i n g
o v e r t h e i r p r e t e s t s c o r e s f rom t h e p o s t - t e s t c f SAPIM-
Bu t t h e s t u d e n t s o f t h e e x p e r i m e n t a l g r o u p who had l o w ,
a v e r a g e and a l i t t l e above a v e r a g e scores i n t h e p re -
test on SAPIM, had i n c r e a s e d s c o r e s i n t h e p o s t - t e s t
on SAPIM, f see Appendix I X , table 2).
Anothe r o b s e r v a t i o n wor thy of n o t e i s t h e s t u d e n t s f
l e v e l o f e f f e c t i v e pe r fo rmance i n Ma thema t i c s before
t h e e x p e r i m e n t , T a b l e 4-6.1 shows t h a t b e f o r c t h e
t r e a t m e n t , t h e s t u d e n t s ' mean a f f e c t i v e pe r fo rmance
s c o r e a s measured by SAPIM i s a p p r o x i m a t e l y 59,45%,
C o n s i d e r i n g t h e w e i g h t s (1, 2, 3, 4, 5, 6 and 7 ) o n e ,
two, t h r e e , f o u r , f i v e , six and s e v e n a s s i g n e d t o t h e
s t a t e m e n t s t h a t make up SAPIM, t h e men score of 59.45%
i s closest t o f o u r o u t of s e v e n o r 57.14%. T h i s i s t o
s a y t h a t on t h e a v e r a g e , s t u d e n t s t a f f e c t i v e pe r fo rmance
toward Mathema t i c s i s r a t e d f o u r o u t o f t h e maximum
and minimum scales of s e v e n and one , From t h e way
v ~ e i g h t s were a s s i g n e d t o t h e items of SAPIM, t h e
a f f e c t i v e b e h a v i o u r o f s t u d e n t s toward P ia themat ics is,
on t h e a v e r a g e , n e i t h e r n e g a t i v e n o r p o s i t i v e , i t i s
r a t h e r n e u t r a l . , T h i s f i n d i n g d i f f crs s l i g h t l y from
e a r l i e r o b s e r v a t i o n s , ( A l e 1 9 8 1 and Lassa 1978) t h a t
s e c o n d a r y s c h o o l s t u d e n t s ' a t t i t u d e t o Mathema t i c s is on
t h e a v e r a g e n e g a t i v e .
N e v e r t h e l e s s , a f t e r a b o u t t w o terms of e x p o s u r e
t o t h e u s e of p r o j e c t s f o r t e a c h i n g , l e a r n i n g and
e v a l u a t i o n o f achievement i n Mathemat ics , t h e a f f e c t i v e
( b e h a v i o u r pe r fo rmance of JSS s t u d e n t ; as measured
by SAPIM improved from a mean score o f 58.95% t o a mean
score of 67,993, The ' u s e o f p r o j e c t s t h e r e f o r e a f f e c t s
s t u d e n t s a f f e c t i v e pe r fo rmance i n Ma thema t i c s
p o s i t i v e l y ,
5.1.6 - E x t e n t o f C o r r e l a t i o n s Among S t u d e ~ , t ~ - ~ Co_qni t i v e , Psychomotor and ~f f s ~ t i v e -.- ---. - -..-- Per fo rmances S c o r e s as ~ o n i t g r e d --.-- by_ ...-a, t h e . Projects -
The c o r e l a t i o n c o e f f i c i e n t s between p a i r s o f ;
s t u d e n t s c o g n i t i v e scores (A), s t u d e n t s ' p s y c h o m o t x
scores ( Y ) and t h e i r a f f e c t ~ v e scores (21, a s c o n t a i n e d
i n T a b l e 4,5,2, are a l l h igh . A l s o t h e r e s u l t s of
t-tests t o a s c e r t a i n i f s u c h c o e f f i c i e n t s a r e s i g n i f i -
c a n t show t h a t a l l o f them are h i g h l y s i g n i f i c a n t
( p L 0.01, see T a b l e 4.8.1). T h e s e r e s u l t s i n d i c a t e
t h a t i n t h e m a j o r i t y , ( w i t h o u t r u l i n g o u t some e x c e p t i o n s
s i n c e none o f t h e i n d i c e s / c o e f f i c i e n t s i s + 1.00 1,
s t u d e n t s who pe r fo rm h i g h l y or a b o u t t h e a v e r a g e or
l o w l y i n o n e domain ( c o g n i t i v e , psychornotor and a f f e c t i v e )
d o per form h i g h l y or a b o u t t h e a v e r a g e or l o w l y
r e s p e c t i v e l y i n t h e o t h e r s , However, t h i s t r e n d /
a s s o c i a t i o n i s h i g h e r between a f f e c t i v e pe r fo rmance
scores and c o g n i t i v e o r psychomotor pe r fo rmance s c o r e s
t h a n i t i s between c o g n i t i v e scores and psychomotor
scores, T h i s s i g n i f i c a n t c o r r e l a t i o n between s t u d e n t s t
c o y n i t i v e pe r fo rmance scores a n d a f f e c t i v e pe r fo rmance
scores i s i n a g r e e m e n t w i t h t h e f i n d i n g s o f S i l v e r m a n
( 1 9 7 3 ) , L a s s a ( 1 9 7 6 ) and Mukher jee and Umar ( 1 9 7 8 )
Fur the rmore , t h e a b o v e r e s u l t s are l o g i c a l and i n
~ i n e w i t h e x p e c t a t i o n s c o n s i d e r i n g t h e v iew t h a t t h c
pe r fo rmance o f an i n d i v i d u a l i n a g i v e n a c t i v i t y /
s i t u a t i o n d e p e n d s l a r g e l y on t h e i n d i v i d u a l ' s i n t e r e s t
and a t t i t u d e toward the a c t i v i t y / s i t u a t i o n ( T y l e r l 9 8 6 ) ,
F i n a l l y i t c a n b e a r g u e d t h a t t h e h i g h correlation
c o e f f i c i e n t s among t h e s t u d e n t s ' c o g n i t i v e , p s y c h o m o ~ o r ,
and a f f e c t i v e p e r f ormarice scores s t a n d t o r e a s o n s i n c e
h a v i n g u n d e r s t a n d i n g and i n s i g h t i n t o a s i t u a t i o n /
a c t i v i t y ( e s p e c i a l l y i f i t i s desirable) e n k i n d l e s
i n t e r e s t a n d p o s i t i v e a t t i t u d e toward t h e s i t u a t i o n /
a c t i v i t y ,
5,1.7 L.i n e a r R e q r e s s i o n C o e f f i c i e n t s .--. -.---.- Among- S tuden-3 , ' --.- .-.-.- Coqni t i v e ~ s ~ c h o r n o t o r and A f f e c t i v e - P... -- Per fo rmance S c o r e s i n Mathemat ics - -----. .
T e c h n i c a l l y s p e a k i n g , a l i n e a r r e g r e s s i o n e q u a t i o n
i s a s p e c i f i c s i t u a t i o n a l model o r r u l e for p r e d i c t i n g
a v a l u e o f a d e p e n d e n t v a r i a b l e f rom a g i v e n ( o b s e r v e d )
v a l u e ( s ) o f some i n d e p e n d e n t v a r i a b l e ( s 1 , A v a l u e o f a
d e p e n d e n t v a r i a b l e c a n be p r e d i c t e d f rom a g i v e n ( a n
o b s e r v e d ) v a l u e of an i n d e p e n d e n t v a r i a b l e . But i t i s
almost a l w a y s better t o p r e d i c t a v a l u e o f a d e p e n d e n t
v a r i a b l e f rom v a l u e s o f more t h a n o n e i n d e p e n d e n t
v a r i a b l e s r e l a t e d t o t h e d e p e n d e n t v a r i a b l e , R pre -
d i c t i o n r u l e i s v a l i d o r more s o , r e l i a b l e t o t h e e x t e n t
t h e r e g r e s s i o n c o e f f i c i e n t ( s ) o f t h e i n d e p e n d e n t
v a r i a b l e ( s 1 i s (are) s i g n i f i c a n t .
From T a b l e 4.8.1 a l l t h e s i m p l e r e g r e s s i o n
c o e f f i c i e n t s a r e s i g n i f i c a n t ( p - / 0,05). T h e s e r e s u l t s
mean t h a t w i t h i n some 95% c o n f i d e n c e l i m i t s , the error
t h a t may be i n v o l v e d i n p r e d i c t i n g :
c o g n i t i v e pe r fo rmance score from an o b s e r v e d
( g i v e n ) psychomotor pe r fo rmance score or an
o b s e r v e d ( a g i v e n ) a f f e c t i v e pe r fo rmance score;
psychomotor pe r fo rmance score f rom a g i v e n
( o b s e r v e d v a l u e o f c o g n i t i v e o r a f f e c t i v e
p e r i o r m a n c e score; and
a f f e c t i v e pe r fo rmance s c o r e from a g i v e n
(observed) v a l u e o f c o g n i t i v e or psychomotor
pe r fo rmance s c o r e , w i l l n o t be s i g n i f i c a n t .
i n o t h e r words i f JSS s t u d e n t s a r e g i v e n a p r o j e c t i n
Ma thema t i c s , a s t u d e n t ' s c o g n i t i v e pe r fo rmance ,
psychomotor p e r f o r m a n c e and a f f e c t i v e p e r f o r m a n c e scores
may be p r e d i c t e d from o n e a n o t h e r w i t h some tolerable
errors i n v o l v e d ,
I t i s a l s o e v i d e n t f rom T a b l e 4.5.2 t h a t t h e
m u l t i p l e l i n e a r r e g r e s s i o n c o e f f i c i e n t s i n v o l v e d i n :
p r e d i c t i n g c o g n i t i v e pe r fo rmance scorz (XI f rom
psychomotor pe r fo rmance score ( Y ) and a f f e c t i v e
pe r fo rmance score ( 2 ) ;
p r e d i c t i n g Y from X and Z ; a n d
p r e d i c t i n g 2 f rom X and Y, a r e a l l s i g n i f i c a n t
beyond an - l e v e l 3f 0.05,
The a b o v e - r e s u l t s i m p l y t h a t fo r more t h a n 95% of
a d m i n i s t e r i n g similar Mathematics p r o j e c t s on simil.ar
JSS s t u d e n t s , t h e error t h a t may be i n v o l v e d i n
p r e d i c t i n g :
(i) X ( s c o r e s ) f rom Y and 2 ;
( i i ) Y f rom X and Z; and
(iii) Z f rom X and Y w i l l be i n s i g n i f i c a n t .
Ln o t h e r words, when t h e v a l u e s of two o f X, Y and Z
are known, t h e n t h e v a l u e s o f t h e t h i r d o n e c a n be
computed, Such a computed score w i l l l i e w i t h i n t h e
95% c o n f i d e n c e i n t e r v a l of t h e ' t r u e score,'
5 -1.8 R e a c t i o n s of t h e S c h o o l s t o t h e U s e o f -.--a- --.*------.-------
t h e Mathemat ics P r o j e c t s . . -_.--
R e a c t i o n s of t h e s c h o o l s t o t h e Ma thema t i c s p r o j e c t s
a re r e p r e s e n t e d , i n t h i s c o n t e x t , by react ions of t h e
p r i n c i p a l s , t e a c h e r s (Ma thema t i c s o n e s e s p e c i a l l y ) and
t h e s t u d e n t s of t h e s c h o o l s i n v o l v e d i n t h i s s t u d y ,
T h e e x p e r i m e n t a l s t u d e n t s o f t h i s s . tudy were keen
f rom t h e o u t s e t . They p a i d g r e a t a t t e n t i o n and p a r t i c i -
p a t e d i n u s i n g b i n a r y ( a n d o t h e r number b a s e s ) numbers i n
c o d i n g let ters, words and p i e c e s o f in for rna . t ion , T h e
s t u d e n t s w e r e k w n a t L e a r n i n g t h e a l g o r i t h m s o f some
simple c o m p u t a t i o n s and p r o c e s s e s , The s t u d e n t s i n a l l
t h e s c h o o l s u sed were e n t h u s i a s t i c i n ] . ea rn ing t o c u t
o u t n e t s o f s o l i d o b j e c t s and f o l d i n g such n e t s i n t o
ho l low objects (cones, truncated c o n e s or b u c k e t s ,
c u b o i d s and t r i a n g u l a r p y r a m i d s ) , These s t u d e n t s t h a t
be longed t o t h e e x p e r i m e n t a l g r o u p s , a l s o took t h e
c o l l e c t i o n o f d d t a s e r i o u s l y . The s t u d e n t s i n v o l v a d i n
t h r e e o u t o f t h e f o u r s c h o o l s r e q u e s t e d t h e i r t e a c h e r s
t o u s e t h e e v e n i n g s t o t e a c h them more o f t h e 'newt
Mathemat ics , Tak ing s u c h a c u e , t h e teachers made t h e
s t u d e n t s c a r r y o u t t h e p r o j e c t s i n t h e e v e n i n g s a t
s c h o o l i n a l l t h e s c h o o l s used f o r t h e s t u d y . Many of
t h e s t u d e n t s i n r e s p o n d i n g t o t h e open ended q u e s t i o n
a t t h e end o f SMALOMP wrote s t a t e m e n t s s u c h as:
(i) t h i s i s e n j o y a b l e b e c a u s e Mathemat ics h a s
p r a c t i c a l s a s Bio logy , Chemis t ry , P h y s i c s etc.;
( i i ) Ma thema t i c s t e a c h e s u s how many t h i n g s are
made; and
( i i i ) t h e s e p r a c t i c a l s (projects) we do , e x p l a i n many
th:ir:cjs w e d o i n Ma thema t i c s and o t h e r s u b j e c t s
among o t h e r s o
The e x p e r i m m t a l s t u d e n t s , ( a s g a t h e r e d from t h e t e a c h e r s
i n v o l v e d i n t h i s s t u d y ) have been w o r r y i n g t h e t e a z h e r s
t o c o n t i n u e w i t h t h e 'new1 . ; a t h e m a t i c s . O f t e n t h e
t e a c h e r s a r e a s k e d s u c h q u e s t i o n s es, how do w e d o t h e
p r a c t i c a l s i n t h i s t o p i c ? by t h e i r s t u d e n t s ,
T h e t e a c h e r s i n v o l v e d i n t e a c h i n g t h e s t u d e n t s f o r
t h i s s t u d y a g r e e t h a t t h e p r o j e c t / p r a c t i c a l a p p r o a c h t o
t e a c h i n g Mathema t i c s i s c h a l l e n g i n g and r eward ing , It
a r o u s e s s t u d e n t s ' i n t e r e s t and p a r t i c i p a t i o n . But t h e
t e a c h e r s compla ined t h a t i t i s o v e r t a s k i n g i n terms o f
l o a d . The t e a c h e r s are o f t h e o p i n i o n t h a t t h e r e i s no
money and s p a c e to accommodate t h e material r e q u i r e m e n t s
and p r o d u c t s , The t e a c h e r s i n v o l v e d i n t h e t e a c h i n g
o b s e r v e d t h a t o t h e r t e a c h e r s were e n v i o u s o f what t h e y
d i d b u t a t t h e same t i m e a c c u s e d them o f i n t r o d u c i n g
u n b e a r a b l e loads to t e a c h e r s , The teachers p o i n t e d o u t
t h a t s c o r i n g r e s p o n s e s t o p r o j e c t s i s t i m e consuming.
The p r i n c i p a l s , o f t h e s c h o o l s u sed f o r t h i s s t u d y ,
saw t h e u s e o f Mathemat ics p r o j e c t s f o r t e a c h i n g and
l e a r n i n g JSS Mathemat i c s as i n n o v a t i v e and c r e a t i v e .
, :-' ,L> 2
One o f t h e p r i n c i p a l s , who i s i n c i d e n t a l l y a n E d ~ l r a t i o n
Mathematxcs y r a d u a t e , i s o f t h e o p i n i o n t h a t t r a i n i n g
workshops s h o u l d be o r g a n i s e d t o t r a i n JSS Mathemat i c s
t e a c h e r s i n t h e a r t / s c i ? n c e of c o n s t r u c t i n g and u y i n g
s u c h project^ f o r t e a c h i n g , ( p r i n c i p a l , O j i k e Memorial
S e c o n d a r y s c h o o l , O r l u 19881, A l l t h e f o u r p r i n c i p a l s
p r a i s e d t h e i r Ma thema t i c s teachers who used the
p r o j e c t s t o t e a c h and e v a l u a t e t h e e x p e r i m e n t a l g r o u p s
f o r b c i n g able t o u s e t h e app roach and s a c r i f i c e some
o f t h e i r e v e n i n g s ,
The above r e a c t i o n s from t h e s t u d e n t s , t e a c h e r s
and p r i n c i p a l s were a n t i c i p a t e d and p r e s e n t e d , as p a r t
of w h a t p r o j e c t s i n Ma thema t i c s c a n do, i n t h e back-
ground t o t h i s s t u d y , A l s o t h e s e r e a c t i o n s a g r e e w i t h
some of t h e views o f t h e a u t h o r s of t h e Handbook on
C o n t i n u o u s Asses smen t (Yoloye, 1985) p r e s e n t e d i n t h e
r e v i e w of l i t e r a t u r e , The r e a c t i o n s of t h e t e a c h e r s
a l s o a r e n o t u n u s u a l b e c a u s e human b e i n g s r a r e l y d o
more work f o r no i n c r e a s e i n payment. Indeed t h e
p r o j e c t s were u n d e r t a k e n b e c a u s e t h e costs were b o r n e
by t h e r e s e a r c h e r o t h e r w i s e t h e s c h o o l s c o u l d n o t and
would n o t h a v e funded t h e u n d e r t a k i n g . T h e r e are n o
accommodat ions i n t h e s c h o o l s f o r t h e f i n i s h e d p r o d u c t s ,
5-1.9 ---- A l t e r n a t e Ranks t Grouping ---a- a n d Randomised . .- .-. - .. -- Groupinq a s Reqa rds t h e Use of -4NCOVP. --. -.-..-*..--..--e
The r e s u l t s p r e s e n t e d u n d e r s ec ,b ion 4 - 6 ( i n
T a b l e s 4.6.1, 4,6,3 and 4.6-4) a s r e g a r d s t h e
d i f f e r e n c e s between t h e means, s t a n d a r d d c v i a t i o n : ,
v a r i a n c e s c o n t r i b u t e d by t h e main e f f e c t , c o v a r i a t e
and e r r o r may be i n t e r p r e t e d i n a number of ways,
T h e s e i n t e r p r e t a t i o n s i n c l u d e d t h a t :
( i) a l t h o u g h t h e homogenei ty o f v a r i a n c e w i t h r e s p e c t
t o t h e c o v a r i a t e i n e a c h of t h e two t y p e s o f
g r o u p i n g s , i s n o t v i o l a t e d , t h e a l t e r n a t e r a n k s '
g r o u p i n g most p r o b a b l y y i e l d e d a more homogeneous
g r o u p i n g ;
(ii) most p r o b a b l y , t h e e r r o r s i n t r o d u c e d a s a r e s u l t
o f r andomisak ion (random s a m p l i n g ) ou twe igh
t h e e r r o r s i n t r o d u c e d by a l t e r n a t e r a n k s g r o u p i n g ;
and
(iii) t h e a l t e r n a t e r a n k s g r o u p i n g t ended to a c c e n t u a t e
t h e more t h e e f f e c t s d u e t o t h e main e f f e c t s t h a n
t h e r andomised g r o u p i n g d i d ,
T h e s e r e s u l t s a g r e e w i t h t h e o b s e r v a t i o n s of Da l ton
and O v e r a l l ( 1977) t h a t t h e a l t e r n a t e r a n k s l g r o u p i n g
o f s u b j e c t s w i t h r e s p e c t t o t h e c o v a r i a t e i s as good as,
if n o t better t h a n randomised g r o u p i n g of t h e s u b j e c t s .
They n o t e d t h a t t h e a l t e r n a t e r a n k s t g r o u p i n g h a s
s l i g h t l y g r e a t e r p r o b a b i l i t y o f ; e n s u r i n g homogenei ty
of v a r i a n c e , a c c e n t u a t i n g t h e v a r i a t i o n d u e t o treat-
ment v a r i a b l e and m i n i m i s i n g t h e ( u n e x p l a i n e d ) error
v a r i a n c e t h a n t h e randomised g r o u p i n g ,
5.2 EDUCATIONAL IM2.?,ICATIONS OF THE -..-A,.--.-.- -.-.-.,.--. .
F I N D I N G S O F THIS STUDY, -.---.--,-. .
The f i n d i n g s of t h i s work h a v e a number of
i m p l i c a t i o n s f o r JSS Mathemat ics i n p a r t i c u l a r and
S e c o n d a r y Schoo l e d u c a t i o n i n g e n e r a l , These i m g l i c a -
t i o n s a re d i s c u s s e d u n d e r t h e s u b h e a d i n g s - i m p l i c a t i o n s f o r ,
( i) L e a r n i n g Mathema t i c s i n s e c o n d a r y s c h o o l s ;
(ii) T e a c h i n g Mathemat ics i n s e c o n d a r y s c h o o l s ; and
( i i i ) Cor . t i nuous Assessment i n s e c o n d a r y s c h o o l s ,
5 .2-1 I m p l i c a t i o n s f o r L e a r n i n g -. -. -- Mathemat i c s -..- -A --.-..-- i n Seconda ry S c h o o l s
Dur ing t h e c o u r s e of t h i s s t u d y , t h e e x p e r i m e n t a l
s t u d e n t s r e a c t e d f a v o u r a b l y t o t h e p r o j e c t s , M a j o r i t y
of t h e s e s t u d e n t s d e s i r e d t o have more e x p o s u r e to
s u c h p r o j e c t s and i n d i c a t e d same a s t h e y r e s p o n d e d t o
SMALOMP. The s t u d e n t s , among o t h e r t h i n g s s t a t e d or
i n d i c a t e d t h a t by p a r t i c i p a t i n g i n a n d e x e c u t i n g t h e
p r o j e c t s , some o f t h e p r o c e d u r e s a n d f o r m u l a e used i n
Ma thema t i c s became clearer and p r a c t i c a l tc them.
155
F i n a l l y , i t i s found t h a t t h e u s e o f t h e p r o j e c t s
enhanced s i g n i f i c a n t l y t h e a f f e c t i v e r e s p o n s e s o f t h e
s t u d e n t s t o Mathemat ics .
The above f i n d i n g s imply , t h e r e f o r e , t h a t :
( i) JSS s t u e e n t s t e n d t o b e t t e r u n d e r s t a n d t h e
p r o c e d u r e s a n d f o r m u l a e t h e y mcet w i t h i n JSS
Mathema t i c s , t h r o u g h t h e u s e o f p r o j c c t s t h a n
t h e y would o t h e r w i s e ; and
(ii) JSS s t u d e n t s ' i n t e r e s t i n , a t t i t u d e toward, and
v a l u e o f Ma thema t i c s c a n m o s t p r o b a b l y be
improved by t h e u s e of p r o j e c t s ,
The enhanced u n d e r s t a n d i n g of and improved a f f e c t i v e
b e h a v i o u r toward Mathemat ics a t t h e JSS l e v e l w i l l most
p r o b a b l y be c a r r i e d o v e r t o t h e SSS ( s e c o n d a r y s e c o n d a r y
s c h o o l ) l e v e l i f t h e p r o j e c t a p p r o a c h f o r l e a r n i n g
Mathema t i c s i s a l so made u s e o f a t t h a t l e v e l ,
5.2,2 I m p l i c a t i o n s for Teach inq Mathema t i c s .---.-A -:- -- I n S e c o n d a r y S c h o o l s
.---
R e a l i z i n g t h a t t h e u s e of p r o j e c t s i n t e a c h i n g
Mathema t i c s a t t h e JSS l e v e l makes s t u d e n t s better
u n d e r s t a n d (as t h e s t u d e n t s claim) and react t o
N a t h e m a t i c s more f a v o u r a b l y t h a n t h e y would o t h e r w i s e ,
t h e s e ~ m i n g l y t h a n k l e s s j ob o f t e a c h i n g Mathemat ics a t
t h e JSS l e v e l c a n t h e r e f o r e be made t h a n k f u l by t h e u s e
of p r o j e c t s . I n o t h e r words, a c o n s c i ~ n t i o u s Mathematics
t e s c h e r c a n u s e p r o j e c t s t o make h i s s t u d e n t s u n d e r s t a n d
and react t o Mathema t i c s much better t h a n t h e s t u d e n t s
would i f t h e t e a c h e r does n o t u s e p r o j e c t s . By t h e use
o f p r o j e c t s , Mathema tics t e a c h e r s c a n t r a i n JSS s t u d e n t s
t o be c a r e f u l i n u s i n g t h e l r hands t o p roduce t a n g i b l e
objects, The o b j e c t s produced f rom the e x e c u t i o n o f
some Mathema t i c s p r o j e c t s c a n be x s e d f o r e x h i b i t i o n s ,
Through s u c h d i s p l a y s o f t h i n g s produced from s t u d e n t s l
r e s p m s e s t o Mathemat ics p r o j e c t s :
t h e Ma thema t i c s t e a c h e r shows e v i d e n c e o f h i s / h e r
e f f o r t s I n t e a c h i n g ;
t h e Ma thema t i c s t e a c h e r a n d t h e s t u d e n t s
i l l u s t r a t e t h e r e l e v a n c e of Mathema t i c s a n d make
p e o p l e l i k e and a p p r e c i a t e Mathemat ics ;
t h e Ma thema t i c s t e a c h e r makes case f o r or
j u s t i f i e s t h e p r o v i s i o n of a Mathema t i c s
l a b o r a t o r y / w o r k s h o p i n a s e c o n d a r y s c h o o l ; and
f u n d s and r e s o u r c e s , f o r t h e t e a c h i n g o f
Ma thema t i c s and t h e reward o f e x c e l l e n c e , may
b e o b t a i n e d from w i t h i n t h e s c h o o l and from
w i t h o u to
However, the Mathematics t e a c h e r w i l l be t a s k e d m o r e
t h a n h e / s h e would i f h e / s h e does n o t u s e p r o j e c t s . The
t e a c h e r h a s t o t h i n k a n d c o n s t r u c t t h e r e l e v a n t p r o j e c t s
t o be u s e d f o r t e a c h i n g t h e c o n t e n t of t h e M a t h e m a t i c s
c u r r i c u l u m a n d t h o s e t o be u s e d f o r e v a l u a t i o n , T h e
t e a c h e r h a s t o p r o d u c e mode l r e s p o n s e s t o t h e p r o j e z t s
a n d a l s o t h e i r m a r k i n g schemes . I f t h e material needs
f o r d o i n g t h e t h i n g s a b o v e are n o t t h e r e , t h e t e a c h e r
h a s t o make s a c r i f i c e s t o p r o v i d e them. H e / s h e h a s t o
make t i m e for s c o r i n g s t u d e n t s r e s p o n s e s t o t h e p r o j e c t s .
Somet imes , t h e t e a c h e r w i l l s p e n d some of h i s / h e r
e v e n i n g times t ,d d e m o n s t r a t e how some a s p e c t s o f t h e
p r o j e c t s c a n be u n d e r k a k e n / c a r r i e d o u t , T h e u s e o f
p r o j e c t s f o r t e a c h i n g M a t h e m a t i c s demands some w o r k i n g
s p a c e ( w o r k s h o p / l a b o r a t o r y ) a n d some s t o r a g e facil i t ies,
for k e e p i n g t h e p r o d u c t s , from s c h o o l s . F i n a l l y , t h e
s c h o o l s h a v e t o p r o c u r e some basic e q u i p m e n t a n d f r o m
t i m e t o t i m e p r o v i d e some w o r k i n g materials.
5,2,3 I m p l i c a t i o n s F o r C o n t i n u o u s Asp~~s,"C-,".t . - ..-- I n G c o n d a r y .- S c h o o l s
T h i s work shows t h a t r e l e v a n t a n d s u i t a b l e p r o j e c t s
i n M a t h e m a t i c s c a n b e c o n s t r u c t e d a n d a d m i r ~ i s t e r e d o n
s t u d e n t s t o e l i c i t c o g n i t i v e , p s y c h o m o t o r a n d a f f e c t i v e
r e s p o n s e s f rom JSS s t u d e n t s , The work a lso shows t h a t
s u i t a b l e marking schemes c a n be drawn u p f o r s c o r i n g
t h e c o g n i t i v e and psychomotor r e s p o n s e s e l ic i ted f rom
t h e s t u d e n t s by t h e p r o j e c t s . It follows, t h e r e f o r e ,
t h a t Mathemat ics p r o j e c t s c a n be used t o e v a l u a t e
s t u d e n t s ' c o g n i t i v e and psychcmotor pe r fo rmances i n
Mathemat ics . T h i s i m p l i e s t h a t i n s t e a d o f r e l y i n g on
some haphaza rd o b s e r v a t i - o n s a n d memory (as most
t e a c h e r s d o ) t o r a t e s t u d e n t s ' psychomotor pe r fo rmances ,
s t u d e n t s e r e s p o n s e s t o Mathema t i c s p r o j e c t s c a n be r e l i e d
on. T h e s e r e s p o n s e s are r e l a t i v e l y permanent, d e p e n d a b l e
and c a n b e croz,-checked. The u s e o f p r o j e c t s t o
e v a l u a t i n g s t u d e n t s f psychomotor pe r fo rmances c a n
e s t a b l i s h f o r t h e t e a c h e r t h e t r e n d of t h e areas o f
s t r e n g t h s and w e a k n e s s e s of p a r t i c u l a r s t u d e n t s , T h i s
w i l l h e l p t h e t e a c h e r t o improve t h e s t u d e n t s and a l so
p r o v i d e , u l t i m a t e l y , d a t a on which some career
c o u n s e l l i n g / g u i d a n c e c a n be based , S t u d e n t s 1 s e l v e s 1
a p p r a i s a l s of t h e i r c o g n i t i v e and psych3motor performan-
ces are p o s s i b l e t h r o u g h t h e u s e of p r o j e c t s , T h i s
i s t h e c a s e s i n c e t h e s t u d e n t s c a n compare and c o n k r a s t
t h e i r r e s p o n s e s ( p r o d u c t s ) w i t h t h a t of o t h e r s / t h e i r
f e l l o w s t u d e n t s - The s t u d e n t s c a n improve t h e i r
psychomotor s k i l l s t h r o u g h p r a c t i c e and h e l p s from
o t h e r people .
T h e r e c e n t p r a c t i c e o r t h e p r a c t i c e i n use, i n o u r
s e c o n d a r y s c h o o l s now, t e n d t o assume t h a t c o g n i t i v e
pe r fo rmance i s s u b j e c t area d i f f e r e n t i a t e d b u t
psychomotor and a f f e c t i v e pe r fo rmances a r e g l o b a l o r
i n d e p e n d e n t o f s u b j e c t areas. Though t h e a s s u m p t i o n
i s n o t d i s p r o v e d by t h f s s t u d y , i t i s more r e a s o n a b l e
t o t reat s t u d e n t s 1 psychomotor and a f f e c t i v e performan-
ces a s t h e i r c o g n i t i v e p e r f o r m a n c e s are t r e a t e d on
t h e b a s i s o f s u b j e c t areas, T h i s caE b e accompl i shed
by t h e u s e o f po jec t s and t h e r e f o r e enhance d i a g n o s i s
and r e m e d l a t i o n o f s t u d e n t s 1 weaknesses i n t h e v a r i o u s
s u b j e c t areas. However, t h e hopes r a i s e d by or t h e
p r o m i s e s g i v e n by t h e f i n d i n g s of t h i s s t u d y c a n o n l y
be r e a l i s e d i f Yiathematics t e a c h e r s i n o u r s e c o n d a r y
s c h o o l s have or are made t o p o s s e s s t h e c o m p e t e n c i e s
f o r c o n s t r u c t i n g r e l e v a n t and s u i t a b l e p r o j e c t s and
t h e i r c o r r e s p o n d i n g mark ing schemes, The hopes
raised by t h e r e s u l t s o f t h i s s t u d y may a lso become
t r u e if e a c h e d u c a t i o n a l zone i n a s tate h a s a team
o f p e r s o n s c a p a b l e of c o n s t r u c t i n g t h e d e s i r e d p r o j e c t s
a n d d rawing u p c o r r e s p o n d i n g and s u i t a b l e / a p p r o p r i a t e
marking schemes f o r t h e common u s e by s e c o n d a r y
s c h o o l s u n d e r t h e zone ,
A n o t h e r i m p o r t a n t i m p l i c a t i o n of t h e r e s u l t s o f
t h i s s t u d y i s t h a t when a n a s s e s s o r / e v a l u a t o r , making
u s e of ( s t u d e n t s t scores d e r i v e d from) p r o j e c t ( s 1 is:
( i) i n a h u r r y ,
(ii) u n a b l e t o a s s e s s t h e s t u d e n t s t p e r f o r m a n c e s
i n o n e o r two o f t h e t h r e e b e h a v i o u r a l domains
o r
(iii) n o t o p p o r t u n e d t o l a y hands on t h e s t u d e n t s t
pe r fo rmance s c o r e s i n o n e o r two o f t h e t h r e e
domains , t h e a s s e s s o r / e v a l u a t o r c o u l d , t o some
r e a s o n a b l e e r r o r l i m i t s p r e d i c t / i n f e r t h e non-
a v a i l a b l e pe r fo rmance s c o r e s f rom t h e one /ones
a v a i l a b l e .
5-2.4 I m p l i c a t i o n s f o r Teache r E d u c a t i o n - .-- and R e s e a r c h i n E d u c a t i o n
C e r t a i n l y t h e r e s u l t s o f t h i s work show t h a t t h e
u s e o f m e a n i n g f u l p r o j e c t s i n t e a c h i n g and l e a r n i n g
( i n c l u d i n g a s s e s s m e n t i n ) JSS Mathemat ics : e n h a n c e s
s t u d e n t s t a f f e c t i v e p e r f o r m a n c e s i n Ma thema t i c s and
makes f o r s t u d e n t s ' better u n d e r s t a n d i n g of Mathema t i c s
( a s t h e s t u d e n t s c la im) , It i s t h e r e f o r e r e a s o n a b l e t o
assert t h a t g i v i n g s t u d e n t t e a c h e r s o f Ma thema t i c s
a d e q u a t e t r a i n i n g and e x p o s u r e t o t h e c o n s t r u c t i o n and
u s e of prajects i n N a t h e m a t i c s w i l l mos t p r o b a b l y enhance
t h e i r z f f e c t i v e n e s s a s s e r v i n g t e a c h e r s when p o s t e d t o
s c h o o l s . The a b o v e a s s e r t i o n d e p e n d s on t h e w i l l i n g n e s s
a n d a b i l i t y of Mathema t i c s t e a c h e r e d u c a t o r s t o e x 2 o s e
s t u d e n t t e a c h e . r s of Mathema t i c s t o t h e c o n s t r u c t i o n and
u s e of p r o j e c t s . ~ l s o t h e e x p o s u r e w i l l be made
p o s s i b l e i f r e l e v a n t and s u i t a b l e c o u r s e s are mounted
by t h e t e a c h e r e d u c a t i o ~ l i n s t i t u t i o n s p r e p a r i n g s e c o n d a r y
school Mathematics t e a c h e r s ,
F u r t h e r more some o f t h e r e s u l t s o f t h i s s t u d y ,
b o r d e r i n g on d e s i g n i n g e d u c a t i o n a l r e s e a r c h , show t h a t
f o r t h e u s e of ANCOVA a l t e r n a t e r a n k s ' g r o u p i n g i s as
r e a s o n a b l c and r t a t i s t i c a l l y v a l i d as randomised g r o u p i n g ,
The above s t a t e m e n t i m p l i e s t h a t a l t e r n a t e r a n k s l d e s i g n
as a non-random d e s i g n c o u l d b e used when i t i s more
e x p e d i e n t o r when r a n d o m i s a t i o n is d i f f i c u l t to come by
i n s t u d i e s i n v o l v i n g t h e u s e o f ANCOVA.
5.3 LIMITATIONS OF THIS STUDY - By t h e d e s i g n o f t h i s s t u d y s u c h s o u r c e s t h a t
j e o p a r d i z e i n t e r n a l v a l i d i t y o f t h e f i n d i n g s o f a s t u d y ,
l i k e h i c t o r y , m a t u r a t i o n , t e s t i n g , i n s t r u m e n t a t i o n ,
r e g r e s s i o n , s e l e c t i o n , m o r t a l i t y and i n t e r a c t i o n s among
t h e s e , have been c o n k r o l l e d , But such s o u r c e s t h a t
j e o p a r d i z e e x t e r n a l v a l i d i t y of a s t u d y , l i k e i n t e r -
a c t i o n o f t e s t i n g and t r e a t m e n t , i n t e r a c t i o n of
s e l e c t i o n and t r e a t m e n t , and r e a c t i v e a r r a n g e m e n t s , may
n o t have been c o n t r o l l e d . So t h e g e n e r a l i s a b i l i t y of t h e
f i n d i n g s 02 t h i s s t u d y to t h e ' U n i v e r s e ' o f t e a c h i r g and
l e a r n i n g Mathema t i c s i n N i g e r i a n JSS o r more so
N i g e r i a n s e c o n d a r y s c h o o l s i s l i m i t e d by some f a c t o r s
r e l a t e d to '&he p r o b a b l y u n c o n t r o l l e d s o u r c e s named above.
These f a c t o r s i n c l u d e o r are i n h e r e n t i n t h e
f o l l o w i n g a r r a n g e m e n t s :
( i ) p r e t e s t i n g / p r e a d m i n i s t r a t i o n o f SAPIM on t h e
sampled s t u d e n t s before t r e a t m e n t s ;
(ii) t h e u s e o f o n l y y e a r 111 JSS s t u d e n t s ; a n d
( i i i ) p u r p o s i v e s e l e c t i o n of s c h o o l s ,
The f i n d i n g s o f t h i s work are d e r i v e d f rom r e s p o n s e s
of s t u d e n t s who were p r e t e s t e d a n d t r e a t e d and t h e n
p o s t - t e s t e d , But t h e g e n e r a l i s a t i c n i s on a non-
p r e t e s t e d u n i v e r s e . However, t h i s i s n c t a s t r o n g
l i m i t a t i o n on g e n e r a l i s i n g f i n d i n g s of s t u d i e s on t e a c h -
and l e a r n i n g s i n c e tests o f such k i n d s abound i n t h e
s c h o o l s 1 set ups , Year I11 JSS s t u d e n t s ' r e s p o n s e s /
r e a c t i o n s t o Ma thema t i c s p r o j e c t s may n o t r e p r e s e n t
t y p i c a l r e s p o n s e s / r e a c t i o n s o f JSS s t u d e n t s t o
Mathema t i c s p r o j e c t s , b u t t h e f i n d i n g s o f t h i s s t u d y
are b e i n g g e n e r a l i s e d on JSS s t u d e n t s . F i n a l l y , s i n c e
t h e s c h o o l s u sed for t h i s s t u d y w e r e p u r p o s i v e l y
s e l e c t e d , t h e s t u d e n 7s' and s c h o o l s ' r e s p o n s e s / r e a c t i o n s
t o t h e p r o j e c t s may be t l p i c a l t o a v e r a g e r e s p o n s e s
o r r e a c t i o n s of JS5 s t u d e n t s and s c h o o l s . These
l i m i t a t i o n s are n o t most l i k e l y b e c a u s e t h e sampled
s t u d e n t s ' respmses and r e a c t i o n s 20 t h e u s e of
Mathema t i c s p r o j e c t s , a s measured by t h e i n s t r u m e n t s ,
were f a i r l y normal , t h e m e a s u r e s were n o t of e x t r e m e
c a s e s , A c l e a r l i m i t a t i o n t o t h i s s t u d y i s t h a t t h e
s t u d e n t s cf t h e e x p e r i m e n t a l g r o u p s a re i n t h e same
s c h o o l w i t h t h o s e o f t h e c o n t r o l g roups . No a r r a n g e -
m e n t s were made to p r e v e n t t h e two i n t e r a c t i n g af ter
s c h o o l m
5 ,4 RECOMMENDATIONS
The r e s u l t s o f t h i s s t u d y show t h a t the u s e of
Mathema t i c s p r o j e c t s i n t e a c h i n g and l e a r n i n g of JSS
Mathema t i c s improves s i g n i f i c a n t l y t h e a f f e c t i v e
b e h a v i o u r o f t h e s t u d e n t l e a r n e r s toward Mathemat ics .
It i s t h e r e f o r recommended t h a t Mathemat ics p r o j e c t s
1 64
s h o u l d be u s e d f o r t h e t e a c h i n g - l e a r n i n g p r o c e s s e s /
a c t i v i t i e s of JSS Mathemat ics .
It h a s been s h ~ w n i n t h i s s t u d y t h a t :
r e l e v a n t p r o j e c t s o n JSS Mathemat ics c a n be
c o n s t r i l c t e d ;
a mark ing scheme c a n be drawn u p f o r e a c h p r o j e c t
and t h e e x p e c t e d s t u d e n t s ' r e s p o n s e s t o e a c h
p r o j e c t c a t e g o r i l e d i n t o c o g n i t i v e and
psychomotor l e a r n i n g o u t c o m e s / s k i l l s ;
when a d m i n i s t e r e d on JSS s t u d e n t s , as e v a l u a t i o n
i n s t r u m e n t s , t h e s t u d e n t s r e s p o n d a p p r o p r i a t e l y
a c c o r d i l l g t o t h e i r c a p a b i l i t i e s ; a n d
t e a c h e r s of d a t h e m a t i c s a t t h e JSS l e v e l c a n
score t h e s t u d e n t s ' r e s p o n s e s t o t h e p r o j e c t s
r e l i a b l y if t h e t e a c h e r s u s e t h e mark ing scheme
f o r t h e p a r t i c u l a r p r o j e c t ,
It i s recommended t h a t s u c h r e l e v a n t p r o j e c t s and
c o r r e s p o n d i n g mark ing schemes be d e v e l o p e d and u s e d f o r
e v a l u a t i n g JSS s t u d e n t s t c o g n i t i v e , psychomotor a n d
a f f e c t i v e p e r f o r m a n c e s i n Mathemat ics , T h i s a p p r o a c h
w i l l e n s u r e t h a t pe r fo rmance scores awarded t o s t u d e n t s
on t h e psychomotor and a f f e c t i v e domains are based on
e m p i r i c a l / c o n c r e t e and r e l a t i v e l y permanent e v i d e n c e
r a t h e r t h a n t h e c u r r e n t p r a c t i c e based on h a p h a z a r d
o b s e r v a t i o n s and r e c o u r s e t o memory.
A f o l l o w u p t o t h e above recommendat ion i s t h a t
when t h e a p p r o a c h above i s a d o p t e d , psychomotor and
a f f e c t i v e p e r f o r m a n c e s c o r e s s h o u l d be r e c o r d e d and
r e p o r t e d on s u b j e c t area basis a s w e r e c o r d and r e p r t
p r e s e n t l y o u r s t u d e n t s ' so to s p e a k B c o g n i t i v e
p e r f o r m a n c e scores. It w i l l no l o n g e r be w o r t h w h i l e t o
b r e a k down on r e p o r t forms o r c a r d s , i n t e l l e c t u a l ,
psychomotor and a f f e c t i v e c a p a b i l i t i e s i n t o component
s k i l l s . R a t h e r s u c h a L reak down s h o ~ l d be embodied i n
mark ing schemes u sed t o s c o r e s t u d e n t s r e s p o n s e s t o
p r o j e c t s ,
S i n c e i t h a s been o b s e r v e d t h a t t e a c h e r s l a c k t h e
c o m p e t e n c i e s t o c o n s t r u c t t h e s e p r o j e c t s and t h e i r
c o r r e s p o n d i n g mark ing schemes , z o n a l e d u c a t i o n boards
or commiss ions s h o u l d e a c h i d e n t i f y w i t h i n i t s t e a c h i n g
f o r c e o r i n s t i t u t i o n s close by, p e r s o n s who a r e c a p a b l e
t o c o n s t r u c t s u c h i n s t r u m e n t s , Each z o n a l e d u c a t i o n
board /commiss ion s h o u l d engage t h e s e r v i c e s of s u c h
p e r s o n s t o c o n s t r u c t t h e i n s t r u m e n t s fo r t h e common u s e
of t h e JSS s t u d e n t s and Ma thema t i c s t e a c h e r s , A l s o
workshops s h o u l d b e o r g s n i z e d t o t r a i n t h e P l a thema t i c s
166
t e a c h e r s to a c q u i r e t h e cr;mpet..:~ncics for d e v e l o p i n q t h e
i n s t r u m e n t s , Above a l l , e d u c a t i o n a l i n s t i t u t i o n s t r a i n -
i n g Mathema t i c s teachers f o r o u r s e c o n d a r y s c h o o l s s h o u l d
d e s i g n and e x p o s e t h e i r s t u d e n t s t o a c o u r s e / l e c i x r e s to
make s u c h s t u d ~ n t s , (would be Mathemat ics t e a c h e r s )
a c q u i r e t h e c o m p e t e n c i e s for d e v e l o p i n g t h e i n s . t r u m e n t s ,
J u d g i n g from t h e r e a c t i o n s o f t h e Ma thema t i c s
t e s c h e r s and s c h o o l s to t h e p r o j e c t s a n d t h e i m p l i c a t i o n s
o f such r e a c t i o n s , c e r t a i n t h i n g s have t o be done i f t h e
b e n e f i t s o f t h e u s e of p r o j e c t s a r e t o b e r e a p e d , These
t h i n g s
( i
i n c l u d e :
t h e provi-s ion o f consumable m a t e r i a l s by t h e
s c h o o l a u t h o r i t i e s on r e g u l a r basis f o r t h e
t e a c h e r and s t u d e n t s t o c a r r y o u t t h e p r o j e c t s ;
t h e p r o v i s i o n of Mathema t i c s workshops and
s t o r e s i n t h e s c h o o l s f o r s t u d e n t s t o work i n and
f o r t h e consumab les and p r o d u c t s t o be s t o r e d ;
Ma thema t i c s t e a c h e r s i n o t h e r t o u s e p r o j e c t s
s h o u l d make s a c r i f i c e s i n terms 3f some of t h e i r
e v e n i n g t i m e s t o be d e v o t e d t o d e m o n s t r a t i o n s ;
and
i f t h e t e a c h e r make s u c h sacrifices s c h o o l
a u t h o r i t i e s and gove rnmen t s s h o u l d r e c i p r o c a t e
by p a y i n g M a t h e m a t i c s t e a c h e r s enhanced
a l l o w a n c e s and a l l o c a t i n g f a i r number of p e r i o d s
of c e a c h i n g p e r i o d s p e r week to I-hem.
5.5 SUGGESTIONS - FOR FURTHER STUDIES -- ---- One of t h e c o n s t r a i n t s t o g e n e r a l i z i n g t h e f i n d -
i n g s of t h i s s t u d y to a l l JSS s t u d e n t s i n N i g e r i a i s
t h a t t h e s ample i s made u p o f o n l y y e a r 111 JSS s t u d e n t s .
F o r a more v a l i d g e n e r a l i z a t i o n of r e s u l t s , t h i s s t u d y
s h o u l d be carried o u t u s i n g a w i d e r s ample i n c l u d i n g
y e a r I, y e a r I1 and y e a r 111 s t u d e n t s of o u r JSS.
H e r e i t h a s been o b s e r v e d t h a t t h e u s e of p r o j e c t s
i n t h e t e a c h i n g and l e a r n i n g o f JSS Ma thema t i c s ,
s i g n i f i c a n t l y a f f e c t s t h e a f f e c t i v e b e h a v i o u r of JSS
s t u d e n t s p o s i t i v e l y toward Mathemat ics . It i s wor th-
w h i l e to i n v e s t i g a t e w h e t h e r t h e u s e o f p r o j e c t s w i l l :
( i ) s i g n i f i c a n t l y e n h a n c e JSS s t u d e n t s 1 c o g n i t i v e
and psychomotor p e r f o r m a n c e s i n Ma thema t i c s ;
( i i s i g n i f i c a n t l y improve s e n i o r s e c o n d a r y s c h o o l
s t u d e n t s 1 a f f e c t i v e , p sychomoto~- and c o g n i t i v e
p e r f o r m a n c e s i n Ma thema t i c s ;
( i i i ) s i g n i f i c a n t l y e n h a n c e s t u d e n t s ' r e t e n t i o n and
t r a n s f e r of knowledge and s k i l l s , l e a r n t i n
Ma thema t i c s a t t h e J S S / l e a r n t i n M a t h e m a t i c s a t
t h e s e n i o r s e c o n d a r y s c h o o l s (SSS) ; and
i r s i g n i f i c a n t l y e n h a n c e s t u d e n t s ' p e r f o r m a n c e s i n
other s u b j e c t a r e a s l i k e I n t r o d u c t o r y Technology ,
I n t e g r a t e d S c i e n c e , Woodwork, Metalwork, P h y s i c s
and C h e m i s t r y ,
T h i s s t u d y h a s shown t h a t p r o j e c t s can b e used
as e v a l u z t i o n i n s t r u m e n t s f o r g e t t i n g a t JSS s t u d e n t s
a f f e c t i v e , c o g n i t i v e and psychomotor p e r f o r m a n c e s ;-n
Mathemat ics , It w i l l be w o r t h w h i l e t o c a r r y o u t
s t u d i e s :
(i) t o see how p r o j e c t s w i l l s e r v e t h e same p u r p o s e
i n o t h e r s u b j e c t areas; and
(ii) to c o n s t r u c t , v a l i d a t e and n o r m / s t a n d a r d i z e some
Mathema t j c s p r o j e c t s f o r d i f f e r e n t a g e s and
l e v e l s o f s c h o o l i n g o r s t u d e n t s i n o u r s e c o n d a r y
s p h o o l s ,
5,6 SUMMARY AND CONCLUSION
O b s e r v a t i o n s and r e s u l t s o f some s t u d i e s show
t h a t , i n many c o u n t r i e s i n c l u d i n g N i g e r i a , t h e t e a c h i n g
and l e a r n i n g o f Ma thema t i c s ( e s p e c i a l l y a t t h e p r i m a r y
and s e c o n d a r y school l e v e l s ) are u n s a t i s f a c t o r y , These
s o u r c e s a l s o i n d i c a t e t h a t s t u d e n t s ' pe r fo rmances i n
t h e c o s n i t i v e domain are l o w and s t u d e n t s 1 a t t i t u d e
toward Mathema t i c s i s on t h e a v e r a g e n e g a t i v e .
However, some s t u d i e s o n s e c o n d a r y s c h o o l s t u d e n t s
performances i n M a t h e m a t i c s i n t h i s c o u n t r y i n d i c a t e
t h a t ; male s t u d e n t s , s t u d e n t s i n u r b a n e n v i r o n m e n t ,
a n d s t u d e n t s i n c o e d u c a t i o n a l s c h o o l s , t e n d o n t h e
a v e r a g e t o d o better t h a n female s t u d e n t s , s t u d e n t s i n
r u r a l e n v i r o n m e n t s a n d s t u d e n t s i n u n i s e x s c h o o l s
r e s p e c t i v e l y ,
Most of t h e w o r k s o n t h e t e a c h i n g of M a t h e m a t i c s
i n o u r s e c o n d a r y s c h o o l s f o u n d t h a t m o s t o f t h e t e a c h e r s
l a c k i n t e r e s t a n d e n t h u s i a s m i n t h e i r work. Most of t h e
s t u d i e s i n d i c a t e d t h a t , i n s t r u c t i o n s g i v e n i n M a t h e m a t i c s :
( i ) l a c k t e a c h i n g a ids ;
( i i ) do n o t e n c o r p o r a t e a d e q u a t e oral a n d w r i t t e n
e x e r c i s e s or tests;
(iii) are n o t re la ted t o t r u e l i f e s i t u a t i o n s / ~ r o b l e m s
a n d t o o t h e r s u b j e c t areas; a n d
( i v ) are devoid of c o n c r e t e a c t i v i t i e s ,
I t w a s a l so observed t h a t l i k e i n m o s t o t h e r
s u b j e c t areas i n o u r s e c o n d a r y s c h o o l s , o n l y t h e
c o g n i t i v e p e r f o r m a n c e s o f s t u d e n t s w e r e e v a l u a t e d .
T h e e v a l u a t i o n of t h e s t u d e n t s c o g n i t i v e p e r f o r m a n c e s
w a s d o n e t h r o u g h t e r m l y a n d y e a r l y or e n d - o f - c o u r s s
e x a m i n a t i o n s , w i t h t h e i n t r o d u c t i o n of C o n t i n u o u s
Assessmen t i n o u r Seconda ry S c h o o l s o b s e r v a t i o n s and
some r e c e n t s t u d i e s show t h a t what i s m o s t l y d o n e i s
c o n t i n u o u s t e s t i n g , which m o s t l y i f n o t o n l y a p p r a i s e s
t h e c o g n i t i v e ou tcomes o f l e a r n i n g Mathemat ics . The
a f f e c t i v e and psychor~lo tor l e a r n i n g outcomes of Mathema t i c s
l e a r n i n g are n e g l e c t e d .
The n e g l e c t o f t h e e v a l u a t i o n of s t u d e n t s '
a f f e c t i v e and psychomotor p e r f o r m a n c e s i n Mathemat ics
i s most p r o b a b l y owing to t h e fac t , a s r e s u l t s of some
s t u d i e s show, t h a t t e a c h e r s l a c k t h e c o m p e t e n c i e s t o /. -
c o n s t r u c t i n s t r u m e n t s f o r s u c h a s s e s s m e n t s , I t i s a l s o
known t h a t s u c h i n s t r u m e n t s a r e n o t a v a i l a b l e t o
t e a c h e r s so t h e t e a c h e r s d o n o t know w h a t t h e i n s t r u m e n t s
are, what t h e y c a n be u s e d fo r and how t h e y c a n be used ,
O f c o u r s e e x p e r t e d u c a t o r s know t h a t p r o j e c t s are
among s u c h i n s t r u m e n t s , E x p e r t e d u c a t o r s are of t h e
o p i n i o n t h a t p r o j e c t s c a n b e used t o p p p r a i s e s t u d e n t s '
p e r f o r m a n c e s i n t h e t h r e e domains , A l s o l i t e r a t u r e a n d
l o g i c a l c o n s i d e r a t i o n claim t h a t p r o jec L s i n Ma thema t i c s
c o u l d : improve s t u d e n t s 1 Mathemat ics ; and relate
N a t h e m a t i c s t o t r u e l i f e p r o b l e m s / s i t u a t i o n s and o t h e r
s u b j e c t areas,
I n t h i s s t u d y , therefore, t h e r e s e a r c h e r :
c o n s t r u c t e d some p r o j e c t s b a s e d on JSS
Mathema t i c s c u r r i c u l u m , and f u r t h e r d e v e l o p e d
t h r e e o f t h e p r o jccts i n t o e v a l u a t i o n i n s t r u m e n t s ;
d rew u p mark ing schemes f o r s c o r i n g e x p e c t e d
s t u d e n t s r e s p o n s e s t o t h e t h r e e p r o j e c t s a n d
u s e d e x p e r t e d u c a t o r s and e x p e r i e n c e d Ma thema t i c s
t e a c h e r s t o v a l i d a t e t h e schemes;
d e v e l o p e d a s c a l e f o r A f f e c t i v e Pe r fo rmance I n
Ma thema t i c s (SAPIM) and a l so a s c a l e for
M o n i t o r i n g A f f e c t?-ng L e a r n i n g 0t;tcomes of
Mathema t i c s P r o j e c t s (SMALOMP) ; and
s o u g h t and g o t e x p e r t e d u c a t o r s and e v a l u a t i o n
e x p e r t s h e l p t o c r i t i q u e , v a l i d a t e and w e i g h t
t h e items of SAPIM and SMkLOMP.
The a b o v e i n s t r u m e n t s were t r i e d o u t i n a p i l o t
s t u d y whLch i n v o l v e d 40 y e a r 111 JSS s t u d e n t s . SAPIM
was found t o h a v e r d l i a b i l i t y c o e f f i c i e n t s of 0,84
t h r o u g h t h e s p l i t h a l f a p p r o a c h and 0.92 t h r o u g h t h e
test-retest app roach . SMALOMP had a r e l i a b i l i t y
c o e f f i c i e n t of 0.80 t h r o u g h t h e s p l i t - h a l f a p p r o a c h ,
The s t u d e n t s r e s p o n d e d f a v o u r a b l y t o t h e p r o j e c t s as
d e v i c e s fo r t e a c h i n g and l e a r n i n g Ma thema t i c s a n d as
172
e v a l u a t i o n i n s t r u m e n t s .
To g u i d e t h e main s t u d y , f i v e r e s e a r c h q u e s t i o n s
and f o u r h y p o t h e s e s w e r e f o r m u l a t e d . T h e s e r e v o l v e d
a round :
t h e a c c e p t a b i l i t y o f t h e p r o j e c t s , by e x p e r t
e d u c a t o r s , as s u i t a b l t . and r e l e v a n t i n s t r u m e n t s
for t h e t e a c h i n g and L e a r n i n g of JSS Mathemat ics ;
t h e a g r e e m e n t of t h e e x p e r t s a n d Mathema t i c s
t e a c h e r s h i t h t h e c a t e g o r i z a t i o n s of t h e e x p e c t e d
s t u d e n t s 1 r e s p o n s e s t o t h e t h r e e p r o j e c t s i n t o
c o g n i t i v e a n d psychomotor ou tcomes as p r e s e n t e d
i n t h e mark ing schemes ;
t h e a b l l i t y o f Ma thema t i c s t e a c h e r s t o a g r e e
among t h t m s e l v e s and a t t h e same t i m e d i s c r i m i n a t e
among t h e s t u d e n t s ' p e r f o r m a n c e s w h i l e s c o r i n g
s t u d e n t s 1 r e s p o n s e s t o t h e t h r e e p r o j e c t s u s i n g
t h e mark ing schemes;
e s t a b l i s h i n g some i n t e r - l i n e a r r e l a t i o n s h i p s
among t h e c o g n i t i v e , psychomotor a n d a f f e c t i v e
p e r f o r m a n c e s o f s t u d e n t s ;
i n v e s t i g a t i n g t h e effects of t h e u s e of p r o j e c t s
on s t u d e n t s f a f f e c t i v e b e h a v i o u r toward Mathemat ics ;
173
d e t e r m i n i n g t h e level of s i g n i f i c a n c e o f
t e a c h e r s 1 marker r e l i a b i l i t y ;
i n v e s t i g a t i n g w h e t h e r t h e s i m p l e a n d m u l t i p l e
c o r r e l a t i o n s among t h e c o g n i t i v e , psychomotor
and a f f e c t i v e p e r f o r m a n c e s o f s t u d e n t s i n t h e
p r o j e c t s are s i g n i f i c a n t ; and
i n v e s t i g a t i n g w h e t h e r t h e s i m p l e and m u l t i p l e
r e g r e s s i o n c o e f f i c i e n t s among t h e s t u d e n t s w
c o g n i tibe, psychomotor and a f f e c t i v e pe r fo rmances
are s i g n i f i c a n t .
Four s c h o o l s and t h e i r 240 s t u d e n t s were s e l e c t e d
f o r t h i s s t u d y , I n e a c h o f t w o s c h o o l s , sampled
s t u d e n t s were random1-y grouped i n t o t w o , w h i l e s t u d e n t s
from e a c h of t h e o t h e r two, were p u t i n t o two a l t e r n a t e
r a n k s o r o u p s based o n t h e i r p r e - t e s t scores f rom SAPIM.
The t w o g r o u p s from e a c h s c h o o l were randomly a s s i g n e d
t o t r e a t m e n t o r c o n t r o l . A l l t h e sampled s t u d e n t s were
t a u g h t t h e same u n i t s of work i n Ma thema t i c s by t h e i r
r e s p e c t i v e t e a c h e r s . But p r o j e c t s were e n c o r p o r a t e d
i n t e a c h i n g and a s s e s s i n g t h e e x p e r i m e n t a l g r o u p s on ly .
A 1 though a l l t h e e x p e r i m e n t a l s t u d e n t s ' r e s p o n d e d to
t h e p r o j e c t s u sed a s e v a l u a t i o n i n s t r u m e n t s , t e n
r e s p o n s e s t o e a c h of t h e t h r e e p r o j e c t s w e r e randomly
174
s e l e c t e d and s u b m i t t e d t o t e n Mathema t i c s t e a c h e r s t o
score and s e n d i n t h e scores to t h e r e s e a r c h e r .
Data a n a l y s e s i n v o l v e d t h e u s e o f :
two way ANOVA f o r examining t e a c h e r s ' r a t h e r
r e l i a b i l i t y and d i s c r i m i n a t o r y a b i l i t y ;
AIYCOVA f o r d e t e r m i n i r y t h e effects o f u s e of
p r o j e c t s on s t u d e n t s 1 a f f e c t i v e p e r f o r m a n c e s i n
Ma thema t i c s ;
t-test s ta t is t ic t o d e t e r m i n i n g how s i g n i f i c a n t
t h e i n t e r - c o r r e l a t i o n among s t u d e n ts' c o g n i t i v e ,
psychomotor and a f f e c t i v e pe r fo rmance scores
a r e ; a n d
r e g r e s s i o n a n a l y s i s t o d e t e r m i n i n g t h e s i m p l e
and i n u l t l p l e l i n e a r r e l a t i o n s h i p s among t h e
s t u d e n t s t c o g n i t i v e , psychomotor a n d a f f e c t i v e
p e r f o r m a n c e s and w h e t h e r t h e s e l i n e a r r e l a t i o n -
s h i p s a r e s i g n i f i c a n t o r n o t .
R e s u l t s show t h a t :
t h e e x p e r t e d u c a t o r s and M a t h e m a t i r s t e a c h e r s
a g r e e t h a t t h e t h r e e p r o j e c t s a r e s u i t a b l e f o r
and r e l e v a n t t o e v a l u a t i n g JSS s t u d e n t s
p e r f o r m a n c e i n Ma thema t i c s ;
t h e e x p e r t s and t e a c h e r s a g r e e w i t h t h e c a t e g o r i e a -
t i o n o f t h e e x p e c t e d s t u d e n t s ' r e s p o n s e s i n t o
c o g n i t i v e and psychomotor s k i l l s or l e a r n i n g outcomes ;
t e a c h e r s ' rater r e l i a b i l i t y and d i s c r i m i n a t o r y
a b i l i t y are h i g h l y s i g n i f i c a n t ;
t h e u s e o f p r o j e c t s fo r t e a c h i n g and e v a l u a t i n g
a c h i e v e m e n t i n Mathemat ics h a s 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 JSS s t u d e n t s ' a f f e c t i v e
pe r fo rmances i n Mathemat ics ;
t h e c o e f f i c i e n t s o f s! .np le and m l ~ l t i p l e
c o r r e l a t i o n s among s t u d e n t s r c o g n i t i v e ,
psychomotor -nd a f f e c t i v e p e r f o r m a n c e s i n t h e
Ma thema t i c s p r o j e c t s are h i g h l y s i g n i f i c a n t ; a n d
t h e c o e f f i c i e n t s o f t h e s i m p l e and m u l t i p l e
l i n e a r r e g r e s s i o n e q u a t i o n s are a l so h i g h l y
s i g n i f i c a n t ,
G e n e r a l l y , it i s o b s e r v e d t h a t s t u d e n t s r e s p o n d e d
f a v o u r a b l y t o t h e p r o j e c t s . However, t h e u s e o f p r o j e c t s
i s more t a s k i n g and demanding on t h e teachers and f r o m
t h e s c h o o l s r e s p e c t i v e l y .
T h e s e r e s u l t s h a v e f a r r e a c h i n g i m p l i c a t i o n s for :
( i ) s t u d e n t s r l e a r n i n g of and pe r fo rmance i n
Ma thema t i c s ;
(ii) t h e t e a c h i n g of JSS Mathemat i c s i n terms of
i n s t r u c t i o n s , t e a c h e r p r e p a r a t i o n , t e a c h e r
pe r fo rmance , s c h o o l p l a n t a n d b u d g e t i n g ; and
i i i i ) c o n t i n u o u s a s s e s s m e n t i n terms o f : a v a i l a b i l i t y
afid u s e o f i n s t r u m e n t s , o b t a i n i n g more v a l i d
d a t a f o r g u i d a n c e , t y p e s o f data and o r g a n i z a -
t i o n of s u c h d a t a f o r r e p o r t i n g , and p e r s o n n e l
p r e p a r a t i o n and r e q u i r e m e n t .
The r e s e a r c h e r , t h e r e f o r e , recommends t h a t :
(i) p r o j e c t s be used f o r the t e a c h i n g a n d l e a r n i n g
of JSS Mathema t i c s ;
(ii) p r o j e c t s be used as i n s t r u m e n t s f o r e v a l u a t i n g
s t u d e n t s 1 c o g n i t i v e , psychomotor and a f f e c t i v e
pe r fo rmances i n Mathemat ics a t t h e JSS l e v e l
a t least ;
(iii) school a u t h o r i t i e s s h o u l d p r o v i d e s p a c e ,
equ ipmen t and materials to make t h e u s e of
p r o j e c t s p o s s i b l e ; and
( i v ) t e a c h e r o r i e n t a t i o n , p r e p a r a t i o n and morale
f o r imp lemen t ing t h e u s e of p r o j e c t s s h o u l d be
pur sued u r g e n t l y .
BIBLIOGRAPHY -
Ahman, J ,S , and Glock, MOD, (19711, --- E v a l u a t a p u ~ i L Growth, ( 4 t h e d , ) Boston: A l l y n and Bacon I n c , --
Aiken , LOR, (19731, "Ability and C r e a t i v i t y i n Mathemat ics , s f Review of E d u c a t i o n a l R e s e a r c h 43, 4, .- - ---. -.. --.- 9
A l e , S.0, ( 1 9 8 0 ) , " D i f f i c u l t i e s F a c i n g Mathemat ics T e a c h e r z i n Deve lop inc C o u n t r i e s - A C a s e S t u d y o f N i g e r i a , " E d u c a t i o n a l S t u d i e s i n Mathemat ics ----- . -. - . - ----. - - -..---.- <-9
Vol, 1 2 ,
H l i , A. (19881, " E f f e c t s o f a T r a i n i n g Programme on Good S t u d y Habi ts , on A t t i t u d e s , M o t i v a t i o n and Achievements Among N i g e r i a n Secondary Schoo l S t u d e n t s , t f -- The N i 9 e r i a n J o u r n a l o f E d u c a t i o n
,-,,,,.,,---a. h... , ...- ,,-, V o l , 1, N o , 1 0
A l i , A , (19891, " E f f e c t s o f a T r a i n i n g Programme on T e s t Taking and T c S ~ V J ~ S ~ ~ ~ S S on N i g e r i a n Secondary Schoo l S t u d e n t s ' Achievement i n Mathemat ics , J o u r n a l o f R e s e a r c h i n Educatiop-, Vol, 1, Noo 1.
Ba j ah , SOTe ( 1 9 8 3 ) , ,Teachinq I n t e ~ a t e d .- ...-. -.-- S c i e n c e --- C r e a t i v e l y , -- I b s d a n : u n i v e r s i t y P r e s s ,
B e l l , M,S, (19791, "Appl ied Problem s o l v i n g as a S c h o o l Emphasis: A;I Asses smen t and Some Recommendations, A p p l i e d M a t h e m a t i c a l Problem -------. -,...a- -- S o l v i n g , V o l , 4, N o , 2,
Bigg, M,L, and Hunt, M,P, (19681, ~c,h,o.loc$ca~. F o u n d a t i o n s of E d u c a t i o n , (2nd e d J - , New York: ----.--.. -- Harpe r and Ruw,
Bloom, B,S, e t a l , ( 1 9 7 1 ) ~ -.-.------a-A.-. Handbook on F o r m a t i v e .--. ..--- and Summative E v a l u a t i o n o f L e a r n i x . New York: -- .--- -.-.--,-- McGraw-Hill Book Company,
Brunner , J .S, ( 1 9 6 0 ) , The P r o c e s s of E d u c a t i o n , .,-. --,--- .-.-- .-.- Cambridge: Harva rd U n i v e r s i t y P r e s s ,
t i r unne r , J ,S, ( 1 9 6 0 ) ~ Toward A Theory Of ... - -.. I n s t r u c t i o n , - ------ Cambridge : Harva rd ~ n < v e r s i ty Pre,ss,
Brunner , J,S., O l ive r , R.R. e t a l e ( 1 9 6 6 ) , S t u d i e s i n C o q n i t i v e Growth, New York: Weley,
Campbel l , DOT, and S t a n l e y , J,C, ( l 9 6 3 ) , Experimefits and QU a s i -ex p e r imernt,ad2 ,~-e.gi~gns:'~og-~.e~se_a-rch. - SKOKIZ Ill,: Rand McNally,
C a r p e n t e r , T.P., Coburn, T.G., Reys, R O E a , and Wilson, J , N , (19751, ' @ R e s u l t s and Implications o f NAEP Mathema t i c s k c s e s s m e n t i n Seconda ry Scho01, '~ The Mathema t i c s .---,-- ---- Teacher-, ---- - 68, 453-470,
C a r p e n t e r , T,B,, C o r b i t , M,K,, Kepner, HeSe, L i n d q u i s t , M,M. and Reys, R,E, (19801, s t K e s u l t s o f t h e Second NAEP Mathemat ics Asses smen t i n Seconda ry school^,'^ The Mathem-a-~ic-~,~T,41,a_cJ,e~, 73, 329-338,
Chacko, I, ( 1 9 8 0 ) ~ "The R e l a t i o n s h i p Between S e l e c t e d T e a c h e r ?er formance V a r i a b l e s and S t u d e n t C o g n i t i v e and Af f s c t i v e Outcomes i n Mathemat ics , " Abacus, Vol, 3_5.
Chauhan, S.S, ( 1 9 7 8 ) , Advanced -- E d u c a t i o n a l .... - -- .- -- - ...- Psychology , -. - - - New D e l h i : Vikas ,
C hauhan, S, S, (1979) , I n n o v a t i o n s .. -.-- i n -- ~eachinq- earning -..A-h - P r o c e s s . New D e l h i : Vikas .
D a l t o n , S, and O v e r a l l (19771, uNon-random Ass ignment i n ANCOVA: The A l t e r n a t e Ranks Design." The J o u r n a l o f E x p e r i m g n t a l Educatip_n, V o l , 46, N 7 e l a ,
Dav i s , G.S. (19701, P ~ ~ c h o l o q y ~ ~ f ~ , p _ $ g ~ ~ ~ e ~ ~ ~ ~ ~ 0 1 ~ ~ p ~ : Theory and Practice, New York: Basic Books. ---
Dayton, C.M. (19701, The Desiqn o f .-- E d u c a t i o n -... .-a -.- -. E x p e r i m e n t s , N e w York: McGraw- ill. -
Demars, R, J, (19711, Compara t ive s t u d y of S e v e n t h Grade Low A c h i e v e r s t A t t i t u d e and Achievement i n Ma thema t i c s Under Two Approaches U/CSM and 1 c a d i t ional , l l Doctorial D e s s e r t a t i o n , U n i v e r s i t y o f Alabama. D i s s e r t a t i o n Abstr-a-c-t- I n t e r n a t i o n a l , 1972, 32, ppe 48324-4833'j-4'1-;1----
Dewey, J, (19661, Democracy and E d u c a t i o n , New York: The F r e e P r e s s .
Downie, H o R a and Hea th , R,W, ( l 9 7 O ) , e~i.sS,ta~stical Methods i n E d u c a t i o n , 3 r d ed, New York: Harpe r and Pocl,..
E b e l , R , L , ( 1 9 5 1 ) , q t E s t i m a t i o n of R e l i a b i l i t y of R a t i n g s , " Psychomet r ika , 16 , 407-424,
Lchemazu, K.L, e t a l , (1984) , wDevelopment and P r e l i m i n a r y V a l i d a t i o n o f S u g g e s t e d I n s t r u m e n t s f o r E v a l u a t i n g t h e Psychomotor Domain ( w i t h A p p l i c a t i o n t o Year I JSS Geometry and M e n s u r a t i o n ) . Unpubl i shed ----.A S.Sc, P r o j e c t , Depar tment o f E d u c a t i o n , U n i v e r s i t y o f N i g e r i a , Nsukka,
E g y l e s t o n , J-F, and K e l l y , PoJ, ( 1 9 7 0 ) ; q tAssessment o f P r o j e c t work i n A- leve l 3 io l0gy .~ ' E d u c a t i o n R e s e a r c h , V o l , 1 2 , N o , 1,
Epunam, B, e t a l , (19831, "An I n v e s t i g a t i o n I n t o t h e E v a l u a t i o n T e c h n i q u e s Used i n Our Seconda ry S c h o o l Ma thema t i c s and I ts I m p l i c a t i o n s t o C o n t i n u o 1s k s s ~ s s m e n t O u Unpubl i shed B,Sc, -- Prr) ject, Dept , of E d u c a t i o n , U o N s N ,
Ewing t o n , E. J. and S p e n c e r , J. (1980 ) , ,Prplects-,& Bio loqy , London: R o u t l e d g e a d hegan ,
E z e i l o , J.O.C, ( l 9 8 8 ) , Keynote Addres s a t t h e 1988 M a t h e m a t i c a l A s s o c i a t i o n of N i g e r i a (MAN) Annual C o n f e r e n c e h e l d a t t h e U n i v e r s i t y of N i g e r i a , Nsukka,
F e d e r a l ivkinistry o f E d u c ~ t i o n ( 1 9 8 0 ) ~ t.andbgc&con C o n t i n u o u s Assessment , Lagos: F e d e r a l M i n i s t r y - .- .- o f E d u c a t i o n ,
F e d e r a l R e p u b l i c o f N i g e r i a (19811, -.-, N a t i o n a l ..-. -...... --,.-- P o l i c y On E d u c a t i o n , Lagos: F e d e r a l M i n i s t r y of In fo rma t i o n .
F f ~ y , ReBo, C r o s s , L-H,, and Lowry, RcS. (19771 , "Random G u e s s i n g , C o r t e c t i o n f o r G u e s s i n g acd K e l i a b i l i t y of M u l t i p l e - C h o i c e T e s t s S c o r e s p I 1 The J o u r n a l of E x p e r i m e n t a l E d u c a t i o n V o l , 46 , - , - - 3 N o , 1,
GLaser, H. ( 1 9 6 2 ) , u P s y c h o l o g y and I n s t r u c t i o n a l Techno logy , T r a i n i n q R z s e a r c h and-,=a&icoc. R , Glaser (ed ,'I, P i t t s b u r g : University of P i t t s b u r g P r e s s .
Glass, G.V, and S t a n l e y , J.C, (19701, -- S t a t i s t i c a l .. ---- Methods i n E d u c a t i o n and Psycholpq41- Englewood - C l i f f s i ' J , J , : P r e n t i c e - H a l l ,
Greenburg , BOG, (19531 , T h e Use of ANCOVA and B a l a n c i n g i n A n a l y t i c a l Su rveys . I' ~ ~ e ; - i ~ - a ~ J o u r n a l o f P u b l i c H e a l t h , 43,
H a r b o r - I b e a j a , V,F, (19831 , u C o n t i n u o u s Asse s smen t : I ts I m p l i c a t i o n s , P rob l ems a n d P r o s p e c t s i n Anembra S t a t e ( w i t h some a p p l i c a t i o n s t o ; l a t h e m a t i c s T e a c h i n and E v a l ~ a t i n l1 A a er p r e s e n t e d a t t h e 2n% ~n t e r n a t l o n a ? 2 d n f e r e W on T e a c h i n g P r a c t i c e a n d T e a c h e r E f f e c t i v e n e s s , U n l v x s i t y of N i g e r i a , Nsukka, A p r i l 6-9, 1983 ,
H a r b o r - I b e a j a , V.F. and Nworgu, BOG. (19831 , "The S y s t e m a t i c and Comprehens ive A s p e c t s of C o n t i n u o u s Assessment , " A p a p e r p r e s e n t e d a t t h e 2 1 s t Annua l C o n f e r e n c e of MAN. U n i v e r s i t y o f C a l a b a r , Sep t ember 18-22, 1983 ,
Has san , B, (19811 , " E f f i c i e n c y i n t h e Use of t h e S c h o o l Day i n T e a c h i n g Ma thema t i c s , " Unpub l i shed Ph,D, D i s s e r t a t i o n , U n i v e r s i t y of C h i c a g o ,
Hays, W,L, (19731 , S t a t i s t i c s for S&aal Sci,e5c,es0 (2nd ed , ) , New York: H o l t , R i n e h a r t a n d Wins ton ,
H o r r o c k s , J ,E, and Schoone r , T.1. (19681 , Measurement for T e a c h e r s , Ohio: C h a r l e s E o Merril Pub, Co,
I gboko , P.M. (19761 , "The P o t e n t i a l F o r T e a c h i n g Modern M a t h e m a t i c s i n S e c o n d a r y S c h o o l s and 'I',T,C, of Iqro and Anambra S t a t e s e s g ,A.acus, V o l . 11,
I n t e r n a t i o n a l C e n t r e f o r E d u c a t i o n a l E v a l u a t i o n (19821, S c h e d u l e of L e c t u r e s fo r a C o n f e r e n c e on . P r i o r i t i e s i n E d u c a t i o n a l R e s e a r c h i n N i g e r i a , h e l d i n 1982 a t U n i v e r s i t y of Ibadan.
I p a y e , T, (1982), C o n t i n u o u s Assessment ig-Scc,h,o,o~ With Some C o u n s e l l i n q A p T l i c a t i o n s , I l o r i n : -- -- I l o r i n U n i v e r s i t y P r e s s ,
Iwuchukwu, 0-B. e t al, ( 1 9 8 4 ) , "Development a n d P r e l i m i n a r y V a l i d a t i o n o f Sugges t ed I n s t r u m e n t s f o r E v a l u a t i n g t h e Psychomotor Domain ( w i t h a p p l i c a t i o n to Year One JSS Geometry and M e n s u r a t i o n ) .I1 Unpubl i shed B,Sc. P ro j ecL , Depar tment of E d u c a t i o n , U n i v e r s i t y of N i g e r i a , Nsukka,
Jack.son, So (19741, A T e a c h e r ' s Gu ide to T e s t s and -....--... .- - T e s t i n q . London : Longman Group Ltdo -.
J a n v i t i r , C , (19811, ttUses o f S i t u a t i o n ? i n Ma thema t i c s E d x a t i o n O u E d u c a t i o n a l S t u d i e s i n Mathemat ics , ---- "-- - -- Boston: U. ~ e l d e l P u b l i s h i n g Coy.
Jonah , C-No e t a l , (19861, ftDeveloprnent a n d P r e l i m i n a r y V a l i d a t i o n o f I n s t r u m e n t s for A s s e s s i n g A f f e c t i v e Outcomes i n Year I, J u n i o r Secondary S c h o o l Mathemat ics , (Geometry and ~ e n s u r a t i o n ) , " Unpubl i shed - B,Sc, P r o j e c t , Dept , of E d u c a t r o n , U n i v e r s i t y o f N i g e r i a , Nsukka,
Kale j a i y e , A - 0 , (19761, "An A n a l y s i s of O b s e r v a t i o n s o f t h e Use of T e a c h i n g P r o c e d u r e s i n Ma thema t i c s lesson^,^^ Abacus, V o l . 11, Nos. 1 and 2, pp, 33-41.
K a l a j a i y e , A,O, ( 1977) , "A Comparison of A t t i t u d e s and O p i n i o n s o f Low a n d High Mathema t i c s A c h i e v e r s i n J u n i o r and S e n i o r C l a s s e s of Seconda ry school^.^^ L a os E d u c a t i o n a l Review Vol. I, N o , 1, pp, 1-6, A ---,-..,..- 9
Kempa, HoFo and J , M , Grough (19771, "A S tudy of A t t i t u d e s Towards Ma thema t i c s i n R e l a t i o n t o S e l e c t e d S t u d e n t characteristic^,^^ B r i t i s h
- 7 . u -
J o u r n a l of E d u c a t i o n a l Psychology , 47, 296-304. - --
Keppal , G, (19731, Des iqn a n d A n a l y s i s : A R e s e a r c h e r ' s -- Handbook, Englewood C l i f f s N , J , : P r e n t i c e H a l l ,
K e r l i n g e r , F,N, (19731, F o u n d a t i o n s o f - -- B e h a v i o u r a l ---. - ,A . .- .--.dm - - Resea rch . (2nd e d , ) , London: H o L t , R i n e h a r t and Nins ton .
K i f e r , E, ( 1 9 7 4 ) , t q R e l a t i o n $ h i p Between Academic Acnieve- ment a n d 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 ; A Quas i - l o n g i t u d i n a l ~ t l ~ d ~ American --. ~ d u c a t i o n a l - - - - --... .. R e s e a r c h J o u r n a l , 1 2 , 2, pp, 191-210. p-
Kreathwohl , D,R, e t a l , (19701, ~axonom~o.f~~,du,c:ational O b j e c t i v e s , The ~ 1 a s s i f i c a T i o n of E d u c a t i o n a l ~ o a l s andb book 11, A f f e c t f v e Doman. London: Longman Group L td ,
K r u s t e t s k i i , V,A, (19761, The P s y c h o l , --.-..-.---.-- of M a t h e m a t i c a l .-- A b i l i t i e s i n Schoo l C h i l d r e n , Chicago: U n i v e r s i t y of C h i c a a o P r e s s ,
Landa, LON, ( l 9 7 4 ) , A l q o r i ~ L h m i z a t i o n i_n_~ea rn ig , a , "d : n s t r u c t i o n , Englewood-Cl i f f s N , J , : E d u c a t i o n a l techno lo^;^ P u b l i c a t i o n s I n c ,
L a s s a , P,?i, (19861, T h e S o r r y S t a t e of Mathema t i c s Educh t ion i n N i g e r i a , An I n a u g u r a l Addres s , U n l v e r s i t y o f Jos, N i g e r i a .
Lassa, P.N, (19781, " T r a i n i n g Rece ived i n Ma thema t i c s by Grade Two Teache r s , " Abacus, V o l , 12,
Lassa, P,N, (19751, " C o r r e l a t i o n a l S tudy Between Mathema t i c s Achievement a n d A t t i t u d e Toward Mathema t i c s Among P r o s p e c t i v e Grade 1Wo Teache r s , " N i q e r i a n E d u c a t i o n a l Forum, 2, 1, pp, 18-24,
Lesh , H, (19811, "Appl ied M a t h e m a t i c a l Problem ~ o l v i n g , ~ E d u c a t i o n a l S t u d i e s i n Ma thema t i c s Vol, 1 2 ,
-, ------, Boston: D , R e i d e l Pub. Co,
Lubin , A, and S, Harris (19651, IqProblems i n Measur ing Change, American J o u r n a l of P s y c h o l o a ~ , 76, --- .-.
McBridge, C.C. ( 1 9 7 4 ) , ('The E f f e c t s o f H i s t o r y of Mathema t i c s on A t t i t u d e Toward Mathema t i c s o f C o l l e g e A l g e b r a S t u d e n t s , q q (Doctorial D i s s e r t a t i o n , 'Texas-A. & - M , U n i v e r s i t y ) . D i s s e r t a t i o n - A b s t r a c t --.-.-- I n t e r n a t i o n a l , 35, p, 1963A.
Mcilemar, Q, ( 1 9 6 9 ) , P s y c h o l o g i c a l ,-.... S t a t i s t i c s .. .,.... ._ , .-.-. ,I N e w York : N i l l e y .
Mgbo, A,M,N, !1980) , "The A ~ a m b r a S t a t e Common E n t r a n c e Examina t ion as a P r e d i c t o r of Academic Achievement ," Unpubl i shed Masters ---. T h e s i s . --.. a -- Depar tment of E d u c a t i o n , U,N,N.
Mkpa, M.A, ( l 9 8 5 ) , I1A S t u d y of t h e Problems of Implement ing t h e C o a t i n u o u s Asses smen t Programme o f E v a l u a t i o n i n Er imary S c h o o l m M A p a p e r p r e s e n t e d a t t h e I n t e r n a t i o n a l C o n f e r e n c e o f P r imary E d u c a t i o n , h e l d a t t h e U n i v e r s i t y o f N i g e r i a , :!sukka March 1985,
Mukher i j ee , A , and F. Umar (19781, " A t t i t u d e s , A t t i t u d e Chanqe and Mathema t i c s A t t a i n m e n t s : A S tudy i n ~ i g e r i a . I n t e r n a t i o n a l Review o f ~ d u c a t i o n .- ---9 22, pp, 518-521.
Neale, DOC, ( 1 3 6 9 ) , "The Role of A t t i t u d e i n mat he ma tic^,;^ A r i t h e m e t i c T e a c h e r , 16, pp, 631-640.
Nwagu, i3 ,K.N. e t a l , ( l 9 8 2 ) , Weed f o r Concept -Or ien ted Programme of Geography i n N i g e r i a Seconda ry school^.^^ Unpubl i shed B,Sc, Pr_o_ject. Depar tment ------..- .----- o f L d u c a t i o n , U n i v e r s i t y o f N i g e r i a , Nsukka.
Obicma, G O O . ( 1 9 8 2 ) , "The Development a n d ~ r e l i m i n a r y VaJ l ida t ion of A D i a g n o s t i c Ma thema t i c s Achievement T e s t f o r N i g e r i a n Seconda ry S c h o o l S t u d e n t s , " Unpub l i shed M,ED, T h e s i s Dept. of E d u c a t i o n , U.N.N. --.d
O b i o m a , G,O, ( 1 9 8 4 ) , "The Development and V a l i d a t i o n of A D i a g n o s t i c Mathemat ics T e s t s f o r JSS -
S t u d e n t s O 1 l Unpubl i shed --- PhoD. - ' 2hes i s -.- - --.- 9 Depar tment o f E d u c a t i o n , U.N.No
Ogomaka, P,M,C, ( 1 9 8 4 ) , "Development a n d P r e l i m i n a r y V a l i d a t i o n o f S t u d e n t s t E v a l u a t i o n o f T e a c h e r E f f e c t i v e n e s s S c a l e (SETESJw Unpubl i shed MOLD, P r o j e c t , Depar tment of E d u c a t i o n , U o N , N ,
Ohuche, H,O, and S,A, Ake ju ( 1 9 7 7 ) ~ , T e s t i n % - a ~ i Evaluation i n E d u c a t i o n , Lagos: A f r i c a n - ------ E d u c a t i o n a l H P S O U ~ C ~ S ~
O k a f o r , F,Co ( 1 9 8 4 ) ' P h i l o s o p h y o f Educa$iofi (2nd e d , ) , Enugu: S t a r P u b l i s h i i i g Co,
Okeke, E,R.C. (1982 1, " E v a l u a t i o n Through Ass ignmen t s and P r o j e c t s , I 1 Unpubl i shed Pape r , Dept of E d u c a t i o n , U,N.N,
O k u n r o t i f a , P.0, ( 1 9 7 7 ) , I f E v a l u a t i o n - . _ . - - - - . - i n Geographx, .. - Ibadan : Oxford U n i v e r s i t y P r e s s ,
O l a t u n j i , S o '. 9 7 4 ) , E v a l u a t i n g L e a r n i n g a t u n i v e r s i t y L e v e l , West A f r i c a n J o u r n a l o f ducat ion . . . - - - - - . -- 4 - - --' Vol, X V I I I , N o , 2,
Onwuka, U, (ea,) ( 1 9 8 1 ) ~ C u r r i c u l u m D e v e a m e n t -. -.-- &-..-- f o r A f r i c a . O n i t s h a : A f r i c a n a P u b l i s h e r s Ltdo - -
Osborn , H , H , i 1 9 8 3 ) , "The Assessment o f Ma thema t i ca l A b i l i t i e s . I 1 E d u c a t i o n R e s e a r c h V o l , 25, No, I. --.-7
Osibodu, M, ( 1 9 7 8 ) , lvMathematics E d u c a t i o n i n N i g e r i a n Seconda ry School,I1 A p a p e r p r e s e n t e d a t t h e I n t e r n a t i o n a i C o n f e r e n c e of Mathema t i c s E d u c a t o r s . I l n i v e r s i t y o f Leeds ,
P i a g e t , J, (19701, " P i a g e t l s Theory" i n Mussen, P,M, ( e d , ) C a r m i c h a e l t s Manual of C h i l d P s y c h 0 1 ~ ~ . New York: Wiley,
P e a r s o n , F o (19671, Choos inq A Voca t ion , N e w York: Agathon P r e s s ,
Po lya , G, ( 1 9 7 5 ) ' H o w To S o l v e It. Garden C i t y N e w York: Anchor Books,
R e m e r s , H,H,, G r a g e , NoL, a n d J - F , Rummel ( 1 9 6 6 ) , h P r a c t i c a l I n t r o d u c t i o n Measurement a@- -...---- E v a l u a t i o n , New York: H a r p e r a n d Row, ------
R u s s e l , S,S.B, (19821 , "The C o n s t r u c t i o n , V a l i d a t i o n a n d llse o f a S t a n d a r d i z e d T e s t t o M e a s u r e A c h i e v e m e n t i n Hiqh S c h o o l M a t h e m a t i c s i n L i b e r i a , ' @ ~ n p ~ b l i z h e d Ph.D, T h e s i s , D e p a r t m e n t of E d u c a t i o n , U n i v e r s i t y of Nigeria, Nsukka.
Sawyer , d ,vd, (136B), f l iblathGmatics i n S e c o n d a r y S c h o o l s , i s - . M a t h e m a t i c s ------- i n .-.- -. Commonwealth ---.-.,..-..--. S c h o o l s , on don : Commonwealth S e c r e t a r i a t e ,
S i l v e r m a n , H , J , (19731 , "Des ign a n d E v a l u a t i o n of a U n i t A b o u t M e a s ~ r c m e n t as V e h i c l e f o r C h a n g i n g A t t i t u d e Toward M a t h e m a t i c s a n d S e l f - c o n c e p t of Low A c h i e v e r s i n I n t e r m e d i a t e Grades,Iq ( D o c t o r i a l D i s s e r t a t i o n , Fordham U n i v e r s i t y ) . D i s s e r t ~ t i o n Abstract ~ n t e x a t i o n a l -,---.-- 9 1 9 7 4 , - 34, p, 4 7 1 7 ~ :
S t r u d e r , M o R o ( 1 9 7 7 1 , "The R e l a t i o i l s h i p of D i s c o v e r y Methods i n M a t h e m a t i c s t o C r e a t i v e T h i n k i n g a n d A t t i t u d e Toward Mathemat ics . " ( D o c t o r i a l 9 i ~ s e r t a t i o r i , O h i o S ta te U n i v e r s i t y ) . P L s s e r t ~ t i o n Abst rac t I n t e r n a t i o - n a L , 1972 , 32 , - - . - - _ I .-. PO 3816A..
T y l e r , R O W o ( 1 9 8 6 1 , "Changing C o n c e p t s of E d u c a t i o n a l E v a l u a t i o n , v r I n t e r n a t i o n a l J o u r n a l o f ----.- E d u c a t i o n a l R e s e a r c h , V o l , 1 0 , No, 4,
Vernon , P,E, ( 1 9 6 4 1 , The S t r u c t u r e of ---- Human -- --- A b i l i t i e s , --.. . L -..-a
London: Metbuen,
WHEC (19761 , T h e C h a l l e n c e - - A--- o f E x a m i n a t i m - - - . 1 r r e g u l % r i ties, Yaba Lagos : WAEC,
WAK (19711 , A n n u a l R e p o r t . Yaba Lagos: WAECo
WHEC ( 1 9 7 3 1 , " T e a c h e r s G r a d e Two C e n t r a l ~ x a m i n a t i o n C h i e f E x a m i n e r ' s R e p o r t s o n t h e P e r f o r m a n c e of " a n d i d a t e s o A r i t h m e t i c P a p e r , J u n e 1973 , L a g o s : WAEC ,
VJicklegeen, W,A, ( 1 9 7 4 ) , How T o S o l v e - p m , b > ~ ~ . San F r a n c i s c o : N O H , Freeman.
W i l l i a m s , JAO, ( 1 9 7 2 ) , "Teaching Emphasis i n P r i m a r y Mathema'cics. l1 E d u c a t i o n a l Keseara, N o . 14,
~ i n e r , B.J. ( l 9 7 l ) , s t a t i s t i c a l P r i n c i p l e s i n E x p e r i m e n t a l Des ign , New York: M c G r a w H i l l . -
wood, R. ( l 9 6 8 ) , " O b j e c t i v e s i n t h e Teach ing of Mathema t i c s f f , E d u c a t i o n a l , R e s e a r c h , N o , 10, ..--
Yoloye, e t a l , (19801, Handbook on - C o n t i n u o u s .-.- -- a-L-... Assessment , Lag7r: F e d e r a l M i n i s t r y of Educa t ion .
Zukoski , J, J, ( l 9 8 O ) , "The I d e n t i f i c a t i o n and A n a l y s i s of A c t u a l and Desirable Teach ing Compe tenc ie s o f Seconda ry Schoo l I n s t r u c t o r s of Academic -
D i s s e r t a t i o n A b s t r a c t I n t e r n a t i o n a l S u b j e c t s o Is - . . , - - . - - . - - ~ - . . . . -. -. . .- Series V o l , 40 , N o , 7 , --?
187
APPENDIX IA
AN INSTRUMENT FOR CkRRYING OUT SOME 'VALIDATION OF SAPIM
( Q u e s t i o n n a i r e f o r L x p e r t E d u c a t o r s -- and T e a c h e r s ) -.
I a m a s t u d e n t , i n t h e Depar tment of E d u c a t i o n , U n i v e r s i t y o f N i g e r i a , Nsukka, d e v e l o p i n g a scale f o r ~f f e c t i v e Per formance i n Mathemat ics ( sAPIM) . B e l o w are s t a t e m e n t s e x p r e s s i n g some f ee1i i ;gs ( p o s i t i v e , n e u t r a l o r n e g a t i v e ) toward Mathema t i c s , K i ~ d l y a s s i g n a w e i g h t t o e a c h of t h e s t a t e m e n t s w i t h r e s p e c t t o a s e v e n p o i n t scale. A s s i g n s e v e n t o s t a t e m e n t s i n d i c a t i n g v e r y p o s i t i v e f e e l i n g s , f o u r t o n e u t r a l s t a t e m e n t s , a n d o n e t o s t a t e m e n t s i n d i c a t i n g v e r y n e g a t i v e f e e l i n g s .
t-- very-ve N e g a t i v e N e u t r a l
_W_t P o s i t i v e v e r y + v e
Use t h e above scale t o a s s i g n w e i g h t s t o t h e items below as you c o n s i d e r a p p r o p r i a t e .
1, The s t u d y of Mathema t i c s is e n j o y a b l s /I-/
2, Mathemat i c s i s a v e r y d i f f i c u l t s u b j e c t ~7
3, Knowledge of Mathema t i c s h e l p s s t u d e n t s t o l e a r n many e t h e r s u b j e c t s /17
4. The t h i n g s l e a r n t or s k i l l s a c q u i r e d from Mathema t i c s are n o t u s e d o u t s i d e s c h o o l - 1-7
5. It i s j u s t i f i e d t o make Mathema t i c s compu l so ry i n s e c o n d a r y s c h o o l s /7 -- -
6. P e o p l e c a n become v e r y r i c h w i t h o u t knob:ing any b i t o f Mathemat ics
M a t h e m a t i c s s h o u l d be s t u d i e d r e g u l a r l y b y J u n i o r S e c o n d a r y s c h o o l s t u d e n t s . -- /7 S e c o n d a r y s c h o o l s t u d e n t s s h o u l d n o t w o r r y i f t h e y h a v e n o t M a t h e m a t i c s t e x t b o o k s . 0 S e c o n d a r y s c h o o l s t u d e n t s s h o u l d b e l o n g to M a t h e m a t i c s c l u b s i n t h e i r s c h o o l s . - /7 M a t h e m a t i c s i n v o l v e s f o r m u l a e w h i c h h a v e l i t t l e or n o r e a l / m e a n i n g f u l e x p l a n a t i o n s .
Ideas a n d s k i l l s l e a r n t i n M a t h e m a t i c s are n o t e a s i l y f o r g o t t e n , /1;7 I M a t h e m a t i c s s h o u l d be d o n e f i v e or f o u r - p e r i o d s p e r class p e r week i n s e c o n d a r y s c h o o l s /_/
P e r f o r m a n c e i n M a t h e m a t i c s e n t i r e l y d e p e n d s o n t a l e n t / g i f t.
O n l y v e r y s e r i o u s or c o g e n t r e a s o n s s h o u l d make s t u d e n t s m i s s ath he ma tics classes. E-7 M a t h e m a t i c s tes t s / examina t ions / a s s ign rnen ts make e v e r y s t u d e n t n e r v o u s . i7 M a t h e m a t i c s p r e s e n t s c h a l l e n g e s ,
W i t h o u t M a t h e m a t i c s t h e r e c a n be n o s c i e n c e and t e a c h n o l o g y .
18, M i t h l i t t l e or n o efforts, I p a s s M a t h e m a t i c s v e r y w e l l .
1 9 . M a t h e m a t i c s makes many p e o p l e mid.
20. Nobody c a n be i n b u s i n e s s w i t h o u t t h e k n o w l e d g e of M a t h e m a t i c s . - 17
21. M a t h e m a t i c s i s n o t p r a c t i c a l . /-'7 - 22. M a t h e m a t i c s i s t h e most i m p o r t a n t subjec t . ~7 1
AN INSTRUMENT FOR CARRYING OUT SOME ASPECT OF VALLDATION OF --- ~ M P -
I am a s t u d e n t , i n t h e Department of Educat ion , U n i v e r s i t y of N i g e r i a , Nsukkn, deve lop ing a Scale f o r Moni tor ing A f f e c t i v e Learning Outcomes from Mathemat ica l Projects (SMALOMP), Below a r e some s t a t e m e n t s e x p r e s s i n g some f e e l i n g t o n e s ( p o s i t i v e , n e u t r a l o r n e g a t i v e ) toward s named Mathematical p ro jec+ ; t h a t has been execu ted by t h e r e sponden t s . Kindly a s s i g n a w e i g h t t o each of t h e s t a t e m e n t s w i t h r e s p e c t t o a seven p o i n t scale, Ass ign seven to s t a t m i e n t s i n d i c a t i n g v e r y p o s i t i v e f e e l i n g s , f o u r t o n e u t r a l s t a t e m e n t s , and one to s t a t e m e n t s i n d i c a - t i n g v e r y n e g a t i v e f e e l i n g s . Assign f i v e o r s i x 2nd t h r e e o r two t o s t a t e m e n t s i n d i c a t i n g r e l a t i v e l y p o s i t i v e and n e g a t i v e f e e l i n g s r e s p e c t i v e l y a s you c o n s i d e r a p p r o p r i a t e .
v e r y v e r y n e g a t i v e n e i a t i v e , n e u t r a l pos i t?ve p o s i t i v e
The p r o j e c t I have now c o m p l e t e d / c n r r i e d o u t and o t h e r s t h a t a r e s i m i l a r t o it:
1, a r e c h a l l e n g i n g and e n j o y a b l e /r7 2. a r e v e r y tiresome and time-wasting /7 3 , h e l p t o t e a c h u s more a b o u t Mathematics _/-7
- 4, consumme a l o t of m a t e r i a l s which are c o s t l y //
5 . h e l p t o e x p l a i n some fo rmulae and t h e i r u s e s - /7 - 6 , have l i t t l e o r no v a l u e t o t h e s t u d y of Mathematics
7 , c o u l d be used t o produce t h i n g s for e x h i b i t i o n s - /7 - 8. l e a d u s t o produce c h i l d i s h and w o r t h l e s s a r t i c l e s //
9. c o u l d be o f much u s e i n i n d u s t r y / w o r k s h o p s / r e n d e r i n g s e r v i c e s / s p e n d i n g l e i s u r e - 17
10. are some o f t h o s e t h i n g s d o n e i n s c h o o l s t o k e e p s t u d e n t s b u s y and n o t h i n g m o r e - 17
11. s h o u l d be made r e g u l a r p a r t o f t h e s t u d y of M a t h e m a t i c s i n s c h o o l s - 1-7
1 2 , s h o u l d n e v e r be u s e d t3 assess a n d r c c o r d s t u d e n t s l p e r f o r m a n c e s i n M a t h e m a t i c s L 7
13. s t a t e how/what you feel a b o u t t h i s p a r t i c u l a r p r o j e c t i n f o u r o r m o r e s e n t e n c e s ~7
P l e a s e delete t h e number b e s i d e a n y o f t h e a b o v e i t e m s t h a t i s n o t r e l e v a n t o r a p p r o p r i a t e . F u r t h e r s u g g e s t i o n s t o i m p r o v e t h i s i n s t r u m e n t w i l l b e welcome,
T h a n k s f o r y o u r h e l p ,
Y o u r s t r u l y ,
APPENDIX I1
SCALE FOR I~F 'FECTIVE PERFORMANCE - - I N PiATHl3lATICS (SAPIM).TO B3 _RESPO'I'\IDED-
BY STUDENTS
P l e a s e f i l l o u t t h e f o l l o w i n g :
N a m e of r e s p o n d e n t : , o . . . . o , O O O O o O O o O Q O ~ a O o O O O O O O o o ~ o o e o o ~ .-
Sex of r e s p o n d e n t : Male E-7 Female -- / /
T i c k ( q ) i n t h e r e c t a n g l e s b e s i d e t h e s t a t e m e n t s a b o u t Ma thema t i c s w i t h which you a g r e e v e r y w e l l / o b s o l u t e l y ,
1, The s t u d y o f Ma thema t i c s i s en j o y a b l e ,
2, Ma thema t i c s i s a v e r y d i f f i c u l t s u b j e c t . = 3 , Knowledge ?f Mathemat i c s h e l p s s t u d e n t s t o
l e a r n many o the r s u b j e c t s . - /-7 4, The i d e a and s k i l l s l e a r n t from Mathema t i c s
a r e n o t u seu o u t s i d e s c h o o l , L7 5, It is j u s t i f i e d t o make M a t h e m a t i c s
compu l so ry i n s e c o n d a r y s c h o o l s .
6. P e o p l e c a n become ve.ry r i c h w i t h o u t knowing any b i t o f Ma thema t i c s , /I/
7. Ma thema t i c s s h o u l d be s t u d i e d r e g u l a r l y by s e c o n d a r y s c h o o l s t u d e n t s .
8, Seconda ry s c h o o l s t u d e n t s s h o u l d n o t w o r r y i f t h e y have no Mathemat ics t e x t b o o k s . n
9. Seconda ry s c h o o l s t u d e n t s s h o u l d be long t o Mathema t i c s c l u b s i n t h e i r s c h o o l s . - 1-7
10. Mathemat ics i n v o l v e s f orrnulae wh ich have l i t t l e o r no r e a l / m e a n i n g f u l e x p l a n a t i o n s .
11, I d e a s o r s k i l l s l e a r n t from Mathemat ics are n o t e a s i l y f o r g o t t e n .
M a t h e m a t i c s s h o u l d be d o n e f i v e o r f o u r p e r i o d s p e r class p e r week i n s e c o n d a r y s c h o o l s .
P e r f o r m a n c e i n M a t h e m a t i c s e n t i r e l y d e p e n d s o n t a l e n t 02 g i f t ,
Only v e r y s e r i o u s o r c c g e n t r e a s o n s s h o u l d make s t u d e n t s m l s s a n y M a t h e m a t i c s class- L 7
M a t h e m a t i c s t e s t s / e x a m i n a t i u n s / a s s i g n m e n t s make e v e r y s t u d e n t n e r v o u s . U
APPENDIX 111
SOME PROJECTS BASED ON THE J U N I O R --.-- SECONUAKY SCHOOL MATHEMATICS ---- CURRICULUM ----,.--.----
1, C o n s t r u c t i o n o f Abacus by s t u d e n t s u s i n g local materials.
2. C o n s t r u c t r o n s of modely of t h e number l i n e w i t h d e v i c e s for c a r r y i n g o u t basic a r i t h m e t i c o p e r a t i o n s ( a d d i t i o n , s u b t r a c t i o n and mu1 t i p l i c a t i o n ) ,
3 . Des ign ing some Mathema t i ca l Games and c o n s t r u c t i n g t h e f r a m e and materials.
4. Developing and d r a w i n g f l o w char t s o f some a l g o r i t h m s for: s o l v i n g s i m p l e l i n e a r e q u a t i o n s , c a l c u l a t i n g p e r i m e t e r , area and volume of r e g u l a r objects.
5, Cc .ns t ruc t i o n of r e c k o n e r s (money, l i n e a r measu res , etc.)
60 C o n s t r u c t i o n o f c o n v e r s i o n g r a p h s , and c o n v e r s i o n t a b l e s (va?-ues or money o f d i f f e r e n t c o u n t r i e s , l i n r q r measu re o f d i f f e r e n t u n i t s , r e a d i n g s from the rmomete r s o f d i f f e r e n t u q i t s ) ,
7, Drawing t o scale t h e p l a n of: t h e s c h o o l b u i l d i n g , t h e s c h o o l farm or o t h e r objects of i n t e r e s t ,
8, C o r t s t r u c t i o n of t a b l e s of s i m p l e t r i g n o m e t r i c rat ios of a c u t e a n g l e s ( b y t h e u s e o f and c o n s t r u c t i o n o f r i g h t a n g l e d t r i a n g l e s i n a q u a d r a n t o f r a d i u s 10cm).
9. De te rmin ing t h e a p p r o p r i a t e v a l u e of ( b y many t r i a l s i n v o l v i n g d rawing c i r c u l e s o f known r a d i i , u s i n g s t r i n g s t o measu re c i r c u m f e r e n c e s of t h e circles and comput ing t h e r a t i o , d i a m e t e r : c i r c u m f e r e n c e as 1: ? 1,
1 0 , De te rmin ing t h e areas o f circles ( b y c u t t i n g e a c h i n t o many small sectors, a r r a n g i n g t h e sectors to f o r m a n object v e r y much l i k e a r e c t a n g l e ) .
I n v e s t i g a t i o n i n t o I g b o numerology or common p l a c c f i g u r e s / s h a p e s ,
F i n d i . ~ g a p p r o x i m a t e l y ( t o o n e decimal p l a c e ) t h e v a l u e s o f t h e s q u a r e roots o f some whole numbers t h r o u g h t h e c o n s t r u c t i o n o f r i g h t a n g l e d t r i a n g l e s .
Group 11
I n v e s t i g a + i o n s i n t o t h e p r o p e r t i e s o f a n i n d i g e n o u s number sys t em o r number naming sys tem. D e s i g n i n g and a s s i g n i n g n u m e r a l s ( s y m b o l s / f i g u r e s ) t o t h e number names.
C o n s t r u c t i n g / d r a w i n g to scale t h e n e t s of a c u b o i d o f g i v e n d i m e n s i o n s ; f o l d i n g o n e o f t h e n e t s i n t o a model c u b o i d ; c o s t i n g t h e m a t e r i a l s and l a b o u r f o r b u i l d i n g t h e g i v e n c u b o i d , a n d s t a t i n g p o s s i b l e u s e r / u s e s and makers .
C o l l e c t i o n a n d p i c t o r i a l / g r a p h i c a l p r e s e n t a t i o n of scme f a r m s t a t i s t i c s o r ou tcomes of s t a t i s t i c a l e x p e r i m e n t s ( c a s t i n g a d i c e a l a r g e number of times),
Group 111
U s i r q B i n a r y numbers to c o d e i n s t r u c t i o n s , i n f o r n i a t i m and a l g o r i t h m s f o r t h e s o l u t i o t ~ of some problems,
Details :
( a ) c o n s t r u c t r e c t a n g l e s o f w i d t h 7 c m a n d a s l o n g as p o s s i b l e on c a r d b o a r d p a p e r , f u l l s c a p d u p l i c a t i n g p a p e r , or t h e p l a i n s i d e o f any o ld c a l e n d a r pape r ,
( b ) Turn ( t h r o u g h c o n s t r u c t i o n ) t h e r e c t a n g l e s i n t o s q u a r e d p a p e r s 1 c m x lcm.
(c) C u t o u t t h e c o n s t r u c t e d r e c t a n g l e s which h a v e been made s q u a r e d p a p e r s , j o i n them o n t o t h e e n d s o f o n e a n o t h e r t o make a w e r y l o n g s t r i p o f paper .
( d l Ass ign t h e le t ters A, B, C, .,.,. Z t o 1, 2, 3 ,,., 26 r e s p e c t i v e l y and c o n v e r t s u c h numbers t o t h e B ina ry sys t em,
( e ) L e t t i n g ; e a c h b i n a r y number r e p r e s e n t i n g :
a l e t te r t o t a k e a row o f t h e s q u a r e d p a p e r ,
a b l a n k r o w t o i n d i c a t e end o f a word, and
a b l a n k row f o l l o w e d by * t o i n d i c a t e end of a s e n t e r c e n w r i t e down t h e a l g o r i t h m s for :
i) comput ing tl-A<: area of a t r a p e z i u m ;
ii) c o n s t r u c t i n g a t r i a n g l e g i v e n l e n g t h s of t h r e e s i d e s ; and
iii) s o l v i n g two s i m u l t a n e o u s l i n e a r e q u a t i o n s i n t w o v a r i a b l e s .
( f ) Using t h e c o d i n g sys t em stat-e: w h e t h e r you i i k e o r d i s l i k e t h e u s e of c o d e d i n f o r m a t i o n and i n w 5 a t areas of human endeavour c o u l d coded i n f o r m a t i o n be u s e d ,
2. Cons t r u c t l fig, c o s t i n g , i d e n t i f y i n g u s e s and p o s s i b l e Takers o f a c y l i n d e r c f g i v e n d i m e n s i o n s ,
Detai is: --- ( a ) Using a s u i t a b l e scale, c o n s t r u c t t w o n e t s of a
cyl:?der c l o s e d a t o n e end , w i t h b a s e r a d i u s a s 1-75m a n 4 c a p a c l t y o f 38.5m making n e c e s s a r y p r o v i s i o n s f o r f o l d i n g o n e of t h e two n e t s i n t o a m2del o f t h e c y l i n d e r , Fo ld t h e n e t s o d e s i g n e d i n t o t h e model of c h e c y l i n d e r . ( M a t e r i a l s h o u l d be c a r d b o a r d p a p e r o r o l d c a l e n d a r p a p e r ) ,
( b ) If t h e a c t u a l c y l i n d e r b e i n g mode l l ed i s t o be b d i i t :
i) f rom a s h e e t metal c o s t i n g W8.00 p e r m2; ii) by l a b o u r cost o f i4100.00; and
iii) from o t h e r m a t e r i a l s c o s t i n g N150,OO; a l l o w i n g f o r a r e a s o n a b l e g a i n how much .would you se l l t h e a c t u a l c y l i n d e r ?
( c ) Write down t h e p r o c e d u r e s you u n d e r t o o k to a r r i v e a t y o u r r e s p o n s e s ,
( d l L i s t t h e p o s s i b l e u s e s / u s e r s o f s u c h a c y l i n d e r and t h e p o s s i b l e makers .
3 . Simple Data c o l l e c t i o n and p r e s e n t a t i o n :
D e t a i l s : C o l l e c t t h e f o l l o w i n g pieces o f -- -- i n f o r m a t i o n (o r d a t a ) ,
( a ) The number of s t u d e n t s i n t h e J.S,S. section of y o u r s c h o o l a c c o r d i n g t o t h e i r :
i) c l a s s e s
ii) a.ges or y e a r s of b i r t h , and
iii) b e s t s u b j e c t .
(b) P r e s e n t t h e d a t a so c o l l e c t e d i n f r e q u e n c y tables.
(c) P r e s e n t ( a ) ( i ) and ( a ) ( i i ) u s i n g :
j ) p i e chart, and
ii) b a r c h a r t s ,
( d l P r e s e n t ( a ) ( i i i ) u s i n g a n i d e o g r a p h .
(e) hihat do y ~ u c o n s i d e r are t h e a d v a n t a g e s and d i s a d v a n t a g e s o f :
i) f r e q u e n c y tables
ii) t h e p i e char t , and
iii) bar ~ h h r t or t h e ~ d e o g r a p h .
A P P E N D I X IVA
H SCALE FOR MONITORING THE A F F E C T I V E -- - --- LEARIVING OUTCOMES OF' MAT1-IED/iA1rICkL -7 --.
PROJECTS ( sMALOMP )
T i c k 4) t h e staternarlts you v e r y w e l l a g r e e w i t h on ly .
N a m e of s t u d e r t : o o o o o e o o o u a o o o . Y D o O . I ~ O D O m . D O D O O D O .
and
1 0
2 0
3.
4.
5 0
6 .
7 ,
8 ,
9 0
The p r o j e c t i have now c o m p l e t e d / c a r r i e d o u t o t h e r s l i k e it:
a r e c h a l l e n g i n g and e n j o y a b l e
are v e r y tiresome and t i m e w a s t i n g
h e l p t o e x p l a i n some f o r m u l a e and t h e i r u s e s
; ,ave l i t t l e o r no v a l u e t o t h e s t u d y of Ma t h e m a t i c s
c a n be o f much u s e i n i n d u s t r y / w o r k s h o p s / rendering s e r v i c e s / s p e n d i n g o n e ' s l e i s u r e
are some o f t h o s e t h i n g s done i n s c h o o l t o keep s t u d e n t s busy and n o t h i n g more
s h o u l d be made s r e g u l a r p a r t o f t h e s t u d y of Mathema t i c s i n s c h o o l s
s h c u l d n e v e r be used to assess and r e c o r d s t u d e n t s ' pe r fo rmance i n Mathemat ics
How d o you feel a b o u t t h i s p a r t i c u l a r p r o j e c t ?
APPENDIX I V C -1
ASPECTS OF MARKING SCHEMES TOR ---. -...-. RESPON-SES TO PROJECTS ----I---._.-
P r o j e c t One: Use o f B i n a r y Numbers t o Co_d-e_ - ---.--.- - - --,---. - - I n s t r u c t i o n s , InfognajLion a n d A a o r i t h m s -- - .- .- . -a -
f o r t h e So111tj-cn o f Some Problems --- ....- \. - -..
Co n i t i v e Outcomes: - - _ - l(a) C o n v e r s i o n of d e n a r y numbers
(;"i f o r e a c h )
(1-26) t o b i n a r y numbers (corrrect 1 M(16) A(13) ( c o n v e r s i o n )
( b ) Correct (one- to-one mapping, A(6 : ma tch ing o f l e t t e r . ; (A-Z) (% fcr e a c h map to) o n t o b i n a r y numbers, ( a max N 6 marks 1
2, W r i t i n g down t h e a l g o r i t h m s f o r :
a.' f i n d i n g t h e ares of a t r apez iu : a ; O(6) C ( 6 ) A ( 5 )
A c t u a l c o d i n g A( 6)
b ) c o n s t r u c t i n g 4 ABC g i v e n l e n g t h s of t h r e sides, (AB AC and BC), O ( 6 ) ~ ( 6 ) ~ ( 5 )
A c t u a l ~ o d i n g A ( 6 )
C ) s o l v i n g t w o s i m u l t a n e o u s l i n e a r e q u a t i o n s ( i n t w ~ v a r i a b l e s ) O ( 6 ) C ( 6 ) A(5 )
A c t u a l c o d i n g A ( 6 )
3 . S t a C i n g t h e u s e s / a r e a s of u s e o f t h i s c o d i n g s y s t e m ( a minimum of two p o i n t s )
M (marks! - Method marks
A ( m a r k s ) - Accuracy marks
0 ( m a r k s ) - O r g a n i z a t i o n marks
C (marks ) - Comprehens iveness marks
APPENDIX I V P1 - PROJECT ONE: PSYCHOMOTOH OUTCOMES -.--.---..-.---.- .-
1. C o n s t r u c t i o n of s q u a r e d p a p e r :
a) c o n s t r u c t i o n of r i g h t a n g l e s S ( 1 5 ) % 6 , l i n e s a n d c u r v e s
b measurnmen t of rec t : ~ n q 1.e and S ( 1 5 ) AL( 6 ) s q u a r e s l i n e s A1 (8)
2. J o i : l i n g t h e s t r i p s :
a) n e a t n e s s / s t r a i g h t n e s s o f c u t e d g e s
b ) c o r r e s p o n d i n g a n d l a p p i n g of e d g e s , l i n e s a n d s q u a r e s
3. U n i . f o r m i t y of t h e e n t r i e s t h e f i g u r e s 1 a n d 0.
L ( 1 0 ) e d g e s L(15) l i n e s a n d
s q u a r e s
S ( m a r k s ) - m a r k s fo r u n i q u e n e s s / s h a r p n e s s of l i n e s o r c u r v e s
AL/Aa ( m a r k s ) - m a r k s f o r a c c u r a c y o f l e n g t h s o r a n g l e s
i\l ( n a r k s ) - m a r k s for n e a t n e s s of c u t e d g e s a n d c o n f o r m i t y of r e q u i r e d s h a p e ( s t r a i g h t o r c u r v e d )
L ( m a r k s ) - m a r k s f o r l a p p i n g o r c o r r e s p o n d e n c e of l i n e s , e d g e s or areas
U ( m a r k s ) - m a r k s f o r u n i f o r m i t y o f e n t r i e s , d i a g r a m s / p i c t u r e s r e p r e s e n t i n g t h e same o b j e c t .
APPENDIX I V C2
Cons t r u c t i n q , C o s t i n q and I d e n -.- t i f y - i n q ,. -V,sSe~ and P o s s i b l e Makers of--?- .C-yAiin$.err pg--Gi,~,e~", -. -
C o q n i t i v e Outcorne_s_
1. C h o i c e o f s u i t a b l e s c a l e .
A c t u a l Pape r
25cm/SOcm ' I c m
2. De td rmin ing t h e d i m e n s i o n s o f t h e n e t of c y l i n d e r :
(a) Leng th o f r e c t a n g l e t o form c u r v e d s u r f a c e a r e a .
Knowing t h a t i t i s O= of base B( 3
( b ) vJiath of r e c t a n g l e
Knowing i t i s h e i g h t of c y l i n d e r B(3) 2 vol . of c y l i n d e r = fp? h M ( 4 )
R h a n g e o f s u b j e c t -- o f f o r m u l a 9
A ( 2 ) Lcorrect s u b s t i t u t i o d -
~ z o r r e c t h a n d l i n g of expressiop7
A ( 3 )
( c ) Determining w h a t r e p r e s e n t s
4m on paper
25cm/50mm b I c m
4m/400cm 16cm/8crn
3. Knowing the precedure f o r c o n s t r u c t i n g 900 a s evidenced hy l ines and a r c s .
4. Determining t o t a l s u r f a c e a r e a
(a) a r e a of c i r c u l a r base
( b ) curved s u r f a c e area
( c ) T o t a l s u r f a c e area of a c t u a l c y l i n d e r
(9,625 + 44) m 2
53,625 m2
5. Cos t ing t h e a c t u a l c y l i n d e r 2 (a) 1 m of s h e e t metal costs N8.00
53.625 m2 of s h e e t meta l c o s t
id8 x 53,625)
(b) Other m a t e r i a l s c o s t H150.00
(c) Labour c o s t s N100,00
( d ) T o t a l c o s t o f f i n i s h e d p roduc t
H429.00 + N150.00 + H100.00
S679,OO
(el Gain ( r e a s o n a b l e ) . Between 20% and 40% o f cost
Range of g a i n H115.80 - d213-60 ( f ) Hange of selling p r i c e id794 -80-M10.60 A ( 2 )
6. R e p o r t of t h e p r o j e c t .
( a ) O r g a n i z a t i o n
(b) Comprehens ivenes s
APPENDIX 1 V P 2 --. PROJECT T W : PSYCHOMOTOR OUTCOMES --- -
C o n s t r u c t i o n o f Nets:
( a ) c o n s t r u c t i o n o f r i g h t a n g l e s
( b ) c o m p l e t i o n o f r e c t a n g l e s
( c ) c o n s t r u c t i o n o f circles t a n g e n t i a l t o l e n g t h s o f r e c t a n g l e s
1 S ( 8 ) ~ ~ ( 4 )
(D iame te r of c i rc le A a r t o l e n g t h o f r e c t a n g l e ,
measurement of r a d i u s
F e a t u r e of c u t e d g e s ~ ( 8 ) N ( G )
( i) s t r a i g h t e d g e s A,(4) HL(4)
(ii) c u r v e d ( c i r c u l a r base)
(iii) r i g h t a n g l e d ( r e c t a n g l e s )
( i v ! l e n g t h , w i d t h and r a d i u s
3. $ e a t u r e s o f f i n i s h e d c y l i n d e r
( i ) t o p and base ( h o r i z o n t a l or f l a t on a smooth p l a n e )
(ii) u n i f o r m c u r v e s u r f a c e
(iii) un i fo rm o v e r l a p s
( i v ) f i r m n e s s o f p r o d u c t
4 , S i z i n g up t h e a c t u a l c y l i n d e r e v i d e n c e d by s t a t i n g correct u s e s ,
APPENDIX I V C? -- PROJECT THREE : COGNITIVE .- ASPECTS
S imple Da ta C o l l e c t i o n .--. and P r e s e n t a t i o n C o q n i t i v e Outcomes
1. C o l l e c t i o n o f d a t a * B(3) f o r e a c h ( i ) - ( i i i )
( a ) T a l l y i n g M(3)
(i) g r o u p i n g w.r . t , class A ( 2 )
(ii 1 I) a g e A ( 2 ) I f I t i t
(iii) g r o u p i n g w e r o t , s u b j e c t H(2) l i k e b e s t
2, P r e s e n t a t i o n of d a t a
( a ) Frequency tables * B ( 5 ) f o r e a c h table
(i) g r o u p s a c c o r d i n g to f o r m s M(3) A(2)
(ii) g r o u p s a c c o r d i n g t o a g e s M(3) A ( 2 )
( i i i ) g r o u p s a c c o r d i n g t o best s u b j e c t s M(3) A ( 2 )
( b ) P i e char ts
( i ) g r o u p s a c c o r d i n t fo fo rms c a l c u l a t i n g r e p r e s e n t a t i v e a n g l e s M ( 3 )
a p p r o x i m a t i n g t o t h e n e a r e s t d e g r e e ( a l l t h r e e M(3) A(2 )
( i i ) g r o u p s a c c o r d i n g t o a g e s M(3)
c a l c u l a t i n g r e p r e s e n - t a t i v e A ( 2 ) ~ ( 2 ) A ( 2 )
a n g l e s and f o r . a n y t h r e e c o r r e c t 6 S O
a p p r o x i m a t i n g t o t h e n e a r e s t d e g r e e ( a n y t h r e e
(c) Bar c h a r t s
(i) groups a c c o r d i n g t o forms c h o i c e o f scale B(3)
c a l c u l a t i n g r e p r e s e n t a t i v e h e i g h t s M(3) A(2)
(ii) groups a c c o r d i n g to a g e s cl-,oice of s c a l e B ( 3 )
C a l c u l a t i n g r e p r e s e n t a t i v e h e i g h t s M(3)
(any three h e i g h t s ) h ( 2 ) A(2) ~ ( 2 )
( d [email protected] tograms s t u d e n t s i b e s t s u b j e c t .
c h o i c e of s c a l e / k e y o r r e p r e s e l ~ t a t i v e diagram
c a l c u l a t i n g r e p r e s e n t a t i v e number of d iagrams M(3)
( a n y t h r e e c o r r e c t o n e s ) ~ ( 2 ) A ( 2 ) A ( 2 )
3 , S t a t i n g advan tages and d i s a d v a n t a g e s ,
(a ) Frequence t a b l e s B ( 3 )
(b) P i e c h a r t s B( 3 )
(c ) B a r c h a r t s B ( 3 )
( d Ideographs /P ic tograms B ( 3 )
( two a d v a n t a g e s and two d i s a d v a n t a g e s fo r e a c h )
*sugges ted marks by a s i g n i f i c a n t m a j o r i t y of v a l i d a t o r s .
APPENDIX I V P3
PROJECT - THREE : PSYCHOMOTOR OUTCOMES I--------
1, P i e c h a r t s
( a ) g r o u p s w , r , t , forms
U n i q u e n e s s / s h a r p n e s s of l i n e s and c u r v e s S ( 9 1 Accuracy of a n g l e s
( b ) g r o u p s w , r , t , a g e s
Un iqueness s h a r p n e s s of l i n e s and c u r v e s S ( 9 )
Accuracy t f a n g l e s
2 , Bar c h a r t s
( a ) g r o u p s w . r , t . forms
( i i ) P e r p e n d i c u l a r i t y of a x e s and bars Aa(2) A,(2) Aa(2) Aa(2)
( i i i ) Accuracy of h e i g h t s o f b a r s AL(2 ) AL(2) AL(2)
( i v ) Accuracy o f w i d t h of bars and g a p s HL(2) AL(2) AL(2) h i s /
h e r s
( b ) g r o u p s w . r , t . a g e s
(1) Unique a n d s h a r p l i n e s ~ ( 9 )
( i i ) F requency a x i s and b a r s a r t o g r o u p ax i s P.a(2) Aa(2) A2(2) Aa(2)
( i i i ) Accuracy o f h e i g h t s of b a r s AL(2) AL(2) ~ ~ ( 2 )
( i v Unif orrni t y and a c c u r a c y of w i d t h of b a r s and g a p s AL(2) AL(2) AL(2)
Uniformity of diagrams and spacing
General Guide
(1) In the measurement of lengths and angies:
error %; / 1% - max, mark - - 1% L error %; g 5% - % (max. mark)
error %; 1, 5% - no mark
(2) Accuracy of figures from calculations
error % L 1% - max, mark - error % \ 1% - no mark -
APPENDIX V - - - VALIDATION ,IN,STRUMENT FOR THE
PROJECTS BY EXPERTT--
Univ. o f Nig, Sec, School, U n i v e r s i t y o f ~ i g e r i a , Nsukka.
J u l y 18, 1987
I n s t r u m e n t Valida_t_i2nn
I am a p a r t - t i m e s c ~ d e n t of t h e Depar tment o f ducati ion, U n i v e r s i t y of N i g e r i a , Nsukka, I am i n v o l v e d i n d e v e l o p i n g some p r o j e c t s based on t h e J u n i o r Seconda ry School Ma thema t i c s C u r r i c u l u m (JsSMC), T h e s e p r o j e c t s when d e v e l o p e d a r e e x p e c t e d t o be used f o r e v a l u a t i n g J u n i o r Seconda ry S c h o o l s t u d e n t s ' a ch i evemen t i n Ma thema t i c s i n t h e t h r e e domains o f e d u c a t i o n a l o b j e c t i v e s ( A f f e c t i v e , C o g n i t i v e and Psychomotor ) .
May you read t h r o u g h my p r o p o s a l s and make comments, s u g g e s t i o n s a n d c o r r e c t i o n s a s you c o n s i d e r n e c e s s a r y , May you a l so go t h r o u g h t h e t a b l e t h a t follows and v_et w h a t i t c o n t a i n s w i t h r e s p e c t t o the criteria therein&- .
Thanks f o r y o u r h e l p and s e r v i c e s .
Yours t r u l y ,
( s g d , ) Ogomaka, P,M.C.
Proposed P r o j e c t s -
1. The u s e of B i n a r y numbers t o c o d e i n s t r u c t i o n s , i n f o r m a t i o n or a l g o r i t h m s for t h e s o l u t i o n of some p r o b l e m s / e x e r c i s e s i n Mathemat ics ,
D e t a i l s : ( a ) C o n s t r u c t r e c t a n g l e s of w i d t h 7 c m and a s l o n g a s p o s s i b l e on c a r d b o a r d pape r .
F u l l s c a p d u p l i c a t i n q p a j e r - ( p r e f e r a b l y ) o r t h e p l a n s i d e of a u sed c a l e n d a r pape r ,
Through c o n s t r u c t i o n or by t h e u s e o f s e t s q u a r e s ; t u rn /maks t h e r e c t a n g l e s i n t o s q u a r e d p a p e r s l c m x l c m .
C u t o u t t h e c o n s t r u c t e d r e c t a n g l e s which have been made s q u a r e d p a p e r s , j o i n them o n t o t h e e n d s of o n e a n o t h e r ( w i d t h t o w i d t h ) t o make a v e r y l o n g s t r i p of s q u a r e d pape r .
A s s i g n 1, 2, 3, ...26 t o t h e le t ters A, B, C, ,., Z r e s p e c t i v e l y and c o n v e r t t h e numbers t o t h e b i n a r y system.
Write down b r i e f l y p r o c e d u r e s f o r (i)
t h e ( ' a l g o r i t h s ' s t e p s or comput ing t h e area of a t r a p e z i u m ;
c o n s t r u c t i n g a t r i a n g l e i f g i v e n t h e l e n g t h s o f t h e t h r e e s i d e s ; and
~ o l v i n g t w o s i m u l t a n e o u s l i n e a r e q u a t i o n s i n t w o v a r i a b l e s ,
L e t i i n g ; e a c h b i n a r y number t h a t r e p r e s e n t s a l e t te r t o t a k e a row o f t h e s q u a r e paper,; a b l a n k r o w t o i n d i c a t e t h e er,d of a work; a b1an.k row f o l l o w e d by t o i n d i c a t e t h e end of a s e n t e n c e ; c o d e i n t h e a l g o r i t h m s i n ( e l ( i ) , ( i i ) and ( i i i ) i n t o t h e zquared p a p e r ,
C o n s t ~ u c t i n g , c o s t i n g and i d e n t i f y i n g t h e p o s s i b l e make r s a n d u s e r s of a c y l i n d e r of g i v e n d i m e n s i o n s ,
Detai ls: ( a ) Choos ing and u s i n g a s u i t a b l e s c a l z , c o n s t r u c t two n e t s o f a c y l i n d e r c l o s e d a t t h e base (me e n d ) w i t h b a s e r a d i u s as 1.75m and c a p a c i t y of 38,5m3 (make n e c e s s a r y p r o v i s i o n s f o r f o l d i n g o n e o f t h e n e t s i n t o a model o f t h e c y l i n d e r d e s c r i b e d ) ,
( b ) N e a t l y c u t o u t t h e n e t s c o n s t r u c t e d and fo ld t h e o n e w i t h t h e n e c e s s a r y p r o v i s i o n s i n t o a model o f t h e c y l i n d e r s o t h a t i t c a n b e h a n d l e d and s e e n a s a c y l i n d e r ,
B h e material t o b e used i s c a r d b o a r d a p e r or t h e p l a i n s i d e of a used c a l e n d a r paperf -
( c ) I f t h e a c t u a l c y l i n d e r b e i n g m o d e l l e d i s t o be b u i l t :
( i ) f rom a s h e e t metal c o s t i n g id8.00 p e r m2;
( i i ) by l a b o u r cost of H100,00, a n d
( i i i ) from o t h e r ~ n a t o r i a l s c o s t i n g H150.00; a l l o w i n g for a r e a s o n a b l e g a i n , ( e x p r e s s e d 2s p e r c e n t a g e of c o s t 1, hob much would you se l l t h e a c t u a l c y l i n d e r ?
( d ) Write down a l l t h e p r o c e d u r e s you t o o k to f in i s11 t h i s p r o j e c t .
( e l L i s t t h e p o s s i b l e : ( i) maker s (ii u s e s / u s e r s
o f c y l i n d e r s t h a t dre o f t h e s i z e of t h e a c t u a l c y l i n d e r you produced i t s model,
3 , Simple Data C o l l e c t i o n and P r e s e n t a t i o n ,
Details: ( a ) C o l l e q t t h e f o l l o w i n g p i e c e s of i n f o r m a t i o n o r i n d o r m a t i o n or d a t a : t h e number o f t h e s t u d e n t s i n t h e JSS s e c t i o n of your s c h o o l a c c o r d i n g t o t h e i r ;
(i) c l a s s e s , (ii) y e a r s o f b i r t h , and
(iii) best s u b j e c t s ,
( b ) P r e s e n t t h e d a t a you c o l l e c t e d i n f r e q u e n c y tables,
(c) P r e s e n t d a t a as i n (a) (j ii) u s i n g an i d e o g r a p h or p i c tog ram,
( d l What d o you c o n s i d e r are t h e a d v a n t a g e s and d i s a d v a n t a g e s o f :
( i ) f r e q u e n c y tables o v e r o t h e r s
(ii) p i e c h a r t s o v e r o t h e r s
(iii) bar c h a r t s o v e r o t h c r s , and
( i v t h e i d e o g r a p h / p i c togram o v e r o t h e r s
i) Measur ing o u t s m a l l d i s t a n - ces ( l c m ) .-.*..
ii) J o i n i n g two p o i n t s by s t r a i g h t l i n e s or u s i n g l l e l l i n e s -
I --. L I iii) Drawing
s h a r p l i n e s I -- -- --- i) Use o f c u t t - I i n g i n s t r u -
men t s t o c u t
1 p a p e r f o l l o w - i n g a l i n e o r
1 i a curve -..-.. - ---- lii) J o i n i n g
s q u a r e d p a p e r / Of so t h a t l i n e s 1 and co lumns of
I s m a l l s q u a r e s I c o i n c i d e and
j i ) Drawing ~ i p a '
1 f r e q u e n c y t a b l e .
ii) A b i l i k y t o o r g a n i z e d a t a i n o r d e r .
i ) A b i l i t y t o draw circle. - --
A b i l i t y t o draw s h a r p
f l i n e s . -- I i i i ) A b i l i t y t o
a n q l e s .
I i v ) ~ b i l i t y t o c a l c u l a t e r e p r e s e n t a t i v e anq1es.- -- -.."
V ) A b i l i t y t o draw pe rpend i - f u l a r l i n e s ,
v i ) A b i l i t y t o I c h o o s e scale
and r e p r e s e n t a - t i v e l e n q t h s . -- --
v i i ) A b i l i t y t o measu re or mark o u t ( e q u a l ) l e n q t h s , ... - -- ..
, as num-
i) Choice of t h e no , of real objects a
] u n i t diagram/ , ~ i c t u r e will 1
APPENDIX V I
TABLE 1 -- RESULTS OF SOMd ASPECTS O F VRLI,DA,T&OJm
OF PROJECTS BY EXPERTS .- A N D TEACHERS -. - - Number o f e x p e r t s and t e a c h e r s who a g r e e . / d i s a g r e e
t h a t :
1, t h e t a s k s s p e c i f i e d t o kt accompl i shed by t h e s t u d e n t s u n d e r each of t h e t h r e e p r o j e c t s a r e :
( i) w i t h i n t h e r e a c h o f the s t u d e n t s and
( i i ) e x p r e s s e d i n a l a n g u a g e -t41ich t h e s t u d e n t s c o u l d f o l l o w ; and
2. t h e s k i l l s a n d knowledqe e x p e c t e d t o be u s e d by khe s t u d e n t s i n e x e c u t i n g t h e p r o j e c t s :
-
( i ) a r e n e c e s s a r y , ( i i ) c o u l d be e x h i b i t e d by t h e s t u d e n t s , and
' i i i ) when e x h i b i t e d c o u l d be s c o r e d o k j e c t i v e l y .
I a, Knowledge and $ $a s k i l l s !: e x p e c t e d t o 0 be used by u h24h t h e s t u d e n t s o ca
Z Q U 1
Y e s No -
Yes-)%-Yes No
28 2 l ( a ) ( i ) 28 2 30 0
( i i ) 30 0 30 0 I 1 ( i i i 1 20 0 30 0
( i v ) 30 L&3A ----. 0
25 5 l ( b ) (i) 29 1 29 1
(ii 29 1 30 0
( i i i ) 25 - --5~-L5 5
2 6 4 1 ( c ) ( i ) 2 6 4 2 6 4
( i i ) 27 3 28 2
Yes No
27 2
30 0
30 0
26 4
29 1
29 1
29 7
25 5
25 5 --
UO I Y e s
PHOPGKTIONS OF VkLIdkiTOi3s AGREEING OR DISAGHLLING THAT EACH P K O J i X T Is S U I T A B L E
FOR hilo HLLLVANT TO JSS MrlTHLMATICS LOUCATION AND COli i lLSPONDING Z-TCST ( O F DIYE'EKENCES
BLTdEEN PHOPOi2TIONS S T A T l S T l C S .
Z - t e s t P r o p o r t i o n i n d i c d t i n c j t h a t p r o j e c t , S t a t i s t i c
P r o j e c t
1
2
3.
i s n o t r e l e v a n t
---
0.178
0.12
0.09
s u i t ab i - l i t y
12.497
7.404
16.959
is i s n o t ' C-- i s su.i t a u l e s u i t a u l e r e l e v a n t
0.d76 1 0.124 0.822 - -----
I 0.78 1 0.22 0.88
r e l e v a n c e - -
9,221
1 2 - 8 1
15.694
--
0.92
-
0 . 0 1 0.91
A P P E N D I X V I I
DETERMINATION O F WEIGHTS FOP THE ITEMS OF SAPIM A N D S M A L ~ ~ P - - - - -- .- .- -=
Table 1: F r e q u e n c i e s of d e i ~ q h t s 1 1 s s i ~ n e $ ~ - -- By E x p e r t s t o the Various Items
------.-----.-..,L--
of SAPIM.
T h e u n s t r u c t u r e d / ~ n r e s t r i c t e d r e s p o n s e i t e m i s
scored o u t 02 a maximum of 28 p o i n t s .
APPENDIX V I I I
Proforma t i c k ( v ) where you a g r e e w i t h p r o j e c t NO, (1, 2 or 3 ) ( C o g n i t i v e or Psychomotor )
r _I
I t e m on t h e A l l o t e d mark i s S u g g e s t e d
A p p r o p r i a t e I n a p p r o p r i a g e i n a p p r o p r i a t e -..--.--
I
A LETTER TO EXPERT MATHEMATICS EXAMINERS (WASC/G.C .E0 ) TO CRITIQUE THE . MARKING _-
SCHEMES, AND CONSEQUENT KES --.-a JLTS
Univ, o f N i g e r l a Sec. Sch., University of N i g e r i a , :;A J.!.~L e
J u l y 1987
k R e q u e s t To C r i t i q u e Some Markinq S c h e n . 3 ~ -.
I a m a p a r t - t i m e s t u d e n t o f t h e Depar tment o f E d u c a t i o n , Univers i . t y o f N i g e r i a , Nsukka, d e v e l o p i n g some ' p r o j e c t s ' t h a t c o u l d be used f o r a s s e s s i n g JSS y e a r 111 s t u d e n t s i n Mathemat ics ,
By v i r t u e o f your l o n g e x p e r i e n c e and s k i l l as a WAEC examiner , you have t h e e x p e r t i s e t o c r i t i q u e "Marking Schemesu o f Seconda ry S c h o o l Mathemat ics tests and o t h e r s im i l a r i n s t r u m e n t s f o r e v a l u a t i o n a t t h a t l e v e l , A t t a c h e d to t h i s letter are c o p i e s o f t h e d e s c r i p t i o n s o f t h r e e m a t h e m a t i c s p r o j e c t s and t h e i r c o r r e s p o n d i n g mark ing schemes.
May YOU, t h e r e f o r e , c r i t i q u e t h e s e mark ing schemes by f i l l i n g o u t t h e proforrna h e r e u n d e r , Thanks.
Yours t r u l y ,
(Sgd.) Ogomaka, P.M.C.
T a b l e 1
Number of Examiners Who Ticked ( d ) i n t h e Var ious Columns A g a i n s t o r Suggested I n d i c a t e d Mark,s f o r t h e Var ious Items o f
t h e Marking Schemes.
P r o j e c t 1 C o q n i t i v e
T o t a l 84 -
--. -, - .- p L 0.3%-
A l l o t e d Fuggested Chi- S i g n i f 1- mark i s mark to s q u a r e inappro- be a l l o t e d
p r i a t e I A4~e-s.d-
1 ~ ( 3 ) - ' 1 11-26 1 Y e s - - 0 t Y e s
3 1 C(4)-3 5.40 1 y e s 2 1 C(0)-2 8.06 Yes
1 - -1 I 15,OO I ye, II.-- -.._..a ,A,,
P r o j e c t 1 Psychomotor -
y7fi-; T o t a l
I t e m of
s k i l l
Sugges ted C h i - mark to be s q u a r e
a l l o t e d c a l c u l a t e d c a n t 1
A l l o t e d mark i s appro- p r i a t e
M a j o r i t y A reed LL.&-- --
I S
A l l o t e d mark i s inappro- p r i a t e
Y e s I 2
1
-
l ( a >
(b)
2 ( a Y e s
Y e s
1 3
14
15
-.--A
Y e s --a ---
224
Pro j'ect 2 Coqni t ive
A l lo t ed Al fo ted - Suggested I t e m mark i s m&kk'.is mark t o of 1 appro- i n a p p r o 1 be skill p r i a t e p r i a t e --A- -. - ...
1
1. 1 1 5 2(a) 1 15 15,OO
( b ) 15 15.GO (c > 13
3 12 5.04 ---. Yes -. ...-A 4(a) 12 5.04 Yes
L P r o j e c t 3 , Coqni t i v e
p L V O V J
",loted / A l l o t e d Suggested Chi- I S i g n i f i - ) Item / mark is ! mark i s mark t o of squa re I- / c a n t I
s k i l l 'ppro- ' inappro- be a l l o t e d c a l c u l a t e d m a j o r i t y j p r i a t e ! priate agreed I :: B ( 3 ) 15.00 1-7 (iii)
2(a> ---- ..- -----
(i>-(iii) 1 B ( 5 ) 15.00 Yes
P r o j e c t 3, Psy_chornotor .--
I tern k l l o t e d A l l o t e d mark i s mark i s of
s k i l l appro- inappro- p r i a t e p r i a t e
l ( a > 1 2 3 ( b ) 1 2
2 ( a ) ( i l - ( i v J ) (b)
( i l - ( i l l , (c)
(i)-(ii>; 1 5
(d ) 1 5
,Suggested mark t o be a l l o t e d
-- --.. -- :hi- p L 0005
square S i g n i f i-
5-04 5-04
15,OO
15.00
15,OO Yes
15,OO -.A. LA.-.-
APPENDIX IX Table 1
Randomised Groupings .L
I Schoo~ i w o - School One - P r e - test -
68 70 78 60 62 63 58 80 62 72 60 54 70 68 82 52 50 56 62 44 70 43 6 5 70 62 75 82 68 52 62 -
-- Post- test
78 70 78 60 50 6 3 58 68 62 72 82 54 70 68 68 52 50 56 62 44 68 43 65 70 62 75 8 2 82 52 62
P r e - test
65 78 64 83 68 48 45 56 7 0 PO 52 43 50 4 6 78 62 52 80 69 53 64 45 48 70 56 48 64 4 9 62 45
?os t - test
6 5 78 64 68 68 58 45 56 58 80 52 43 50 4 6 78 62 52 80 82 53 64 45 48 52 56 48 56 49 62 45
- ?re- t e s t 78 78 73 70 70 67 66 65 62 62 62 58 57 57 57 56 54 53 53 52 52 52 48 48 48 47 4 6 46 4 6 42 -
Post- test
78 78 73 70 7G 67 66 78 62 62 62 80 68 57 57 56 54 53 42 52 42 52 48 62 48 47 46 4 6 4 6 42 - --
3chool Four -- P r e- t e s t 83 83 76 70 68 68 67 66 65 62 60 60 58 58 57 56 54 54 53 53 52 52 48 48 48 48 47 4 6 45 43
APPENDIX I X T a b l e 2
andmi mi s e d G r o u p i n q s
S c h o o l One
test I test
- S c h o o l Two
P r e - test
70 43 58 64 48 40 60 36 40 50 42 4 6 44 60 40 52 44 72 60 66 44 50 56 42 49 68 58 4 6 64 48 -
P o s t - t e s t
80 52 68 64 57 56 70 52 48 62 57 4 6 58 83 40 67 58 72 80 80 67 68 65 58 66 40 58 66 80 72
-- A1 t e r n a t e Ranks Groupins;
T ---,---.-.-.
S c h o o l T h r e e S c h o o l F w r -* -- ? r e - test -. 83 83 75 72 68 68 67 67 65 62 62 60 60 58 57 56 54 54 53 53 52 52 48 48 48 48 47 4 6 43 43
.-
-.....--
P o s t - test -- .-
83 83 80 78 68 80 75 78 78 7 ii 72 60 75 7 0 65 60 6% 60 68 64 G 2 68 58 62 60 64 64 68 48 50
APPENDIX X Table 1
1 1 Coc;ni t i v e =e Rver- aqe
69 6 5 6 5 70 5 1 62 6 7 78 56 57 78 47 74 65 85 60 85 67 83 66 6 1 50 65 80 57 69 78 68 73 50 -
P e r f 7-- ce Aver-
-29% 52 73 58 60 64 6 7 43 87 65 75 69 56 63 64 8 2 69 78 6 1 72 62 59 57 72 86 70 74 7 0 57 76 40
erf - 2
-.-
7 1 75 67 70 65 70 62 90 60 72 8 2 65 8 6 78 88 70 89 82 8 6 75 78 56 76 88 74 8 6 83 70 76 50
APPENDIX X T a b l e 2
PERCCi\ITHGE SCORES ANAKDED TO STUDENTS' P
COGNITIVE , PSYCHOMOTOR AND AFFECTIVE RESPONSES TO THE THREE PROJLCTS AND THE
AVERAGES OF THL SCORES (SCHOOL TWO)
r ------- -1--. .-a
C o q n i tive I P s y c h o m o t o r A f i'ec t i v e
t P :,e- -- k v e r - 1 a g e
77 75 38 36 67 48
48
83 76 42 i 52 5 1 1 42 64 54 54 42
40 4 6 51
44 8 1 74 62 5 5 49 40 6 3 52 67 60 40 42 79 70 73 65 63 49 30 58 56 44 52 48 78 74 7 0 62 52 50 74 62
660 46 ----.--- - - -*..
nance -. A v e r -
75 40 5 2 1 60 48 , 56
46 ( 50 40 4 6 42 50 76 78 60 67 40 45 57 60 66 65 40 50 74 80 65 69 60 60
74 60 50 50 65 52 56
APPENDIX X Table 3
1 I Cogni Live 1 A f f e c t i v e I -:
I n q e j [eEf Aver-
a e
P e r f o r m a n -- 1 rJ-Y
r m a :
3 - 85 70 84 87 65 62 8 1 79 8 1 68 90 87 62 83 87 74 67 50 58 65 58 4 1 55 35 44 44
6 1 44 63 --
APPENDIX X -1-
Table 4
- - - ~ S V C t ~VIIIO tor ) Cogn i t i ve
ze -- -- Rver- .2!=&.-
77 82 67 66 63 87 80 72 68 6 1 7 1 57 74 53 66 62 56 6 1 62 58 64 63 6 1 48 45 6 5 58 6 1 42 99
-I- U
r f ormance IAver- 1 aqe
APPENDIX X I
SCCIKES AWARDED i3Y TEN .RANDOMLY Sr:T,ECTED - - - . . - - . . . . . . . . - - - - - - - - - - MATHEMATICS TEnCHERS TO THE COGNITIVE
ASPYCTS OF RESPONSES TO PEOJECT I BY TEN RANEOMLY SELECTED STUDENTS - I - 4 - . - - -
S t u d e n t s 1 R e s p ~ s e s ( r a t e e s )
4
M a t h e m a t i c s ' i e a c h e r as M a r k e r s ( r a t e r s ) 1
T a b l e 2
I tudentS M a t h e m a t i c s T e a c h e r s R e s p o n s e s I--. . -- M a r k e r s ( r a t e r s )
9 - -- 8 1 53 60 46 68 57 85 72 63 65 ----
10
8 2 54 64 48 69 58 88 73 64 68
T a b l e 3
R e s p o n s e s ( r a t e e s 1
4
T a b l e 4
SCORES AWARDED RY THa TEN RANDOMLY SELECTED - - WlTHLMnTICb TEtiCHERS TO THE PSYCHOMOTOR -
ASPECTS - OF TEN RANDOMLY SELECTED STUDENTS1 RESPONSES TO PROJECT I
R e s p o n s e s -1 T e a c h e r s a s - M a r k e r s ( r a t e r s )
6 .
Table 5
SCORES ANAHDED BY THE TEN RANDOMLY -- SELECTED MATHEMATICS. TEACHERS TO THE PSY1HCMOTOR
ASPECTS OF T E I ~ RANDOMLY SELECTED STUDENTS' -- RESPONSES TO PROJECT 11
R e s p o n s e s ( r a t e e s )
I r I 1
Table 6
SCORES AWARDED BY THE TEN RANDOMLY SELECTED - MATHEMATICS TEACHERS TO THE FSYCHOMOTE
ASPECTS OF TEN RANDOMLY SELECTED STUDENTS' - RESPONSES TO PROJECT 111,
I -.
Students' I M a t h e m a t i c s T e a c h e r s a s M a r k e r s ( r a t e r s ) R e s ~ o n s e s .
APPENDIX X I 1
A DRAFT OF SAPIM CRITIQUED BY EXPERT EDUCATORS
I am a s t u d e n t , i n t h e Depar tment o f E d u c a t i o n , U n i v e r s i t y of N i g e r i a , Nsukka, d e v e l o p i n g a S c a l e f o r A f f e c t i v e Per formance i n Mathemat ics ( S A P I M ) . B e l o w are s t a t e m e n t s e x p r e s s i n g some f e e l i n g s toward Mathema t i c s , K ind ly i n d i c a t e by:
a t i c k (#I b e s i d e s t a t e m e n t s c o u l d ; b e u n d e r s t o o d by JS3 y e a r I11 s t u d e n t s , and a p p r a i s e s t u d e n t s J a f f e c t i v e b e h a v i o u r toward Mathemat ics ,
w r i t i n g ( s m ) or(om) b e s i d e s t a t e m e n t s t h a t are same o r o p p o s i t e i n meaning w i t h o t h e r s t a t e m e n t ( s 1 i n t h e d r a f t ,
Thanks f o r your a s s i s t a n c e .
Ma thema t i c s i s an i n t e r e s t i n g s u b j e c t ,
The s t u d y o f Ma thema t i c s i s en joyab le .
I s t u d y Mathema t i c s on my own e v e r y day .
M a t h e n a t i c s i s a v e r y d i f f i c u l t s u b j e c t .
Knowledge of Ma thema t i c s h e l p s s t u d e n t s t o l e a r n many o t h e r s u b j e c t s ,
Ma thema t i c s i s n o t p r a c t i c a l ,
Ma thema t i c s makes many p e o p l e made.
Many s u b j e c t s c a n be s t u d i e d w i t h o u t t h e knowled2e of Mathemat ics .
The t h i n g s l e a r n t i n Ma thema t i c s or s k i l l a c q u i r e d f rom i t a r e n o t used o u t s i d e s c h o o l ,
It i s j u s t i f i e d t o make Mathemat ics compulsory i n s e c o n d a r y s c h o o l s ,
I a m a f r a i d o f Mathemat ics ,
1 2 , Ma thema t i c s p r e s e n t s c h a l l e n g e s ,
1 3 , Ma thema t i c s i s t h e most i m p o r t a n t s u b j e c t ,
1 4 , Ma thema t i c s s h a r p e n s t h o u g h t s ana r e a s o n s of p e o p l e ,
1 5 , With l i t t l e or no ef for ts I p a s s Ma thema t i c s v e r y w e l l ,
16, N o matter t h e efforts I make I f a i l Mathemat ics .
17, P e o p l e c a n become v e r y r i c h w i t h o u t t h e knowledge of any b i t o f Mathematics .
18, Mathemat ics s h o u l d be s t u d i e d r e g u l a r l y by J u n i o r Secondary S c h o o l s t u d e n t s ,
19, Seconda ry s c h o o l s t u d e n t s s h o u l d n o t w o r r y i f the:! have no Mathema t i c s t e x t books.
20, Nobody c a n ( s t a y ) b e i n b u s h i n e s s w i t h o u t t h e knowledge of Mathema t i c s ,
,lo Withou t Ma thema t i c s t h e r e c a n be no s c i e n c e and t echno logy .
22, Ma thema t i c s s h o u l d be made compul so ry f o r s t u d e n t s who i n t e n d to s t u d y s c i e n c e / t e c h n o l o g y f u r t h e r ,
23. Seconda ry s c h o o l s t u d e n t s s h o u l d b e l o n g to M a t h e m a t i c s c l u b s i n t h e i r s c h o o l s .
: 4, Mathematics s h o u l d b e t a u g h t t o and l e a r n e d by al .1 s t u d e n t s b u t i t s e x a m i n a t i o n s s h o u l d be o p t i o n a l ,
25, Mathemat ics i n v o l v e f o r m u l a e which h a v e l i t t l e or no r e a l / m e a n i n g f u l e x p l a n a t i o n s .
26, Idea and s k i l l s l e a r n t f rom Mathema t i c s are n o t e a s i l y f o r g o t t e n ,
27, Mathemat i c s s h o u l d be d o n e f i v e or f o u r p e r i o d s p e r ciass p e r week i n s e c o n d a r y s c h o o l s ,
28, Pe r fo rmance i n Ma thema t i c s e n t i r e l y d e p e n d s on t a l e n t / g i f t e d n e s s ,
29, Only v e r y s e r i o u s or c o g e n t r e a s o n s s h o u l d make s t u d e n t s m i s s Ma thema t i c s l e s s o n s / c l a s s e s ,
30, Ma thema t i c s tests/examinations/assignments make e v e r y s t u d e n t n e r v o u s ,
APPENDIX X I 1 1
- -- - --a
A Units o f Work
v and VI - Revision of Idumber sys- tems. The Binary courk t i n g sys tem r e l a t e d im: punch c a r d s , c o d i n g and d e c o d i n g i n f o r m a t i o n ,
UNITS OF WORK AND LESSONS THWHT AN-D-THC APPROACHES/STRATEGIES ADOPTED I N TEACHING THE STUDENTS INVOLVED I N THE STUDY-
Lesson T o p i c s
1. Revis ion o f t h e proper- t ies and characteristics of Hindu-Arabic and Roman Number sys temso
2, R e v i s i o n of t h e I g b o number naming system,
3, Comparisons and c o n t r a s t s of t h e s y s t e m s
4, Bases of some known number sys tems,
5. Binary system of c o u n t i n g and i t s c h a r a c t e r i s t i c s ,
6. Convers ion of Denary numbers t o Binary numbers.
7. Convers ion of Binary numbers to Dinary numbers,
8, Coding of i n f o r m a t i o n u s i n g b i n a r y number as i n punch cards and t a p e s ,
1 i
I 1 I
9, Decoding i n f o r m a t i o n f p r e s e n t e d as b i n a r y a
umbers.
Approaches/S trategies and and/or t h e Control Groups
G l a s e r * s basic t e a c h i n g method/approach i n v o l v i n g c lass room a c t i v i t i e s as: q u e s t i o n i n g , exposition, d i s c u s s i o n , and perf orrnance a s s e s s m e n t ( i n t h e form of paper and pen m u l t i p l e choice and essay t y p e test items),
Qlaser's basic t e a c h i n g approach i n v o l v i n g class room a c t i v i t i e s : e x p o s i t i o n , d i s c u s s i o n , i l l u s t r a t i o n s and ~ e r f o r m a n c e assessments , An eben ing class of one to one and a h a l f hours u s i n g G l a s e r ' s basic t e a c h i n g approach f o r more- l e s s o n s on Binary numbers and cod ing and decoding,
A c t i v i t i e s Undertaken by t h e Teachers , a n d l o r - - t h e Exper imenta l Groups
Lesson 1: Glaser's basic t e a c h i n g approach i n v o l v i n g c lass room a c t i v i t i e s as; q u e s t i o n - i n g , e x p s i t i o n , d i s c u s s i o n and performance a s s e s s m e n t (pen and paper free r e s p o n s e test i t e m s ) , Then sequenced assf --mments (done o u t of class or home) i n v o l v i n g : i n v i e w i n g p a r e n t : g u a r d i a n s or rnetrons, and w r i t t e n report on t h e r e s p o n d e n t s c o n c e p t i o n and knowledge of t h e lgbo number naming system, Followed by d i s c u s s i o n s and i l l u s t r a t i o n s of t h e charac- teristics of t h e t h r e sys tems, C o n t i n u a t i o n of t h e sequenced ass ignments (done o u t of c l a s s ) i n v o l v i n g d e s i g n i n g numeroals (symbols, f i g u r e s / c h a r a c t e r s ) having s p e c i f i e d charac - t e r i s t i c s / p r o p e r t i e s f o r r e p r e s e n t i n g Igbo number names ( p r e v i o u s and p r e s e n t ) up to hundred. 9
G l a s e r ' s b a s i c t e a c h i n g approach i n v o l v i n g c l a s s room: e x p o s i t i o n , d i s c u s s i o n , i l l u s - t r a t i o n s and performance assessment (based on t h e i n d i v i d u a l s tudent! s r e p o r t s , d e s i g n e d numerals and some free ' r e sponse test items, An hour to one hour and h a l f evening l e s s o n s on Binary numbers and cod ing and decoding,
- --
A U n i t s of dork
V I I - - -. I n t e r p r e t i n g word problems i n t o a l g e b r a i c e x p r e s s i o n s or e q u a t i o n s and v i c e . versa; and s o l v i n g t h e s i m p l e o r s i m u l t a n e o u s l i n e a r e q u a t i o n s r e s u l t i n g from t h e former ,
B Lesson Topice
-
1, T r a n s l a t i n g word pro- ~lems ( s u i t a b l e o n e s ) to s imple a l g e b r a i c e x p r e s s ions and e q u a t i o n s and s o l v i n g t h e e q u a t i o n s .
3. T r a n s l a t i n g t h e 3 l g e b r a i c e x p r e s s i o n s or simple equation to word ?roblems,
3 , T r a n s l a t i n g a p p r o p r i a t e f o r d problems to l i n e a r l x p r e s s i o n s or two s imul tb- ~ e o u s l i n e a r e q u a t i o n s zach i n two v a r i a b l e s .
3, S o l u t i o n s of two l i n e a r e q u a t i o n s each i n t w o v a r i a b l e s . OR s o l u t i o n s of t w o s imul taneous l i n e a r 2quat ions.
- .. C
A c t i v i t i e s of t h e C o n t r o l Grocps
Glaser's basic t e a c h i n g model i n v o l v i n g ; q u e s t i o n i n g , expos i - tion, d i s c u s s i o n , i l l u s t r a t i o n and performance assessment (free r e s p o n s e test items) for b o t h l e s s o n s , Then one even ing class for f u r t h e r d r i l l i n g on s o l v i n g s imple e q u a t i o n s ,
G l a s e r ' s b a s i c t e a c h i n g model i n v o l v i n g ; q u e s t i o n i n g , e x p o s i t i o n , i l l u s t r a t i o n and performance assessment (free r e s p o n s e test items) and one even ing l e s s o n s on backward t r a n s l a t i o n ,
Glaser s basic t e a c h i n g model i n v o l v i n g classroom a c t i v i t i e s : q u e s t i o n i n g , e x p o s i t i o n , i l l u s t r a t i o n s (of e l i m i n a t i o n and s u b s t i t u t i o n methods) , and performance assessment (free r e s p o n s e test items). Two evening l e s s o n s f o r f u r t h e r d r i l l on s o l v i n g s imul taneous l i n e a r e q u a t i o n s i n two v a r i a b l e s ,
D A c t i v i t i e s of t h e Exper imenta l Groups
Glaserts basic t e a c h i n g m o d e l i n v o l v i n g c l a s s r o o m a c t i v i t i e s ; q u e s t i o n i n g ? e x p o s i t i o n , d i s c u s s i o n , illustrations and performance assessment (free r e s p o n s e test i t e m s ) for b o t h l e s s o n s , Followed by two even ing l e s s o n s , one on s o l v i n g s i m p l e e q u a t i o n and the other on A l g o r i t h m i s a t i o n OR w r i t i n g f low-char t s of how to s o l v e s i m p l e equa t ions .
G l a s e r l s b a s i c t e a c h i n g model i n v o l v i n g a c t i v i t i e s a s ; ques t ion ing , exposition, i l l u s t r a t i o n and performance assess- ment ( i n v o l v i n g free r e s p o n s e test items). One even ing c l a s s on back- ward t r a n s l a t i o n .
Glaserl s basic t e a c h i n g model i n v o l v i n g c lass room a c t i v i t i e s as : q u e s t i o n i n g , e x p o s i t i o n , i l l u s t r a t i o n s (of e l imina- t i o n and s u b s t i t u t i o n methods) and performance assessment (free r e s p o n s e test items), A d m i n i s t r a t i o n of P r o j e c t 1 on t h e g roup t h r e e l i s t of p r o j e c t s .
Osqau q3yq o; saDuaJaJa3 6uryeu sauo~ pue 's~apuT1A~ so Sean a~e~~ns osp pue spyue~Ad myqo pue sauo~ 30 sqau
30 6ur~e~p oyq sqmpnqs 6ury~eaq uy pa -Ao~dma sr Iapou 6u~y~eaq qseq s
(suraqy qsaq asuodsa~ paqm~qs/paqzq~qsa~ 6ursn)
q uaznssasse a~uern~og~ad *suorss -n~s~p pue uorq~sodxa '6uyuorqsat-?
Volumes of c u b i d s
Uni t s of Work 1x1
c y l i n d e r s and cones.
D
Lesson Topics
IV Rough estimates of magnitudes, s i z e s , q u a n t i t i e s c a p a c i t i e s , etc.
I 1, Volumes of cuboids and cubes,
2. Vo)umes of cy l inde r s .
3 , Volumes of cones.
1. Revis ion on estimates of va lues r e s u l t i n g from mu1 ti- p l i c a t i o n and d i v i s i o l of mixed numbers, 2. Rough estimates of linear measures of objects,
3, Rough e-sthates of Linear measures of / ob)Rts.
Cont ro l Groups
Glaser's basic m o d e l invo lv ing : d i s c u s s i o n and d e f i n i t i o n of t h e u n i t s of volume; i n d u c t i v e or deduc t ive e s t ab l i shmen t of the formula for f i n d i n g / c a l c u l a t i n g t h e volume of a cube/cuboid (us ing packets of cubes of sugar or b l o c k s of wood); i l l u s t r a t i o n s of t h e u s e of formula and Performance assessment,
Glaser's basic teaching model is employed for l e s s o n t o p i c s (2) and ( 3 ) , Each l e s son involves: d i scu- ssion; es t ab l i shmen t of formula for finding volume as, Base a r e a x height i n (2) and base a rea x h e i g h t i n (3) j ill a s t r a t i o n s of use of formulae and Performance assessment.
Glasert s basic teaching model involv ing ; ques t ion ing and i l l u s - t ra t i o n s . Performance assessment i s by speed test, ,
Glaser's basic teaching model i nvo lv ing classroom d i scuss ions , i l l u s t r a t i o n s and practical works (u s ing c a l l i p e r s , screw callipesq srrew guages and squared papers. Also Per f ormmce assessment involv ing some p r a c t i c a l work,
D Experimental Groups
Glaserts basic' t e ach ing model i nvo lv inp : d i s c u s s i o n and C e f i n i t z o n of t h e u n i t s o f volume; . i nduc t ive or deduc t ive establish- ment of the formula for f i n d i n g t h e volume of a cub/cu&id as LxWxH or base area m u l t i p l i e d by h e i g h t (using packe t s of cubes of sugar or blocks of wood) ; i l l u s t r a t i o n s of t h e u s e of b o t h approaches and Performance assessment,
Glaser s basic ' teaching &el is employed for lesson topics ( 2 ) and (31. Lesson t o p i c (2) involved d i s c u s s i o n and esta- bl ishment t ha t , vo lume of cylinder is base a r e a m u l t i p l i e d by height, Lesson t o p i c ( 3 ) involvcd us ing a c y l i n d e r and a cone { teache r made) .of equal base area and h e i q t to establish t h a t volume of cone i s - base area, x height, I l l u s t m t i o n s of the use of formulae, and Performance assessment.
Glaserc s basic teaching d e l i nvo lv ing : ques t ion ing and i l l u s t r a t i o n s , Perform- a n c e assessment is by speeded m u l t i p l e cho ice test i t e m s .
PrmzUcal works: i nvo lv ing use of t a p e s , handspans, callipers, screw guages and squared papers/graph papers. Performance assessment i s i n b u i l t t o t h e p r a c t i c a l s of measuring and e s t i m a t i n g sizes, a r e a s and capacities of some common/school objects, b
Adrn&niskpflbn of Pro jec t 2 used as eva2uatian Pnstrumen t .
Urxtsof hfork 1 D
VIII - Lesson T o p i c s ---..--n-
R e v i s i o n of s t a t i s t i c s ,
3 e v i s i o n l e s s o n s on: 1, What i s statistics and why s t u d y - s t a t i s t i c s ?
2. C o l l e c t i o n and o r g a n i z a t i o n of data ( T a l l y i n g and grouping zccord ing to type, s i z e . class, magnitude, a l p h a b e t i c a l o r d e r etc)
3 , P r e s e n t a t i o n o f d a t a u s i n g f r e q u e n c y tables and p ic tograms/ ideographs .
4, P r e s e n t a t i o n of d a t a u s i n g b a r c h a r t s and i i n e g r a p h s ,
5 , R e p r e s e n t a t i o n of d a t a from f requency t a b l e s u s i n g p i e c h a r t s and v i c e v e r s a ,
C Con tml Groups
31aser8s b a s i c t e a c h i n g model i s employed, A c t i v i t i e s i n c l u d e
q u e s t i o n i n g and answering, e x p o s i t i o n and d i s c u s s i o n , and Performance assessment ,
Same a p p r o a c h e s / s t r a t e g i e s and a c t i v i t i e s as above i n c l u d i n g I l l u s t r a t i ~ o n s of t a l l y i n g and g roup ing ,
Same a s for l e s s o n 1 above and i n c l u d i n g d i s c u s s i o n s on r e p r e s e n - t a t i v e unit pic ture /d iagrarn and f requencyjnumber to be r e p r e s e n t e d , and i l l u s t r a t i o n s on f requency t a b l e s and pic tograms.
Same as for lesson 1 above i ~ c l u d i n g i l l u s t r a t i o n s of b a r c h a r t and l i n e g r a p h s and performance a s s e s s m e n t it? form of t a k e home ass ignment ,
G l a s e r ' s basic t e a c h i n g model, A c t i v i t i e s i n c l u d e q u e s t i o n i n g and answer ing, e x p o s i t i o n , d i s c u - s s i o n and i l l u s t r a t i o n s of c o n s t r u c + i n g ; p i e cl--rt from f requency ~ d m e , ~ I I U V I L ~ versa . Followed by t a k e home ass ignment ,
- - -
D Exper imenta l Groups
G l a s e r g s basic t e a c h i n g model, Ac t iv i t i e s i n c l u d e : q u e s t i o n i n g and answering; e x p o s i t i o n and d i s c u s s i o n , and performance assessment i n form of essay- type test i tems,
Execut ion of project 3 of the Group I1 l i s t o f p r o j e c t s by t h e t e a c h e r s and s t u d e n t s , The d a t a i n v o l v e d include: number of r i d g e s l b e d s c u l t i v a t e d by each class i n t h e schoo l , number of maize p l a n t s per r i d g e , number cassava s tems p e r r i d g e , q u a n t i t y of f e r t i l i z e r expended on t h e farm and garden accord- i n g to t y p e , cost of f e r t i l i z e r a p p l i e c i n t h e farm a c c o r d i n g to crop. OR number of heads and t a i l s i n t o s s i n g t h e c c i n t ' n r i ce and n o t i n g t h e order o f o c c u r r e n c e , c a s t i n g t h e d i c e t w i c e and n o t i n g t h e score as t h e y o c c u r and a l s o t h e sur of t h e p a i r or' s c o r e s , Holding some even ing c l a s s e s i n a d d i t i o n t o s c h o o l hour mathemat ics classes t o c o l l e c t t h e n e c e s s a r y data and u s e t h e a p p r o p r i a t e mcdes of d a t a presentat!-cn t o r e p r e s e n t t h e d a t a c ~ l l e c tcd .
The e x e c u t i o n of t h e p r o j e c t c o v e r s l e s s o n t c p i c s 2 tnough t o 5 and a s p e c t s of l e s s o n 7,
Y I I L . . _ . _ . - I 4
dnit 81- Work VIII
---.a -_Y
3 e v i s i o n ok statistics zon t inued
_ Y_...--- --I_--
B 1 C Lesson T o p i c s I C G ~ trcl Groups -
i 5, R e p r e s e n t i n g d a t a fro-? Same as for l e s s o n 5 above ( o v e r f r equency t a b l e s using page). b u t i l l u s t r a t i o n s of h i s t o - His tograms and v i c e ve r sa . grams and i n t e r p r e t i n g h is tograms,
also eveninq s e s s i o n f o r f u r t h e r
7, Comparisons and c o n t r a s t s of t h e d t f f e - z e n t modes of p r e s e n t i n g d a t a , i n terms of a u d i e n c e , t y p e of d a t a , and f e a t u r e s of t h e p r e s e n t a t i o n s .
8, Mode of ungroupe8. d a t a -
9. Median of ungrouped d a t a ,
10. Mean of s i n g l e o r ungrouped numer ica l d a t a ,
I -
d r i l l s ,
' G l a s c r v s basic t e a c h i n g model, i A c t i v i t i e s i n c l u d e : q u e s t i o n i n g
I '
and answer ing, e x p o s i t i o n , d i s c u - s s i o n , and i l l u s t r a t i o n s . Perform- a n c e a s s e s s m e n t i n t h e c lass room u s i n g free r e s p o n s e tesi: items.
Glaser* s basic t e a c h i n g model. A c t i v i t i e s i n c l u d e : q u e s t i o n i n g and answer ing, e x p o s i t i o n and d i s c u s s i o n , d r i l l i n g i n i d e n t i f y i n g r ~ o d e l / s o f data u s i n g ample examples and Performance assessment (using mu1 t i p l e c h c i c e test items,
1 Same I same
a s f o r l e s s o n tckic 8 above,
a s f o r l e s s o n t o p i c 8 abcve performance assessment i s
c t i t e m s , 1 th rough e s s a y t y p e te-
I
D Filvnari m ~ n tal C r n r i n ~
Glaser ?1, A c t i v i l i n g and a n s w s . t l u s t r a - t i o n s jrams ar?d i n L';L p r L l l r r j ' I A a L-L a t L 3 . Some even ing c l a s s e s f
Same as f o r t h e c o n t r o ~ group b u t re-err f o. m c e s to -the J ted ,
Same a s for t h e c o n t r o l group b u t t l e d a t a i n v o l v e d are d e r i v e d from t h e e x e c u t i o n of p r o j e c t 3 of group 11 of the l ist of projects,
Same as f o r t h e c o n t r o l g roup i n l e s s o n t a p i c 9.
Same a p p r o a c h / s t r a t e g y and a c t i v i - t i es i n v o l v e d i n t e a c h i n g and l e a r n i n g of l e s s o n t o p i c 10 a s w i t h the c c n t r o l group,
A d m i n i s t r a t i o n of p r o j e c t 3 of g r o u p XI1 of t h e l i s t of p r o j e c t s , - . ._. . .- ._ . - --IIL---UII&-.-I--