HIROSHIIIAJOL'R\:ALOF IIATHEII.ITICS EDL'C,'LTIO\'
7: l7-30,1999
t7
STTTDENTS'
INTLIITIVESTRATEGIES
II{ JUDGII{GASSOCIATION
WHENCOMPARING
TWOSAMPLBS
Antonio ESTEPA
L ' n i v e r s i t io' f J a e n ,S p a i n
Carmen BATANERO
L , n i v e r s i t )o f G r a n a d a .S p a i n
F. T. SÁ¡icHnz
U n i v e r s i t v o f J a é n ,S p a i n
(Receit,ed
.January5, 1998)
Abstract
T h i s p a p e ri s a n e r p e r i n e n t asl t u d ¡ ' o ns t u d e n t s ' s t r a t e g i a
en
sdassociationjudgements
whenfacedwith the comparison
of a numericalr,ariable
in trvo dilferentsanples(comparison
of two samples).
The strategiesare
point of view. r.vhichallolvsus to identif¡'
classifiecl
from a mathematical
theorenlsin actionand two tJ'pesof misconceptions
aboutassociation.
STUDENTS'
INTUITIVESTRATEGIES
IN JUDGINGASSOCIATION
1VHENCONTPARII{G
TWOSAMPLES
S t a t i s t i c a la s s o c i a t i o ni s a r e l e t ' a n ta r e a i n m a t h e m a t i c se d u c a t i o n b
. c c a u s ei t e x ,
t e n d s f u n c t i o n a l d e p e n d e n c ea n d h a s s i g n i f i c a n t c o n n e c t i o n su , i t h p r o b a b i l i t v a n d
p r o p o r t i o n a l r e a s o n i n g .S t u d : " i n ga s s o c i a t i o na m o n g v a r i a b l c s i s f u l d a n e n t a l i n e r , iologr.e
p e r i m e n t a lr e s e a r c hi n d i f f e r e n t s c i e n c e s( e . g . b
' . c o n o m i c sn. l e c l i c i n ee. d u c a t i o n )
t o f i n d c a u s a l e x p l a n a t i o n s ,t h a t p e r m i t u s t o u n d e r s t a n do u r e n v i r o n m e n t .H o r v e v e r .
t h e e x i s t e n c eo f a s s o c i a t i o nb e t w e e nt w o v a r i a b l e sd o e s n o t n e c e s s a r i l l ' i m p l l ' a c a u s a l
r e l a t i o n s h i p ,b e c a u s ei t i s p o s s i b l e t o f i n d a h i g h c o e f f i c i e n to f a s s o c i a t i o na m o n g
v a r i a b l e s w h e n t h e r e i s n o c a u s a l l i n k ( s p u r i o u s c o r r e l a t i o n ) ,d u e t o t h e c o n c u r r e n c e
o f d i f f e r e n t f a c t o r s a f f e c t i n g t h e r e s p o n s ev a r i a b l e .
B e ¡ . ' o n dt h e s e e p i s t e m o l o g i c apl r o b l e n r s ,p s v c h o l o g i c a lr e s e a r c h h a s s h o w n t h a t
ls
A n t o n i o E , s r ¡ p , qC, a r m e nB l r . q x p R oa n d F . T . S . r N c H E z
j u d g i n ga s s o c i a t i oi ns n o t a n i n t u i t i v ea b i l i t v .A d u l t s s o m e t i m euss eo n l ¡ ' p a r to f t h e
a t ' a i l a b l ed a t at o j u d g ea s s o c i a t i oann d b a s et h e i r j u d g m e n to n t h e i r p r e v i o u sb e l i e f s
aboutthe type of associationthat ought to exist betweenthe variablesstudiedrather
t h a no n t h e e m p i r i c a d
l a t a .C h a p m a n& C h a p m a n( 1 9 6 9 ) h a v ed e s c r i b e dt h i s p h e nomenonas "illusory correlation"becausepeoplemaintaintheir beliefsin spite of evid e n c eo f d a t a .J e n n i n g sA, m a b i l ea n d R o s s( 1 9 8 2 ) ,W r i g h t a n d M u r p h v ( 1 9 8 4 ) ,a n d
A l l o i ' a n d T a b a c h n i k( 1 9 8 , 1 )a, m o n go t h e r s ,h a v es t u c l i e dt h e e f f e c tt h a t p r e v i o u s
theories,aboutthe contextof a problem,haveon judgingassociation.
The generalconc l u s i o ni s t h a t w h e n t h e r ei s a g r e e m e nbt e t w e e nd a t a a n d p r e v i o u se x p e c t a t i o npse o p l e i n c r e a s et h e i r c o n f i d e n c ien t h e a t t r i b u t i o no f c o - v a r i a t i o nO. n t h e o t h e r h a n d ,
w h e n d a t ad o e sn o t c o i n c i d ew i t h t h e s ee x p e c t a t i o n tsh, e r ei s a c o g n i t i v ec o n f l i c ta n d
t h e a c c u r a c l ' o ft h e p e r c e p t i o n
of co-variatiod
n e p e n d so n t h e r e l a t i v es t r e n g t ho f t h e
tlvo sourcesof information.
D e t e r m i n i n gt h i s p r e c o n c e p t i oins h i g h l y r e l e v a n tf o r t h e t e a c h i n go f a s s o c i a t i o n .
A s C o n f r e y( 1 9 9 0 )h a s p o i n t e do u t , t h e n e e do f r e s e a r c ho n s t u d e n t s ' c o n c e p t i olni ess
in the fact that sometimestheseconceptions
differ in fundamentalwa1'sfrom the scientific conceptsthat we try to teach.Furthermore,students'conceptions
are resistant
to changein spite of instruction.Although many studentsin our study demonstrated
c o r r e c to r p a r t i a l l l 'c o r r e c tj u d g e n e n t sa n d s o l u t i o n ss t r a t e g i e st,h e m i s c o n c e p t i o n s
a n d i n c o r r e c ts t r a t e g i e p
s r e s e n t e di n t h i s a r t i c l ei n d i c a t ea g a p b e t w e e nt h e m e a n i n g
o f a s s o c i a t i otnh a t w e t r ) ' t o t e a c h a
, n d t h e s u b j e c t i v em e a n i n gt h a t s t u d e n t sm i g h t
a t t r i b u t et o t h i s c o n c e p(tG o d i n o& B a t a n e r o1, 9 9 4 ) .
Background
T h i s s t u d y ' c o n t i n u eosu r p r e v i o u sr e s e a r c ho n s t a t i s t i c aal s s o c i a t i o(nB a t a n e r o
et
a l . , 1 9 9 6 , E s t e p a& B a t a n e r o1 9 9 6 , B a t a n e r oe t a l . , 1 9 9 7 ) , w h e r e w e s t u d i e d
judgementsand solvingstrategiesin contingerrcy
students'preconceptions,
associatron
t a b l e sa n d s c a t t e rp l o t s a n d t h e i r e v o l u t i o na f t e r a t e a c h i n ge x p e r i m e nbt a s e do n t h e
use of computers.In the aforementioned
papersthe follorvingmisconceptions
concerning statisticalassociation:
were described:
a) Deterministconception
of associatlor¿:
Somestudentsdo not admit exceptionsto the
existenco
ef a relationship
b e t w e e nt h e v a r i a b l e sT. h e y e x p e c ta c o r r e s p o n d e n ct hea t
a s s i g n so n l v o n e v a l u ei n t h e r e s p o n s ev a r i a b l ef o r e a c hv a l u eo f t h e e x p l a n a t o r y
r a r i a b l e .\ \ ' h e nt h i s i s n o t s o , t h e y c o n s i d e rt h e r e i s n o a s s o c i a t i o n
b e t w e e nt h e
r a r i a b l c s .F o r e x a m p l es, o m es t u d e n t sa r g u et h a t c e l l s o u t o f t h e d i a g o n a il n a 2 x 2
c o n t i n g e n c ) ' t a bol eu g h tt o h a v ez e r of r e q u e n c yi n c a s eo f a s s o c i a t i o n
I n. s c a t t e rp l o t s
S t u d e n t sI' n t u i t i v e S t r a t e g i e si n J u d g i n gA s s o c i a t i o n
l9
theyexpectno spreadin the plot and sonetimes
thel'evenlook for an algebraic
equation
relatingthe two r,'ariables.
t s i n go n l ) '
b ) L o c a lc o n c e l t t i oorfta s s o c i a t i o rSl :t u c l e n t os f t e n f o r m t h e i r j u d g e m e n u
p a r t o f t h e d a t ap r o r , i d e d
i n t h e a s s o c i a t i opnr o b l e n rI .f t h i s p a r t i a li n f o r n a t i o ns e r v e s
t o c o n f i r ma g i v e nt y p e o f a s s o c i a t i o n
t h. c r r v o u l da d o p tt h i s t y p e o f a s s o c i a t i ornn
t h e i r a n s w e r .F o r e x a m p l ei,n c o n t i n g e n c tva b l e s .s t u d e n t ss o m e t i m ebs a s et h e i r
j u d g e m e not n o n l ¡ ' o n ec o n c l i t i o n d
a il s t r i b u t i o no r . e v e no n o n l y o n ec e l l ,u s u a l l ; t"h e
c e l l f o r w h i c h t h e f r e q u e n c ¡ ' i sn r a r i n r u mT. h e s es t u d e n t sf a i l e dt o g r a s pw h a t i n
o p i n i o no f l ( o n o l de t a l . ,( 1 9 9 7 ) i s t h e e s s e n coef t h e s t a t i s t i c apl e r s p e c t i v ea:t t e n d i n g
t o f e a t u r e so f a g g r e g a t eass o p p o s e dt o f e a t u r e so f i n r l i r . i d u a l sa,s d i s t r i b u t i o n sa n d
t h e i r p a r a m e t e r sa r e f e a t u r e sb e l o n g i n gn o t t o a n ¡ ' i n d i v i cul a l e l e n l e n tb, u t t o t h e
$'hichther comprise.
aggregate
c ) L i t t i d i r e c t i o t rcuol n c e p t i ootfta s s o c i a t i o u
S:o m es t u d e n t sp t ' r c r i v et h e a s s o c i a t i o n
o n l y w h e nt h e i r s i g n i s p o s i t i v ec, o n s i d e r i n tgh e i n v e r s ea s s o c i a t i oinn d e p e n d e n c e
a n d i n o t h e r c a s e st h e ¡ ,h a v et r o u b l ew i t h t h e i n t e r p r e t a t i o no f n e g a t i v cc o r r e l a t i o n
coefficient.
d) Causalcurception
o.fassociation:
Somestudentsonly considerthe association
betlveen
t h e v a r i a b l e si f t h i s c o u l db e a t t r i b u t e dt o a c a u s a lr e l a t i o n s h i pb e t w e e nt h e m .T h i s
m i s c o n c e p t i owna s p e r s i s t e nitn o u r t e a c h i n ge x p e r i m e n(tB a t a n e r o
e t a l . , 1 9 9 7 ) ,a n d
the theoreticalstud¡rof the relationships
betweencorrelationand causalityduring the
teachingwas insufficientto changethe students'conception.
I n t h i s a r t i c l e w e r e p o r t o n a n e x p e r i n l e n t asl t u d y w h o s ea i m w a s t o f i n d o u t
w h e t h e rs o m eo f t h e a f o r e s a i dd e s c r i b e dp r e c o n c e p t i o nc so n c e r n i n a
g s s o c i a t i ow
nere
f o u n dw h e n s t u d e n t sc o m p a r et w o s a m p l e so n a n u m e r i c avl a r i a b l e S
. o m ea u t h o r s ,
s u c ha s K o n o l de t a l . ( 1 9 9 7 ) h a v ee x p l o r e ds t u d e n t s ' a b i l i t yt o c l e t e r m i nw
eh e t h e r
t h e r ei s a d i f f e r e n c e
b e t w e e ng r o u p s .A l t h o u g hj u d g i n gg r o u pd i f f e r e n c eiss f o r m a l l l '
c o m p a r a b l teo j u d g i n ga s s o c i a t i o nt h, e s et w o t y p e so f j u d g e m e n t ps r o b a b l l 'd e s c r i b e
d i f f e r e n tc o g n i t i v et a s k si n o p i n i o no f K o n o l de t a l . ( i 9 9 7 ) w h o s u g g e s t tsh a t t h e
methodsusedby studentsin theseproblemswould dependon the \\¡a)'the questionis
posedand the reasonfor doingthe comparison.
Our data was takenfrom a sampleof pre-univer-sit1'students,
throughtheir written
r e s p o n s etso a q u e s t i o n n a i rien c l u d i n gt w o p r o b l e n s .A f t e r c l e s c r i b i ntgh e p r o b l e m s ,
perceivedbv the stuclents
we, firstly, discussthe type of association
in the two different
t a s k s .W e , s e c o n d l ya, n a l 1 ' steh e s t u d e n t s ' s o l v i nsgt r a t e g i e sa,n d p r e s e n ta n o r i g i n a l
point of view.in whichwe identifl'
classification
of thesestrategies,
from a nrathenatical
(1982).Finally,we describe
and theoremsin actionas describedb¡-'Vergnaud
concepts
AntonioEsrsp,r.CarmenBrr.rsonoandIr. T. SrxcHEz
20
t h e d i f f e r e n t m i s c o n c e p t i o n sf o u n d c o n c e r n i n g s t a t i s t i c a l a s s o c i a t i o n .
METHOD
Sample
T h e s a m p l ec o n s i s t e do f 2 1 3 s t u d e n t si n t h e i r l a s t 1 ' e a ro f s e c o n d a r ys c h o o l( 1 7 1 B
y e a r - o 1 ds t u d e n t s )a t t e n d i n gt h r e e d i f f e r e n t h i g h s c h o o l si n t h e c r t l ' o f J a e n ( e i g h t d i f f e r e n t s t u d e n t g r o u p s w e r e u s e d ) .I t i s a t t h i s l e v e l t h a t t h e t o p i c o f a s s o c i a t i o ni s t n t r o d u c e d i n t h e S p a n i s hc u r r i c u l u m . A b o u t h a l f t h e s t u d e n t s( 1 1 3 ) w e r e m a l e s a n d
h a l f w h e r e f e n r a l e s( 1 0 0 ) . T h e q u e s t i o n n a i r ew a s g i i - e n t o t h e s t u d e n t s b e f o r e t h e
t e a c h i n go f a s s o c i a t i o nb e g a n ,s o t h a t t h i s s t u d i ' m a 1 ' b e c o n s i d e r e da s r e s e a r c hi n t o
s t u d e n t s 'p r e c o n c e p t i o n cs o n c e r n i n gs t a t i s t i c a la s s o c i a t i o n .
O n e h u n d r e d a n d t w e n t l ' f o u r s t u d e n t sh a d f o l l o w e d a s c i e n t i f i c a l l y - o r i e n t e cdu r r i c t r l u n ra n d e i g h t v n i n e s t t L d e n t sh a d f o l l o l v e d a h u m a n i t i e s - o r i e n t e dc u r r i c u l u m .
A l t h o u g h b o t h g r o u p s o f s t u d e n t sh a d s t u d i e d m a t h e m a t i c st h r o u g h o u t t h e i r p r e v i o u s
s t u d i e s ,i n t h e f i r s t g r o u p a s t r o n g e r e m p h a s i sw a s g i v e n t o t h e s u b j e c t a n d t h e y h a d
m o r e o p p o r t u n i t i e st o a p p l l ' m a t h e m a t i c si n o t h e r s c i e n t r f i cs u b j e c t s ,s u c h a s p h v s t c s
. o n c e r n i n gs t a t i s t i c s ,b o t h g r o u p s r v e r e t a u g h t f l e q u e n c y d i s t r i b u t i o n ,
o r c h e m i s t r ' ¡ 'C
g r a p h i c a l r e p r e s e n t a t i o n ,m e a s u r e so f l o c a t i o n , a n d s p r e a d o f u n i v a r i a t e s t a t i s t i c a l
v a r i a b l e s .T h e v h a d a l s o b e e n i n t r o d u c e dt o p r o b a b i l i t v i n t h e f i r s t y e a r o f s e c o n d a r y
s c h o o l ( w h e n t h e 1 " , r ' e r e1 . 1 1 5 1 ' e a r s o l d ) . T h i s s t u c l ¡ ' h a s a q u a s i - e x p e r i m e n t acl h a r a c t e r , b e c a u s eo f t h e n o n - r a n d o mn a t u r e o f t h e s a m p l e ' so f s t u d e n t s a n d p r o b l e m s .
Questionnaire
T h e s t u d e n t sw e r e g i v e n t h e t w o q u e s t i o n sc o n c e r n i n gt h e c o m p a r i s o no f t w o s a m p l e s
t h a t a r e s h o w n i n f i g u r e s 1 a n d 2 . I n t h e f i r s t i t e m t h e s a m p l e sa r e r e l a t e d w h e r e a s
t h e l ' a r e i n d e p e n d e n ti n t h e s e c o n di t e m . T h e t - t e s t o f m e a n d i f f e r e n c e sb e t w e e nt h e
t u ' o s a m p l e sw a s c o m p u t e c fl o r e a c h p r o b l e n r ,g i v i n g a p - r ' a l u eo f p = 0 . 0 0 0 9 2 . t = 3 . 3
i n t h e f i r s t i t e m a n d D = 0 . 1 5 1 0 . t = 1 . 5 i n t h e s e c o n do n e .
Data analysis
I n e a c h i t e n l w e a n a l 1 , ' s etdh e t y p e o f a s s o c i a t i o np e r c e i r , ' e d
b y ' t h e s t u d e n t s( d i r e c t
: L s \ ( ) c i a t i o ni n, v e r s e a s s o c i a t i o n o
, r i n d e p e n d e n c e )I.n a d d i t i o n , t h e s t u d e n t s ' w r i t t e n
: , . : i ) ( ) n s e st o t h e q u e s t i o n n a i r ew e r e c a t e g o r i s e da n d , a f t e r s e v e r a i s u c c e s s i v er e v i s i o n s ,
. . - h . ' n r t ' u . a sc l e v e l o p e cf ol r c l a s s i f ¡ , ' i n sg t u d e n t s ' s o i u t i o ns t r a t e g i e sb a s e do n \ i e r g n a u d ' s
. ' r ' ) r r r o t i o no f t l r c o r e n itt t a c t i o n .
in JudgingAssociation
Students'
IntuitiveStrategies
2l
F i g u r e1
g eb l o o dp r e s s u rteo a g r o u p o f
I t e m 1 . T h ef o l l o w i n g
d a t ar v e r eo b t a i n e tchl e n n r c a s u r i nt h
10 lvomen,
beforeandafterapplvingnredical
treatment:
N I r s . A\ l r s . B \ l r s . C \ l r s . D I I r s E \ l r s . F N I r s . GN I r s . Hl \ l r s . l N l r s l
B l o o d p r e s s u r e1 1 5
b e f o r et r e a t m e n t
112
10;
11 9
11 5
138
126
105
10.1 1r5
Blood pressurel28
after treatment
115
106
r28
1'¿2
115
132
109
102
11i
L ' s i n gt h e i n f o r m r t i o nc o n t a i n e di n t h i s t a b l e ,d o ) ' o u t h i n k t h a t t h e b l o o d p r e s s u r ei n t h i s
s a m p l ed e p e n d so n t h e t i n r eo f m e a s u r e n r e n( tb e f o r eo r a f t e r t h e t r e a t m e n t ) ?
E x p l a i n) ' o u r a n s \ \ ' e r .
F i g u r e2
u ' h e nn r e a s u r i nt g
I t e m 2 . T h e f o l l o r v i n cgl a t an ' e r eo b t a i n e d
h e s u g a rl e v e li n t h e b l o o do f
m a l ea n df e n r a lsec h o ocl h i l d r e n :
o
]U
Prnil
ABCDEFG
Gender
l\l
l\l
\l
\l
\l
\l
l\l
Sugarlevel 9
0
9
8
tl
7
l
]\I
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\I
F
F'
F'
F'
T
O
F'
F'
F'
F'
F'
F
98-r)6070836773
NI:male. F:female.
U s i n g t h i s i n f o r m a t i o n .d o v o u t h i n k t h a t l h e s u g a r l e v e l i n t h e b l o o d i n t h i s s a m o l e
d e p e n d so n t h e s e x ?
Explain )'our answer.
A p i l o t s t u d ¡ r v i t h a n a d d i t i o n a ls a n r p L o
e f 5 l s t u d e n t sw a s u s e d t o c h e c k t h e r e l i a b i l i t y a n d t h e c o d i n g s ) s t e n l f o r t h e s t u d e n t s 'a n s w e r s .S i n c e f a c t o r a n a l ¡ ' s i so f
s t u d e n t s ' a n s w e r sh a d a m u l t i d i n r e n s i o n aslt r u c t u r e ,a g e n e r a l i z a b i l i t l ' s t u d r ' ( B r e n n a n ,
1 9 8 3 ) w a s d o n e ,i n s t e a do f c o n p u t i n g a n i n t e r n a l c o n s i s t e n c l ' r e l i a b i l i t fi n d e x . \ \ ' e
o b t a i n e d a g e n e r a l i z a b i l i t ¡ irn d e x , G = 0 . 8 6 , a s a n e a s u r e o f t h e p o s s i b i l i t v o f e r t e n d i n g
o u r c o n c l u s i o n st o t h e h y p o t h e t i c a l i t e m p o p u l a t i o n a n d a n o t h e r i n d e r , G = 0 . 9 4 ,
o f g e n e r a l i z a b i l i t yt o t h e s u b j e c t s ' p o p u l a t i o nT
. h i s p i l o t s a m p l er , l a sa l s o u s e d t o p e r f o r m
a q u a l i t a t i v e s t u d y o f t h e i n t e r r e l a t i o n b e t w e e n s t r a t e g ) - a n c lt h e t ¡ ' p e o f a s s o c i a t i o n
p e r c e i v e d b 1 ' t h e s t u d e n t s ,i n o r d e r t o c h a r a c t e r i s es t u d e n t s 'n i s c o n c e p t i o n sc o n c e r n i n g
s t a t i s t i c a la s s o c i a t i o nT. h e r e s u l t s o f a l l o f t h e s e a n a l l ' s e sa r e p r e s e n t e di n t h e f o l l o w i n g
sections.
))
A n t o n i o E s r e p , qC
, a r m e nB . r r . A N e Raon d F . T . S . r ¡ c H r z
DISCUSSION
I n t a b l e 1 f r e q u e n c i easn d p e r c e n t a g eosf a n s r v e r sc o n c e r n i n g
theexistence
of assoc i a t i o nb e t w e e nt h e t w o v a r i a b l e si n e a c hi t e m a r e s h o w n .I n b o t h c a s e st h e m a j o r i t l o f s t u d e n t sg a v ea c o r r e c tr e s p o n s es.o t h a t j u d g i n gt h e a s s o c i a t i oinn t h i s n ' p e o f
problemwas not a difficult task for our students.
T a b l e I . F r e q u e n c i easn d p e r c e n t a g eosf a s s o c i a t i oj u
n d g e m e n ti sn t h e c o m p a r i s oonf
two samples.
Itenl
I
)
Association
Independence
No ansrver
*t
2 0 ( 9, 1 )
l5 (7.0)
44(20.7)
t2 (5.6)
z8(93.0)
*r-"7
73.7)
correctanswer
H o w e v e r t, h e c o r r e c tj u d g e m e not f t h e e x i s t e n c e
o f a s s o c i a t i obne t w e e nt h e t w o
s a t t h e s t u d e n t sh a v e a b o u t
s a m p l e si s n o t e n o u g ht o d e t e r n l i n et h e p r e c o n c e p t i o nt h
t h i s t o p i c ,s i n c e ,a s d i f f e r e n tr e s e a r c hh a s s h o w n ,w e c a n g e t a c o r r e c ts o l u t i o nf o r a
g i v e np r o b l e n ur s i n ga n i n c o r r e c tp r o c e d u r eF. r o m t h e p e r s p e c t i voef M a t h e m a t i c s
Education,both a correctprocedureand a correctJudgmentof association
are needed
to correctlyunderstandthe ideaof association.
Makinga judgementconcerningthe association
of a numericalvariablewith regards
t o a d i c h o t o m iq
c u a l i t a t i v ev a r i a b l e( a si n t h e c o m p a r i s oonf t w o s a m p l e si )s a t a s k
t h a t m u s tb e s o l v e di n t h e n u m e r i c aflr a n t e( D o u a d y1, 9 8 6 ) .F u r t h e r n o r ea, s t h e r e s p o n s ev a r i a b l ei s q u a n t i t a t i v ew, e c a n c o m p a r ei t s d i f f e r e n ts t a t i s t i c ss, u c ha s a v e r . r a p h i c a rl e p r e s e n t a t i o n s
a g e sa n d r a n g e si,n t h e t w o s a m p l e st,o s o l v et h e p r o b l e m G
of the two distributionscould also be used.
In studyingthe strategies,we noticedthat somestudentsprovedtheir correctintuitive
conceptions
as thel'solvedthe problemb1'usingnormativestrategies.
But the conceptof
a s s o c i a t i oins n o t s i m p l ea n d s o m es t u d e n t sm i g h th a v ec o r r e c ti n t u i t i v ec o n c e p t i o n s
c o n c e r n i n sg o m ep r o p e r t i e so f a s s o c i a t i om
n i r e d r v i t ho t h e r i n c o r r e c tc o n c e p t i o n s .
T h i s w o u l d l e a d t h e mt o c h o o s ea n i n c o r r e c to r p a r t i a l l y c o r r e c ts t r a t e g y A
. ll these
conceptions
needto be identifiedin order to plan adequateinstruction.
I n o r d e r t o i d e n t i f yw h i c h o f t h e p r o p e r t i e sl i n k e dt o t h e c o n c e p ot f a s s o c i a t i o n
w e r e i n t u i t i v e l yu s e dc o r r e c t l yo r i n c o r r e c t l vb i . o u r s t u d e n t sw, e d e v e l o p e ad c l a s s i f i c a t i o no f s t u d e n t s ' s t r a t e g i easc c o r d i n gt o t h e m a t h e m a t i c aclo n c e p t si n v o l v e da n d
t h e o r e m si n a c t i o ni m p l i e di n t h e d i f f e r e n tp r o c e d u r e us s e da n d t h e t ¡ ' p eo f e r r o r s .
S t u d e n t sI'n t uj t i v e S t r a t e g i eisn J L r d g i nAg s s o c i a t i o n
z3
S u c h a c l a s s i f i c a t i o ni s . i n \ - e g n a t r c l '(s1 9 8 2 ) o p i n i o n .e s s e n t i a lf o r c l i c l a c t i c aaln a l ¡ - s i s
I n t h e f o l l o w i n g p a r a g r a p h su ' t ' d e s c r i b et h i s c l a s s i f i c a t i o n .
Correet Strategies
1 . C o m p a r i t r gn t € a t l sS: o m e s t u d e n t s c o m p a r e d t h e t w o m e a n s a n r p l e st o d e c i d e t h e
e x i s t e n c eo f a s s o c i a t i o nb e t w e e n t h e r , a r i a b l e s .s u c h a s S t u d e n t , l 3 i n t h e s e c o n c l
item'."Becauseu,ltettyou add the sugar leuels in all the males yott obtaitt 69, tltut dit,ided
b y 1 ( ) i s 6 . 9 . u , l t i l e t h e s u u ro . fs u g a r l e t t e l s i r t f e m a l e s i s 4 7 t h a t d i l i d e d b t , l 0 i s o n l t '
4.7. Therefore,there is depettdence
ott the sex, os sugar leuel is ltigher in nnles". TIte student
n t i g h t i n p l i c i t l y b e w s i r t g t h e f o l l o u t i r t gl h e o r e mi n o c t i o n : " Í I ' h e t t t h e r e i s a s u f f i c i e n t l l '
high difference betweetttJrc nrcansof the conditional distribwtiotts of a Enrúitatitte uariable
with regards to the ualues of attother qualitatiue uariable, there is an associatiln betueetl
t h e t u , ou a r i a l t l a s "( T 1 7 .T h i s i s a n e x a n r p l eo f a c o r r e c t a n s w e r f o r t h e i n s t i t u t i o n a l
r e l a t i o n i n t r o d u c e d t o t h e s t u d e n t , w h i c h d o e s n o t i n c l u d e t h e u s u a l p r o c e d u r e sf o r
t e s t i n g t h e e q u a l i t v o f m e a n s .w h e r e i t i s n e c e s s a r ] ' t oc o n s i d e r t h e v a r i a b i l i t l ' o f t h e
t w o d a t a s e t s , t h e s a m p l e s i z e s a n d t h e s i g n i f i c a n c el e v e l , i n a d d i t i o n t o t h e m e a n
values.
2 . C o n Q a r ü t gt o t a l s :S o m e s t u d e n t s c o m p a r e d t h e t o t a l s o f t h e r e s p o n s e v a r i a b l e
v a l u e s i n e a c h s a m p l e : " \ \ ' h e n a d d i t r g a l l t h e s u g a r l e u e l si u t u a l e su t eo b t a i t ta g r a t e r
n t t n t b e r t h a t tw h e t ta d d i u g a l l t l t e s u g a r l e u e l si t t , f e n t a l e s("S t u d e r t t3 1 ) . T h e v m i g h t b e
u s i n g t h e f a c t t h a t d e p e n d e n c ew o u l d i m p l y t h e v a r i a t i o n o f t h e s u m o f . u ' a l u eisn t h e
t w o d i s t r i b u t i o n s ,w h i c h i s a m o d i f i c a t i o no f t h e p r e v i o u s t h e o r e mi n a c t i o n T 1 .
3 . C o n t p a r ü t gp e r c e n t a g e sI:n I t e m 1 , s o m e p u p i l s c o m p a r e d t h e t w o v a l u e s o f b l o o d
p r e s s u r e i n e a c h w o m a n a n d b a s e d t h e i r J u d g m e n to n t h e p r o p o r t i o n s n r p e r c e n t a g e
o f c a s e s w h e r e t h e p r e s s u r e i n c r e a s e d o r d e c r e a s e d ." \ \ - e c a n o b s e r t ' ct l t a t t l t c p r e s s u r ( l
decreaseitt 200,4o.fwoutertattd ütreases üt the renninirtg B0o/o",and, therafore,the treatnrcnt
ittcreasesthe presswre"(Student 8).
I n t h i s c a s e ,a g r o u p i n g o f w o m e n i n t w o c a t e g o r i e s- a c c o r d i n g t o w h e t h e r t h e
p r e s s u r e i n c r e a s e do r d e c r e a s e d - w a s p r o d u c e d .T h e s e s t u d e n t s r e l a t e d t h e s o l u t i o n
t o a p r o b a b i l i t y p r o b l e n la n d u s e d L a p l a c e ' sr u l e . c o n s i d e r i n gt h a t i f t h e r e w e r e n o
d e p e n d e n c et,h e s e c a t e g o r i e ss h o u l d h a r , el h e s a m e f r e q u e n c r -T
. h i s p r o c e d u r ei s s i m i l a r
t o t h a t u s e d i n p a r a m e t r i cm e t h o d sb a s e do n r a n k c o m p a r i s o n si n a s e r i e s .T h e r e f o r e ,
t h e s t u d e n t s m i g h t b e u s i n g t h e f o l l o w i n g t h e o r e m i n a c t i o n : " l . f w e t a k e c o n t n t o nt a l u e s
i t t b o t h s a m p l c s ,t h e p e r c e t t t a g e os , fc o s e si u w l t i c h t h e t a l t t c o . it h e r c r i a b l e i n c r e a s e so r
decreaseswould be the same in cese o,f ittdependertce"(72).
, 1 . C o m p a r i n gt l ' t et w o d i s t r i b u t i o t t s ;O t h e r s t u d e n t s c o m p a r e d t h e c o n r p l e t ed i s t r i b u -
AntonioEsrep,r,
CarmenBrr,rxenoandIr. T. SrlcHaz
24
t i o n s o f f r e q u e n c i e si n b o t h s a n t p l e s .a s i n t h e f o l l o w i n g e x a n r p l e :" | \ - h e nu ¡ ec o t n P a r e
ntalesattd fentales, tlte lettet'haue got a lotuerswgar leuel. That is proued by the fact that
ttL,g
o i r l s l t o u e 0 , t u h i l e t h e r e í s o t t l yo b o t , u , i t h 0 : T h e r e a r e 2 g i r l s u i t h , ? ,u , h i l e t h c r e
i s n ' t a b o y u i t h t h a t p r e s s u r e ; t h e r e i s o r t l y 1 b o 1 ' w i t h t ) a I L L4e. T h e r e a r e 2 g i r t s w i t h
r a l u e 6 a n d o r t l y o r t eb o y w i t h 6 . T l t e r e a r e 3 g i r l s w i t h 7 a t t d o t t l v 1 b o y u i t t t 7 . T l t e r e
a r e 2 b o y sw i t l ' t B a n d o n l y 1 g i r l u , i t h B . T h e r e a r e 4 b o y sw i t h g a n d n o g i r l h a d 9 "
( S t u d e r r t1 2 ) . . T h e ¡ ' m i g h t b e u s i n g t h e f o l l o w i n g t h e o r e n ri n a c t i o n : " T h e r e i s i ¡ d e p e n d ettcewhett both santpleshaue the samefrequertcydistribu,tions"(73).
Partially
Comeet Strategies
5. ConQaring the ualues oi the responseuariablefor each case in related sautples.These
s t u d e n t sc o m p a r e dt h e i ' a l u e sc o r r e s p o n d i n gt o t h e s a m e s u b j e c t i n b o t h s a n p l e s .I t i s
v e r y s i m i l a r t o s t r a t e g y 3 , t h o u g h h e r e p r o p o r t i o n s o r p e r c e n t a g e sw e r e n o t u s e d ,
a n d , t h e r e f o r e ,w e c o n s i d e r t h i s s t r a t e g ¡ ' t o b e o n l y p a r t i a l l l ' c o r r e c t . S o m es t u c l e n t s
f a i l e d i n t h e i r a s s o c i a t i o nj u d g e m e n t w h e n u s i n g t h i s s t r a t e g y . b e c a u s et h e y e x p e c t e d
t h e d i f f e r e n c e t o h a v e l h e s a m e s i g n i n e v e r - vc a s e , w h i c h i n d i c a t e s a f i t n c t i o n a l o r d e tennútist conception of the association: "l ltaue contparedthem and I foutd owt that qfter
the treatnrcnt bloodpressuretlecreasedüt some ulnLen, whereas in other casesblood Dress(Student 30).
urc increased.There is no depandereca"
6. Taking üúo accoutttexceptionalcas¿s.Sometimes,a correct Judgment was obtained
b i ' c o n s i d e r i n g t h e c a s e sw h e r e t h e g e n e r a l r u l e w a s n o t f o l l o r . v e d
( e x c e p t i o n acl a s e s ) .
\ \ ' e c o n s i d e r t h e s t r a t e g y p a r t i a l l y c o r r e c t , b e c a u s ei t w a s u s e d m a i n l v t o i n d i c a t e a n
associationA
. s t h e s t u d e n t s d i d n o t u s e p r o p o r t i o n s o r p e r c e n t a g e sa n d b a s e d t h e i r
j u d g e m e n ti n o n l - vp a r t o f t h c ' d i s t r i b u t i o n .a l o c a l c o n c e p t i o nc a n b e d e d u c e di n t h e s e
students. "Blood pressureincreasesin all the wlutelt exceptitt ltlrs C, s0 ),0u can sa1'tltat
bloodpressutredependson treatnrcnt"(Student 29).
7 . I r i r t d i t t g o u t t h e d i f f e r e n c e sT
. he students computed the differences between the
v a l t r e so f t h e r e s p o n s ev a r i a b l e i n e v e r v s u b j e c t ,i n r e l a t e d s a m p l e s ,b a s i n g t h e a s s o ciation judgement on these differences: "l haue comptttedthe diJferencesin pressurebefore
arul after treatntetttitt cuerr ulntelt and ntostof thesedi-ffarences
arc positiue"(Student. 5).
These pupils might have used the theorem in action: "ln order to fütd the associatiott
bctweetta contittttottsatd o dicltotomicrariable the dífferencein the correspotttlirtgtalues
ot' tlte cotttittttottsuariable nutst be large enough" (7-,1).
8 . C l o b a l c o n p a r i s o t tW
. h e n t h e a s s o c i a t i o nj u d g e m e n t w a s c a r r i e d o u t b y m e a n s o f
r r t ¡ u a l i t z r t i v eg l o b a l c o m p a r i s o no f b o t h s a m p l e s ,r v i t h o u t s p e c i f . v i n gw h i c h s t a t i s t i c s
i \ . r ' r ' cL r S € dF. o r e x a n t p l e , " l n t h e d a t s w e c a l r s e e t h , a tt l t e t t t a l e sh a u eg o t a h i g h , e t ' s t L g a r
S t u c l e n t sI 'n t u i t i v c S t r a t e g i e si n j u d g i n g . { s s o c i a t i o n
25
l e t , e lu i t l t t ' a g a r d st o t l t ¿, t ' c u t u l t ' s( S
" tudent29).
Incorreet
Strategies
A s i n d i c a t e cbl v N i s b e t ta n d R o s s( 1 9 8 0 ) , s u b j e c t sd o n o t u s r - r a l l l . e n r p l oa¡n o r m a t i v e
s t r a t e g ¡ . i n j u d g i n g a s s o c i a t i o nw i t h o u t b e i n g g i r e n s p e c i f i ci r r s t r L r c t i o nbse c a u s et h e s e
s t r a t e g i e s a r e h i g h l y s o p h i s t i c a t e d .\ \ ¡ e f o u n d t h e f o l l o w i n g i n c o r r e c t s t r u t e g i c s i n o u r
students:
. o m e t i n e s s i m i l a r v a l u e s i n a l l t h e c a s e sf o r e a c h s a m p l e
9 . E x p e c t i r t gs i n t i l a r t t a l u e s S
w e r e e x p e c t e d .A s t h a t d i d n o t h a p p e n ,b e c a u s eo f t h e d a t a r a n d o m n e s s t. h e v c o n c l L r c l e d
t h a t t h e b l o o d p r e s s u r e w a s i n d e p e n d e n to f t h e t r e a t m e n t ,s h o w i n g a d e t e r n i u i s t r ; o t t r : c p '
t í o n o f t h e a s s o c i a t i o n . " l ' o t t c a t t s e e a l o t o , ft , a r i e t l t i t t t l t ¿ 't q b l e : s o m e u ' ¡ n t e t ru ' i t l t a
high pressureltaue increasedand other haue decreasedand uice-uersa"(Student51).
10. Conrparínghigltcst and lowestualuesof.both distributions, and basing thc assocrmion
j u d g e m e n to n t h e n l. w h i c h i n d i c a t e sa l o c a l c o n c e p t i o no f t h e a s s o c i a t i o n ." l t d o e st r c t
d e p e r t d0 n s e f a s s 0 i l r em a l e s h a t e t l t e h i g l r c s t s u g a r l e u e l a t t d o t h e r sh a u e t h e l o t ¡ s t . "
( S t u d e n t2 0 8 ) .
1 1 . C o m p a r i n gr a u g e s .S o m es t u d e n t s c o n p a r e d t h e r a n g e o f b o t h s a m p l c ' ss. r . r c ha s
student 66: "Because!0u catl see tltat females got tlte lowesttttnttber(0) as u,all as ntules.
How¿t:e1thet' do rtot get the ltighest ttuntber(9) as nrulesdo".
1 2 . A s s c s s i r t gc o i t t c i d e i r c ¿Sso. n r es t u d e r t s j u s t i f i e d i n d e p e n d e n c eb e c a L t s oc f t h e
c o i n c i d e n c eo f s o m e c a s e s j, u s t a s t h e s t u d e n t w h o a n s w e r s : " B e c a t t s se o n t ¡ ' u t u l t l' tsu t ' ¿ ,
ltarc got tltc suntt,
the same sugar leuel as slnu,femalcs, suclt as tlte pupils F uttd -\' ¿r'l¡¿r
sttgar leuel oJ 7 in spite of beirtgdiiferent, ntale attd.fennle" (Student 133). Iionolcl et al.
( 1 9 9 7 ) i n d i c a t e st h a t c r - r n r p a r i nt ug' o o r n l o r e i n d i v i d u a l si n r e s p c ' ctto s o u r r ' ¿ r t t r i b u ti rs
a f r e q u e n t l y u s e d n o n - s t a t i s t i c a l c o m p a r i s o n .I n t h i s e x a m p l e .w e c a n a d d e t i e t ¿ r n t i t t i s t
r o n c ! ' pi o
t ltof association.
1 3 . B a s ü t g o t t p r e t t i o u st h e o r i e sS
. o m e s t u d e n t s b a s e d t h e i r a s s o c i a t i o nj u c l g e m e n lo n
t h e i r k n o u , l e d g ea b o u t t h e c o n t e x t ( p r e v i o u s t h e o r i e s )i n s t e a c lt - r Iu s i n g t h e d a t a g i v e ' ni n
the problem like this student's answer: "lt dependson wltetlter\olt ur(, cruntütedby a nmle
o r . f e t t t a l ed o c t o rw h e r t¡ o t t d o t t ' t f e e l t e r y u , r : l l .A t t l t a t m o t n t ' n.l t o u t u i g l t t g c t a b i t et c i t e d
'.
a n d t h a t ' s t l t e r e a s o t ft o r t h e i n c r e a s eo f b l o o d p r e s s : : ' r t ,T h c r e f o r e . t h e p h e n o m e n o n o f
a n d C h a p m a n ,1 9 6 9 ) .
i l l u s o r ¡ ' c o r r e l a t i o n .n r i g h t b e a c t i n g i n t h e s e s t u c l e n t s( C h r t p n t a n
1 , 1 .O t h e r s . l n t h i s c a t e g o r ¡ 'o t h e r k i n d s o f p r o c c c l u r - easn d s t r a t e g i e sh a v e b e e n
i n c l u d e d ,s u c h a s s t u d e n t s u , h o c o n r p a r e ct lh e n l e a u s ,u ' h i c h \ \ ' a sc o m p u t e dw i t h o u t
t a k i n g i n t o a c c o u n t n u l l v a l u e s ( p h e n o m e n o nt h a t h a v e b e e n d e s c r i b e d b 1 ' M e v a r e c h ,
1 9 8 3 ) , o r u s i n g t h e v a l u e s o f t h e v a r i a b l e s a s i f t h e l ' w e r e p e r c e n t a g e sc. o n f u s i o n
Antonio EsrEp.r.Carmen B,rr.qleRoand Ir. T. S.r:qcsez
?6
VETV
c o m m o ni n s t u d e n t sa c c o r d i n g
t o H a w k i n se t a l . ,( 1 9 9 2 ) .
of two sam
T a b l e 2 . F r e q u e n c i eas n d p e r c e n t a g eosf s t u d e n t s ' s t r a t e g i ei ns t h e c o m p a r i s o n
ples.
I T E N I1
STRATEGIES
ITEltl 2
Correct
1 . C o m p a r i n gm e a n s
' 2( 0 . 9 )
26 (12.2\
2 . C o m p a r i n gt o t a l s
1(0.5)
35 (16.,1)
3 . C o m p a r i n gp e r c e n t a g e s
41(19.3)
1(0.5)
34 (16.0)
4 . C o m p a r i n gt h e d i s t r i b u t i o n s
Partia)l¡ correctstrategies
i
Cnmnerino
rll
¡rses
6 . A s s e s s i n ge x c e p t i o n a cl a s e s
7. Finding out differences
7 3( 3 1 . 3 )
3 (1.1)
63 (29.6)
6 (2.8)
6 , 1( 3 0 . 0 )
8. Globalcomparison
I n c o r r e c ts t r a t e g i e s
9. E-xpecting
sinilar values
3 (1..1)
1(0.5)
1 0 . C o m p a r i n gh i g h e s ta n d l o r , v e svt a l u e s
6 (2.8)
1 1 . C o m p a r i n gr a n g e s
3 (1.,1)
1 2 . A s s e s s i n gc o i n c i d e n c e s
1(0.5)
3 (1..1)
1 3 . E x p r e s s i n gp r e ' " ' i o u st h e o r i e s
1(0.ó)
8 (3.'1)
14 . O t h e r s
8 (3.8)
1 ' 1( 6 . 6 )
1 , 1( 6 . 6 )
15(7.0)
I 5. No answer
Correct and Partiully Comect Conceptions
In Table 2 we presentthe frequencies
and the percentages
of thesestrategiesin the
s t u d e n t s ' a n s w e r sC.o r r e c ts t r a t e g i e sw e r e m o r ef r e q u e n ti n i n d e p e n d e nsta m p l e s
(item2), while we can find a higherpercentage
of partially correctstrategiesin related
s a m p l e s( i t e m 1 ) . I t i s w o r t h w h i l em e n t i o n i n g
t h a t s o m es t r a t e g i e ss,u c ha s s t r a t e g i e s
3 , 5 , 6 , 7a, n d 9 , w e r ec h a r a c t e r i s toi cf r e l a t e ds a m p l e sw, h i l es t r a t e g i e1s , 2 , 4 , B , 1 0 ,
1 7 , 7 2 , a n d 1 3 a p p e a rm a i n l ya s s o c i a t ew
d i t h i n d e p e n d e nsta m p l e sT. h e r e f o r ee, v e n
thoughthe studentshad not formally studiedthe differencebetweenindependentand
r e i a t e ds a m p l e st,h e y w e r e a b l et o d i s t i n g u i s ht h e mw h e n c o m p a r i n gt h e s a m p l e sf o r
studf in the existenceof association.
Studentswho comparedthe neans in independent
samples,conceivedthe meanas a
v a l u er e p r e s e n t a t i voef t h e d a t a( P o l l a t s eak n d c o l s ,1 9 8 1 , M e v a r e c h , 1 9 B 3L)e. o na n d
S t u d e n t sI' n t u i t i v e S t r a t e g i e si n J u d g i n gA s s o c i a t i o n
27
Z a w o j e w s k i( 1 9 9 1 ) c o n s i d e r t h a t t h i s i n d i c a t e sa r e l a t i o n a lk n o r i ' l c d g co f t h e n t e a n
( S k e n r p 1, 9 7 8 ) , c o n s i s t i n gi n h a v i n g s u i t a b l es c h e n t e so r e n o u g hc o n c e p t u a sl t r u c t u r e s
a v a i l a b l e t o s o l v e a w i d e r a n g e o f p r o b l e m s w h e r e t h e c o n c e p t i s n e c e s s a r l ' .T h e s a m e
c a n b e s a i d o f s t u d e n t s c o m p a r i n g t o t a l s ( I 6 . 4 o / o ) s, i n c e t h e s e t o t a l s a r c v a l u e s r L ' p r e s e n t i n g t h e s e t o f d a t a . M o r e o v e r ,c h o o s i n gb e t w e e nv a r i o u s s u n l m a r i u si n a c o n c r e t e
c o n t e x t r e q u i r e s k n o w l e d g eo n r e l a t i o n s b e t w e e nd i s t r i b u t i o n s a n d s u n r m a r i c sa n d
f u n c t i o n a l i n t e r p r e t a t i o n s o f s u m m a r i e s( B i e h l e r , 1 9 9 7 ) .
T h e s t r a t e g y b a s e d o n c o m p a r i n g p e r c e n t a g e s( s t r a t e g y 3 ) t r a n s f o r m e d t h e o r i g i n a l
p r o b l e m i n t o a c o n t i n g e n c yt a b l e 2 x 2 , w h e r e i t w a s p o s s i b l e t o u s e a p r o b a b i l i i l
c o m p a r i s o n .T h e r e f o r e , t h e s e s t u d e n t s a p p h e d g e n e r a l h e u r i s t i c s p r o b l e m - s o l v i n g .
O n e o f t h e m o s t f r e q u e n t q u e s t i o n st h e r e s e a r c h e rf a c e s w h e n t w o s a n p l e s a r e
c o m p a r e d ,i s w h e t h e r t h e s e s a m p l e sc o m e f r o n t h e s a m e p o p u l a t i o no r n o t . T h i s i s
r v h a t t h e s t u d e n t s w h o c o m p a r e dd i s t r i b u t i o n s d i d i n t u i t i v e l v ( s t r a t e g ) ' 4 ) . F r o n r r , ' u r
r e s u l t s w e c a n i n f e r t h a t t h e s t r a t e g y o f c o m p a r i n g c o r r e s p o n d i n gv a l u e s i n r e l a t e d
samplescan be found, at different levels of elaboration in the 87.9 per cent of
s t u d e n t s ( S t r a t e g i e s5 , 6 a n d 7 ) .
W h e n t h e d a t a a r e g l o b a l l y c o m p a r e d( s t r a t e g yB ) t h e s t u d r ' n ti s i m p l i c i t l r a c ( r p t i n g t h a t a l l t h e d a t a a r e r e l e v a n t ,w h i c h i s a b a s i c s t e p i n t h e a s s o c i a t i o nj u c i g e n t r n t
( C r o c k e r ,1 9 8 1 ) .
Misconceptions Found
T h e d e t e n n i n i s tc o r t c e p t i oot ft a s s o c i a t i o na p p e a r e d w h e n t h e s t u c l e n te r l r e c t c r lt h e
v a r i a t i o n i n t h e d a t a t o a l w a y s h a v e t h e s a m e s i g n ( 2 5 ( I l . 7 7 o l o f s t u d r n t s r rs i n g
s t r a t e g y 5 ) o r w h e n h e ¡ s h e e x p e c t e d s i m i l a r v a l u e s o f t h e d e p e n d e n tv a r i a b l c ' i n r ' a c h
sample(strategl' 9).
T h e l o c a l c o r t c e p t i o ct ta n b e n o t i c e d w h e n t h c s t u d e n t u s e d i s o l a t e c lc a s e s t o d e c i d e
h i s / h e r a s s o c i a t i o nj u d g e m e n t ;f o r e x a n p l e , w h e n u s i n g e x c e p t i o n a ci ¿ r s c (ss t r a t e g ] ' 6 )
o r c o m p a r i n g e x t r e n r e si n b o t h s a m p l e s ( s t r a t e g y 1 0 ) . T h r s e s t u d e n t s h a ' u ' en o t
a d o p t e dt h e p r o p e n s i t l ' p e r s p e c t i v ed e s c r i b e db y K o n o l d e t a l . ( 1 9 9 7 ) a n d m a k e s c ' n s e
o f s t a t i s t i c s b y u s i n g m o r e f a m i l i a r , b u t i n a p p r o p r i a t e c o n r p a r i s o ns c h e m e s .
28
AntonioEstep¡..CarnenB.rrrsenoanclF. T. S.rr-cHrz
CONCLT]SIONS
O u r r e s u l t s s h o w a m i x t u r e o f c o r r e c t a n d i n c o r r e c t p r e c o n c e p t i o n sa n d s t r a t e g i e s
i n t h e J u d g m e n to f a s s o c i a t i o nn ' h e n c o m p a r i n gs a m p l e s .\ \ ' e a l s o d e d u c et h a t c o m p a r i n g
s a n p l e s t o j u d g e a s s o c i a t i o ni s a h i g h l v s k i l l e d a c t i i i t y ' e \ : e na t a n e x p l o r a t o r v l e v e l ,
r e q u i r i n g a v a r i e d s t a t i s t i c a l k n o w l e d g e .I t a l s o i m p l i e s s e l e c t i n gt h e b e s t s t a t i s t i c a l
p r o c e d u r e ,a d e q u a t ei n t e r p r e t a t i o no f r e s u l t s a n d r e l a t i n g t h e m t o t h e r e s e a r c hq u e s l i o n s c o n c e p ta n d u s i n g i t i n p r o b l e n is o l r i n g . A l i o u r f i n d i n g s s h o r vt h e c o m p l e x i t l ' o f
a t o p i c s i m p l e i n a p p e a r a n c eD
. u e t o t h i s c o m p l e x i t l ' a n d t h e r e l e v a n c eo f u n d e r s t a n d i n g
the idea of associationw
, e c o n c l u d et h a t t h e r e i s a n e e d t o e r t e n d t h i s r e s e a r c h a n d
r e i n f o r c e t h e t e a c h i n go f t h i s t o p i c a t u n i r e r s i t l ' l e v e l a n d i n t h e f i n a l 1 ' e a r so f s e c o n d a r ¡ '
s c h o o l . T h i s w i l l r e q u i r e i n f o r m i n g t e a c h e r s a b o u t t h e m a t h e m a t i c a l ,p s ¡ , ' c h o l o g i c a l .
a n d e d u c a t i o n a al s p e c t so f t h e t o p i c , i n c l u d i n g i n f o r m a t i o na b o u t s t u d e n t s 'p r e c o n c e p t i o n s
as set out in this article.
A c k n o w l e d g m e n tT: h i s r e p o r t f o r m s p a r t o f t h e P r o j e c t sP B 9 6 1 . 1 1 1 a n d P B g 7 0 8 5 1 ,
g r a n t e d b 1 ' S e c r e t a r i a d e E s t a c l od e L l n i v e r s i d a d e s ,I n v e s t i s a c i o n t ' D e s a r r o l l o
(Nf.E.C.M
). adrid (Spain).
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snd
p r e c o n c e p t i o nasb o u ta s s o c i a t i o inn c o n t i n g e n ctya b l e s J. o u n n lf o r R e s e a r cuht
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of statisticalassociation
in a computer-based
teachingenvironment.
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on the Rr,¡le
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a t t d l e e r n i n gS t a t í s t i c s1. 9 9 6 I A S E R o u t t dT a b l eC o n f e r e n cPea p e r s(.p p . 1 6 9
S t u d e n t s I' n t u i t i r . ' eS t r a t e g i e si n J u d g i n gA s s o c i a t i o n
29
1 9 0 ) , I n t e r n a t i o n a lS t a t i s t i c a lI n s t i t u t e .\ - o o r b u r g .
B r e n n a n , R . L . , ( 1 9 8 3 ) . E I e n r c t ú so . i O c n c r a l i z u b í l i t .Tt 'l t e c t r yA. C T P L r b l i c a t i o n sI,o w a .
C h a p m a n ,L . J .i ' C h a p n r a n J. . P . .t 1 9 6 9 1 .I l ) u s o r - 1 ' c o l ' r ' e l a t :i o
r sna n o b s t a c l ct o t h e u s e
o f v a l i d P s v c h o d i a g n o s t isci g n s .. [ o u n n l o , i. f l t t n r n t o l P s l ' c l t o l o gi r '+. . 2 7 1 2 8 0 .
C o n f r e l ' ,J , ( 1 9 9 0 ) . A r e v i e u ' o f t h e r e s e a r c ho n s t u d e n t sc o n c e p t i o r . ri rsr \ l a t h e n a t i c s ,
S c i e n c e ,a n d P r o g r a m m i n g .R e u i e wo J ' R e s e o r c il t t E d u c a t i o n ,1 6 , 3 5 5 .
C r o c k e r . J , ( 1 9 8 1 ) . J u d g m e n to f c o v a r i a t i o n b ¡ . 's o c i a l p e r c e i v e r s .P s , r ' c f t o l o g i cBt ttlt l l e t t t t .
90.2,',¿72-',¿q2.
D o u a d ¡ ' ,R . ( 1 9 8 6 ) , L e j e u d e C a d r e s e t d i a l e c t i c ¡ u oe u t i l o b j e t d a n s l ' e n s e i g n e m e n t
. e t o f p a t t e r n s a n d t o o l - o b j e c ti n t e a c h i n gm a t h e n r a t i c s ] .
d e s M a t h e n r a t i q u e sI S
R e c l t e r c l t eest t D i d o c t i q u ed e s m a t l t t , n u t t i q u e s . 7 . 25, 3 1 .
E s t e p a ,A . c t B a t a n e r o ,I v l .C . . ( 1 9 9 6 ) , J u d g e m e n t so f c o r r e l a t i o n i n s c a t t e r p l o t s :
s t u d e n t s ' i n t u i t i v e s t r a t e g i e s a n d p r e c o n c e p t i o n s .H i r o s h i u t aJ. o u n n l o f M a t h e n r u '2I-4I.
t i c s E d u c a l i o t t ,4 ,
G o d i n o ,J . D . & B a t a n e r o ,N f . C . , ( 1 9 9 , 1 ) ,S i g n i f i c a d oi n s t i t u c i o n a l1 ' p e r s o n a ld e l o s o b j e t o s m a t e n l a t i c o s[.n s t i t u t i o n a l a n d p e r s o n a l m e a n i n go f t h e n a t h e n l a t i c a lc o n ,
c e p t s . l R e c l t a r c l t eesu D i d a c t i q u e
, . 3, 325 355.
s c l e sl l a t l t t , n t a t i q u e sI 4
H a w k i n s , 4 . , J o l l i f f e ,F . y ' G l i c k n a n , L . , ( 1 9 9 2 ) , T e a c h ü t gs t a t í s t i c a cl r n t c e p tLs o. n g n r a n ,
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D a t a - b a s e dv e r s u s t h e o r l ' - b a s e dj u d g e n e n t s .I n D . K a h n e m a n ,P . S l o v i c & A .
T v e r s k v ( E c l s ) J, u d g n e n t u n d e r t t n c e r l e i t t t t ,H: e u r i s t i c sa t t d b i a s e s(, p p . 2 1 1 2 3 0 ) .
C a r n b r i d g eL l n i r , e r s i t ) 'P r e s s , N e w Y o r k .
K o n o l d ,C . , P o l l a t s e k A
, . . \ \ ' e l l . A . & G a g n o n ,A . ( 1 9 9 7 ) . S t u d e n t sa n a l ¡ , ' z i ndga t a :R e s e a r c h
o f c r i t i c a l b a r r i e r s . i n j . G a r f i e l d & G . B u r r i l l ( E d s ) , R e s c c t r col t t t l t eR o l eo . f
Tecltttologltin Teachittgard leanúng Sfaflstics. 1996 IASE Rrnuul T'ublt'CotLierettce
P a p e r s( p p . 1 5 1 1 6 7 ) , l n t e r n a t i o n a l S t a t i s t i c a lI n s t i t u t e ,\ ' o o r b u r s .
L e o n , M . R . a n d Z a w o j e w s k i ,J . S . ,( 1 9 9 1 ) , L i s e o f t h e a r i t h m e t i c m e a n :a n i n v c s t i g a t i o n
o f f o u r p r o p e r t i e s i s s u e s a n d p r e l i n i n a r ¡ - r e s u l t s . E n D . \ ' e r - e - J o n e (sE, d . ) ,P r o
c e e d i t t g so - it l t e T h i r d I n t e n n t i o n a l C o u " f e r e n coer t T e a c l t i u gS l a l i s f l c s .\ ¡ o l . 1 , ( p p .
3 0 2 3 0 6 ) . O t a g o L . l n i v e r s i t yP r e s s , N e u ' Z e l a n d .
N l e v a r e c hZ. . R . , ( 1 9 8 3 ) . A d e e p s t r u c t u r e n l o d e lo f s t u c l c n t ss' t a t i s t i c a ln r i s c o n c e p t i o n s . E d ¡ ¿ ctri¿o t r u l S t u d i e sü t , \ l a t l t ¿ n a t i c s .I l . I I á 4 ' 2 9 .
N i s b e t t , R . & R o s s , L . , ( 1 9 8 0 ) , H u m a n i n - i e r e t t c es :t r a t e g i c sa u d s l t o r t c o w í r t go f s o c i a l
j u d g e m e r t t sP. r e n t i c e H a l l , N e w J e r s e ¡ ' .
P o l l a t s e k ,4 . , L i m a , S . a n d \ \ ' e l l , 4 . D . , ( 1 9 8 1 ) , C o n c e p to r c o m p u t a t i o n :S t u d e n t s '
30
A n t o n i oE s r e p ¡ . .C a r m e nB l r l x ¡ R o a n d F . T . S A N c u s z
understandings
of the mean.Educational
Studiesitt Ma"tltematics,
12,191 20-1.
S k e m pR
, , ( 1 9 7 8 ) ,R e l a t i o n aul n d e r s t a n d i nagn d i n s t r u m e n t aul n d e r s t a n d i n sA, r. i t h
metic TeacJtsr,
Nov.,9 15
V e r g n a u dG
, . ,( 1 9 8 2 ) .C o g n i t i v ea n d d e v e l o p m e n t aPls y c h o l o g ya n d r e s e a r c hi n
M a t h e m a t i c sE d u c a t i o n s: o n e t h e o r e t i c a la n d m e t h o d o l o s i c ai sl s u e s .F o r t h e
leanúttgof ntathenntics,
3, 2, 3 1-41.
W r i g h t ,J . C . & M u r p h l ' ,G L , ( 1 9 8 4 ) ,T h e u t i l i t y o f t h e o r i e si n i n t u i t i v es t a t i s t i c s :
the robustnessof theory-basedjudgements.
Psychology,
Jounnl o-iExperirnental
1 1 3 ( 23) ,0 r 3 2 2 .
Antonio Estepa Uniuersity oi Ja'en
Facwlty-o.i Htunanities and of Sciencesof the Educatiott
Departnterrto.fDidactic of the Sciettces
Campwsof the Lagwrtillqs, s/tt. Buildirtg D.2
23071 Jabn
Spain
e -tnuil: aestepa@ wjaen.es
Carmen Batanero
LIrúuersit1,of Granada
Facwlty of Sciencesof the Edwcatiorr
Departruentoi X4athentaticsEducatiott
Cantpus o,i Cartuja
18071 Grattada
SPaitt
e-ntail: batanero@ gotiat.wgr.es
Francisco T. Sánchez Linit'ersity of ]abn
School Polytechrtic
Departntent of Mathematics
Auentte of Madrid, s/n.
23071 Jabn
Spain