PME2001 Utrecht - Dipartimento di Matematica

‘something called attitude plays a crucial role in learning
mathematics’ [Neale, 1969]
What does ‘positive’ attitude really
mean?
Pietro Di Martino & Rosetta Zan
Department of Mathematics
University of Pisa - ITALY
RESEARCH ON ATTITUDE
AFFECTIVE DOMAIN
Mc Leod, 1992
EMOTIONS
De Bellis & Goldin, 1999
BELIEFS
VALUES
The affective domain in M.E.
Research on affect in M.E. nasce from the need
of taking into account the strong interaction between
cognition and affect (s. Adams & Mc Leod, 1989)
…man mano che questa ricerca cresce viene sottolineato
the need of:
Clarifying the meaning
of the
A theoretical framework
words and the constructs
for affect
A deeper understanding of the interactions:
between cognition and affect
among the affective factors
In this communication we will face these problems
A theoretical framework
for affect
Clarifying the meaning
of the
words and the constructs
A deeper understanding of the interactions:
between cognition and affect
among the affective factors
attraverso l’analisi del costrutto di ATTITUDE,
e più in particolare di POSITIVE ATTITUDE
1
In social psychology:
SOCIAL PSYCHOLOGY
- Mancano per lo più definizioni esplicite
- Quelle che ci sono sono molto varie
- L’enfasi è sul measurement
However there is agreement in seeing attitude as a
ATTITUDE
To face problems such as:
- acquisto di beni
- votazioni
predisposition to respond in a favorable /unfavorable
way to a certain object (cercare citazione)
FAVORABLE
UNFAVORABLE
PREVISIONE DI COMPORTAMENTI
…in situazioni di SCELTA
TRIPARTITE MODEL
(Eagly & Chaiken, 1998):
•a cognitive component
•an affective component
•a behavioural component
ATTITUDE
A predisposition to respond...
FAVORABLE
UNFAVORABLE
Anche in Mathematics Education...
L’enfasi è sul measurement piuttosto che
sulla theory
In most studies the tripartite model is
present, even if not explicitly, in the
observing / measuring instruments
Mancano per lo più definizioni esplicite
• quando ci sono sono molto varie
• fanno spesso esplicito riferimento al
tripartite model
2
TRIPARTITE MODEL
(Hart, 1989):
• beliefs
• emotions
• behaviour
TRIDIMENSIONAL DEFINITION
SIMPLE DEFINITION:
Attitude is a general emotional
disposition(Haladyna, …)
ATTITUDE
BIDIMENSIONAL MODEL
(Daskalogianni & Simpson, 2000
• beliefs
• emotions
BIDIMENSIONAL DEFINITION
- has been giudicata inadequate to face the
complexity of mathematical activity
- anche chi assume tale definizione di fatto poi
misura / osserva anche beliefs e behaviour
SIMPLE DEFINITION:
Attitude is a general emotional
disposition (Haladyna, …)
MEASUREMENT
In Mathematics Education
FAVORABLE
UNFAVORABLE
Cosa vuol dire nel contesto dell’attività matematica
rispondere in a
favorable / unfavorable way?
Ad esempio nel problem solving?
Non è una dicotomia ‘efficace’, e difatti è sostituita da
POSITIVE
NEGATIVE
…che non fa riferimento esplicito ai comportamenti
…But:
Is the notion of
‘positive’ attitude toward mathematics
capable of taking into account:
- the deep interaction among beliefs /
emotions / behaviour that is the basis of the
tripartite model?
- more generally the deep interaction
between affect and cognition che ha fatto
nascere research on affect in M.E.?
3
In M.E., just like in social psychology, the characterization
of attitude as POSITIVE is for the most part linked to a
MEASUREMENT and therefore SCORE.
Thurston or Likert scales,
…semantic differential technique
In qualitative type studies the characterization of attitude
as positive / negative is not so frequent
When it is present, it is generally accompanied by a
description of the factors (behaviour, beliefs, emotions) from
which is inferred.
‘positive’ can be referred:
- to a single item
- to a set of items (subscale)
- to the entire questionnaire
[Fennema, 1989, p. 208, quoting Fennema-Sherman
scales]
‘One positive item was
“I am sure that I can learn mathematics”
A negative item was:
“I’m no good in math.”…’
Traditional measurement of attitude
Using questionnaires (Likert scales, …):
• for every item a different score is assigned to
every possible answer
(total agreement: 5; ...
• a total score is obtained adding the scores
corresponding to the single items
• ‘positive’ is associated to a high score
In the case of ‘attitude’ the observation /
measurement (i.e. the items) usually refers to:
• Emotions
i.e.: ‘I like problem solving’
• Beliefs
i.e.: ‘Mathematics is useful’
• Behaviour
i.e. ‘I avoid maths because I am not very good
with figures’ CERCARNE UNO ‘PURO’,cioè in cui c’è
SOLO comportamento. Mi ricordo uno tipo: ‘Mi capita di
pensare alla matematica anche fuori dalla scuola. DOVE???
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Our claim:
The meaning of the adjective ‘positive’ is different when it is
…when referred to EMOTION
referred to
EMOTIONS
Normally an emotion is considered ‘positive’
if it is perceived as
pleasurable
BELIEFS
…in this sense is considered a
ANXIETY
‘negative’ emotion
BEHAVIOUR
…when referred to beliefs
Normally a belief is considered ‘positive’
when it is shared by the experts
i.e. in a questionnaire a positive score is
assigned to the agreement with this belief
A ‘positive’ behaviour
• A behaviour is considered ‘positive’ when
it is thought as successfull
In the school context, a successful behaviour
is generally identified with a ‘high achievement’
HOW to assess achievement
Mathematics is useful
??cercarne uno
v.sopra
In standard tests
In problem solving
?
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BUT...
…the different meanings of ‘positive’ can be attached
to a same construct
In the case of beliefs...
POSITIVE
EMOTION
POSITIVE BELIEF
POSITIVE as emotion,
if it is pleasurable
can be considered
positive, if it is thought
to be associated to
is associated to
elicits
shared by experts
POSITIVE
EMOTION
POSITIVE
BEHAVIOUR
POSITIVE
BEHAVIOUR
PROBLEM 1:
The different meanings of ‘positive’, when attached
to a same construct, can give raise to different evaluations
If the researcher does not make explicit his/her choices, it becomes
problematic to interpret results obtained within a study, and to
perform comparisons with different studies.
ANXIETY
NEGATIVE as emotion,
in that it is unpleasurable
can be associated to
Need to make explicit:
What is the meaning of ‘positive’? (anche nella narrativa)
In the case of questionnaires:
POSITIVE
BEHAVIOUR
Why a positive / negative score to a certain item?
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COMMENTO: forse questa va tolta, riguarda le CONCLUSIONI
E anche questa (va tolta)
Ma questa esplicitazione non risolve altri problemi:
3. Ma soprattutto la valenza (positive / negative) di un’emozione
1. The criteria used to evaluate the emotional component of
associata ad un belief dipende dal soggetto
ATTITUDE are different from those used to evaluate its
cognitive or behavioural component,
and can result in contradictory evaluations
2. In particolare the emotion associated to a certain belief (that,
Descrizione della ricerca
according to the tripartite model, ‘fa parte’ dell’attitude)
can be ‘opposite’ rispetto al belief
E andrebbero sostituite con queste due
(segue)
But even if this ambiguity is overcome by making explicit the choices
…thus accepting a cause-effect model, according to which
made, in our opinion other problems remain
the same belief ‘causes’ the same emotion or behavior in all
individuals .
1. The separate observation of beliefs and emotions does not allow
researchers to pick up the emotional dimension of beliefs and the
cognitive dimension of emotions
The inadequacy of this model has emerged from a recent study
(Di Martino & Zan, 2002) in which we set the problem of
investigating the uinteraction between beliefs and emotions.
2. Apparently this interaction is considered when a belief is evaluated
(also??) according to the emotion that it elicits. But in this way it is
assumed not only that a certain belief has an emotional component,
but also the valence (positive / negative) of this component.
In this study, we considered some beliefs typically used to assess
attitude toward mathematics, and considered ‘positive’ in that
they are (thought to be) shared by experts
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We used two different questionnaires:
1 and 2
• The two questionnaires were administered
to different students
• The first 4 questions were common to
questionnaires 1 and 2
• The other questions were about various
beliefs
• For every belief, questionnaires 1 and 2
formulated the question differently
The formulation of the questions about
beliefs used in questionnaire 1
In mathematics there is always a reason for
every thing
It is not true that in mathematics there is
always a reason for every thing
And...
I like
I don’t like
I find
indifferent
…this characteristic of mathematics
Data from questionnaire 1
Questionnaire 2
• Was administered to 205 students
• The formulation of the questions about
beliefs:
In mathematics there is always a reason for every thing
It is not true that in mathematics there is always a reason
for every thing
In mathematics there is always a reason for
every thing
It is not true that in mathematics there is
always a reason for every thing
B
non B
I like
Why?
Can you give same examples?
+
I don’t like
-
I find
indifferent
0
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Two combinations appear particularly significant:
Data from questionnaire 1
B
In mathematics there is always
a reason for every thing
–
211 students from high school
(B, +)
(B, -) (nonB, +) (nonB, -)
(B, 0) (nonB, 0)
And I don’t like this
It isn’t true that in mathematics there is
always a reason for every thing
TOT
nonB +
95
27
17
B
14
38
20
211
-
And I like this
nonB
+
This case would be considered:
This case would be considered:
‘positive’ according to the cognitive component
‘positive’ according to the emotional component
‘negative’ according to the emotional one
‘negative’ according to the cognitive one
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CONCLUSIONS
• The reduction to a positive / negative
dichotomy brings up many types of
problems both on a theoretical and
didactical level
From a didactical point of view:
• The positive / negative dichotomy hides
extremely different situations and needs
under the same label
The negative emotional disposition of a pupil
who sees mathematics as made of nonsensical rules
The negative emotional disposition of a pupil
who has an epistemologically correct view of mathematics,
but a low sense of self-efficacy
At a theoretical level:
A problem of ambiguity:
• there can be different meanings of the
word ‘positive’ relating to attitude
the difficulty in comparing different studies
the necessity that the researcher makes explicit
his / her choices
A problem of coherence (??and completeness)
• the criteria used to evaluate the emotional and
the cognitive components are different, and can
result in contradictory evaluations
the limits of a positive / negative evaluation which
appears inadequate to confront the complexity that
characterizes the interaction between affect and cognition
The need of a multiple approach to assess attitude,
privileging the description in context
COMMENTO: Sono mescolate due cose: 1) il multiple approach
2) l’assessment invece della misura
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A problem of subjectivity:
• the cognition / emotion interaction depends
on the individual
The last conclusion refers to a question posed
by Ruffell et al.:
citazione
The limits of a normative model for attitude,
searching for universal laws and rules
for behaviour and for cause and effect explanation
The opportunity of an interpretive approach, searching for
understanding motives of intentional actions, rather than
to explain causes of behaviour
A quality
of the observed...
We distinguish between constructs:
- that are to be taken as a quality of those
observed
- and constructs created by researchers to
account for observations
An observer’s
construct ...
ATTITUDE
There is a shift, from seeing attitude as a construct… to
...that explains causes of
behaviour, thus enabling
researchers to predict behaviour
…useful to describe the interaction
between affect and cognition,
enabling researchers to understand
motives of intentional actions
Predicting behaviour
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• OPPURE::::
FINE
Further? conclusions derive (require to
make explicit??)
In realtà ogni osservazione:
- evidenzia dei limiti
- suggerisce qualcosa
Dovrei cercare di farlo vedere mettendo in
ordine
A parte fare il discorso epistemologico E dei
costrutti
1. From epistemological assumption for ME,
seeing ME as an autonomous field of
research, che prende i costrutti da altre
discipline ma li adatta ai propri problemi
2. From a distinction between constructs that
are to be taken as a quality of those
observed and constructs created by
researchers to account for observations
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Se attitude in psicologia sociale è nato per
spiegare e quindi predire i comportamenti in
situazioni semplici di scelta…
…le considerazioni presentate evidenziano i
limiti :
of the positive / negative dichotomy
di un approccio normativo all’attitude,
searching for universal laws and rules of
behaviour
nel contesto della ME, caratterizzata da
decisioni complesse piuttosto che da scelte
NO!!! Sono 2 discorsi diversi!!
• The positive / negative dichotomy appears
inadequate to confront the complexity that
characterizes the interaction between
emotions and beliefs, and between affect and
cognition
• In order to account for this complexity, it is
necessary:
• - to overcome this reduction, using a multiple
approach to assess attitude, privileging the
description in context
• C’è una doppia complessità:
• -la complessità delle DECISIONI nell’attività
matematica, che suggerisce una definizione
multidimensionale (QUESTO FORSE PUO’
ANDARE ALL’INIZIO), e coerentemente
una descrizione invece che una misura (i cui
limiti abbiamo visto prima: è il focus di tutta
la relazione)
• e per di più la SOGGETTIVITA’
dell’interazione fra affect e cognition
suggerisce un approccio interpretativo
According to this interpretive approach
Attitude become an observer’s construct:
- useful to take into account the interaction
between affect and cognition
- in particular to understand motives of
intentional actions (rather than to explain
causes of behaviour)
- cos’è
• come si osserva
•
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The limits of a normative model for attitude,
searching for universal laws and rules
for behaviour and for cause and effect explanation
ME as an autonomous field of research,
che prende i costrutti da altre discipline ma li adatta ai propri problemi
FINE
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