Beliefs and attitudes in mathematics teaching and learning
Trygve Breiteig, Barbro Grevholm, Kirsti Kislenko
Introduction
Beliefs play great role in mathematics learning and teaching. The learning
outcomes of students are strongly related to their beliefs and attitudes about
mathematics (Furinghetti & Pehkonen, 2000). Thus assessing or evaluating of
students’ mathematical knowledge must be made in awareness of their beliefs.
During the last twenty years the research area about beliefs and attitudes has
grown considerably, and many different countries have been included in the
research - the work of Erkki Pehkonen who has made several investigations in
Finland; Peter Kloosterman from United States of America, Günter Törner from
Germany, Gilah C. Leder from Australia, and many others. They have been
looking for answers to several different questions. For example, Kloosterman
asked (2002): “What do students think mathematics is and how does one learn
mathematics?”; Pehkonen & Törner (2004) asked: “How well does information
from different methodological sources and using different methodological tools
to investigate teachers’ beliefs of mathematics fit together?” and “Which method
is best suited to which aspect?”.
Before introducing our study we would like to emphasise the work of Alba G.
Thompson who has been investigating teachers’ beliefs extensively. She has
looked into teachers’ conceptions of mathematics and their relationship to
instructional practice. The issue of changing teachers’ beliefs, comments on
methodology, and descriptions of theoretical frameworks, are included in her
work (Thompson, 1992).
In our project the issue about students’ beliefs are of interest because we
want to find answers to questions about what Norwegian and Estonian students
think about mathematics as a school subject, about the way one learns
mathematics, and what is important in the study of mathematics. In this paper
we report from a small pilot study carried out in Norway and an earlier study in
Norway, which it is based upon.
Since 1995 the project KIM has collected national data on students’
understanding of key concepts in the national mathematics curriculum.
Students’ performances related to one particular area of mathematics, named
Measurements and units, were linked to their attitudes. 105 grade six classes
(2106 students) and 90 grade nine classes (2150 students) took part. Amongst
those students approximately 900 were selected according to their birthdays.
The study is based on data from 891 grade 6 students and 893 students in
grade 9. The same schools were asked to participate in the attitude study later
in that school year, which made it possible to compare the mathematics test to
the questions about their thoughts of mathematics and the teaching and learning
mathematics. The conclusions of this project were following: those pupils who
state a positive interest towards mathematics, on average, performed better on
the mathematics test than their fellow students. There was a strong significant
connection between the performance of the test and the self-confidence in both
grades (Streitlien, Wiik & Brekke, 2001).
Definition of beliefs
Leder and Forgasz claimed that
In everyday language, the term “belief” is often used loosely and synonymously
with terms such as attitude, disposition, opinion, perception, philosophy, and
value. Because these various concepts are not directly observable and have to
be inferred, and because of their overlapping nature, it is not easy to produce a
precise definition of beliefs. (Leder & Forgasz, 2002, p. 96).
Different researchers associate belief with motivation and conception.
Kloosterman (2002) sees the direct connection between belief and effort.
‘Student’s belief is something the student knows or feels that affects effort – in
this case effort to learn mathematics’ (p. 248). Moreover, Kloosterman (2002)
argues that student’s choices are on one hand based on beliefs and on the
other hand on personal goals. Thus, there is a close connection between beliefs
and choices. But sometimes the personal goals and the beliefs are at variance.
One major example is the learning of mathematics. Many students believe that
mathematics is boring, and strong effort is needed to learn it, but still find it
important for life. This is a paradox. The reason for seeing mathematics as
important can be practical – needs for a better profession and to some degree
for a better life. ‘Most youngsters know, as an empirical and sociological fact,
that mathematical competence – even if for unclear reasons – is a key to
attractive education and job opportunities’ (Niss, 1994, p. 377). Jens Højgaard
Jensen has marvellously expressed this idea in one sentence ‘Mathematics is
useless to me, but at the same time I know that I am useless without
mathematics’ (Niss, 1994, p. 377).
There have been two different notions about beliefs and conceptions in
literature. In one case the beliefs are understood as a subclass of conceptions
(Hart, 1989; Thompson, 1992) and on the other hand the conceptions are a
subset of beliefs (Pehkonen, 1994). One can explain the concept of
“conceptions” as an originations, comprehensions, ideas, rules, images etc.
The easiest to understand and thus the widest, is the definition given by
Rokeach (1972). He says: ‘a belief is any simple proposition, conscious or
unconscious, inferred from what a person says or does, capable of being
preceded by the phrase ‘I believe that…’’ (p. 113).
Rejecting the McLeod (1989) idea to watch the person’s affective domain as
an aggregate of beliefs, attitudes and emotions, there can be found the ideas
which claim that belief is only one part of attitude in different researches (Aiken,
1980; Rokeach, 1972). More or less they see three aspects within an attitude: a
cognitive component (beliefs and knowledge), an affective component
(emotions, motivation, feelings) and a performance/behavioural component
(actions). Here, the emotions are one component of attitude, and beliefs with
knowledge are seen as a cognitive component of attitude.
As long as there are different people there will be dissimilar views about
belief, attitude, emotions, meanings, mental images, concepts and so on. The
definition does not play a major part in research, and thus every scientist will
ascribe the importance of different aspects related to particular investigations. It
means that the definition is affected by the questions and the motive of the
research. Hence one cannot say that some definition is wrong and the other is
right, they can be considered to be more or less suitable.
Relationship between beliefs and knowledge
The two parts of the individual – the affective domain and the cognitive domain -
are inseparable and in complex connection. ‘The main difficulty has been the
inability to distinguish beliefs from knowledge, and the question is still
unclarified’ (Pehkonen, 1994, p. 27).
Thompson (1992) points out two ways to distinguish knowledge from belief –
‘degree of conviction’ (p. 129) and consensus. Firstly, beliefs can be held weakly
or strongly. One can claim: “the new mathematics teacher is nice but she has
not assessed us still so it can be changed” or “I know that the test in
mathematics will be hard”. Beliefs can compartmentalize as something uncertain
or certain, important or not so important. But one cannot say that one knows the
fact weakly or strongly. Water starts boiling at a hundred degrees Celsius at sea
level, and that’s it! Secondly, it is possible to believe something despite the
awareness that the others do not agree with it and think about it differently. For
example, “I believe gold can be found in North Pole”.
Underhill (1991) uses the word ‘knowing’ and takes the position that ‘knowing
is believing’ (p. 20). Whatever one knows or does not know is simply the same
that one believes and does not believe.
A. whenever we say we know something, we are simply asserting that we
believe something, whether it is about quality called “red” or the being
called “God” or the relationship “3+4=7”;
B. whatever we do not know, we may not know passively (we have no
belief; we have never been exposed to it!), or negatively (we believe it’s
opposite or some anti-belief or substitute belief; we believe something
other than that);
C. all that we know reflects our beliefs based on empirical data or reason or
faith; these might be thought to exist continuum.
Saying that “the delta parameter of the option price shows how quickly the
option price changes when the asset price is changing” only means that there is
general belief in that. It can be because some trustable person (expert) has said
so or because it has worked very well before, and why should it not work now
(importance of the long experiences), or one does not believe that there is
something else that will work better. Most scientific discoveries have started
from the point that somebody believes in its validity and universality. If one does
not believe in what one wants to prove then it is impossible to do that. We think
the belief in rightness of doing research is one of the basic components of being
scientist. To constructivists ‘knowledge without belief is contradictory’ (Confrey,
1990, p. 111).
One does not have to evaluate or justify the beliefs, it is something which
belongs to the person. But there is definitely a need to explain the ideas related
to knowledge because without justification it will not be accepted as knowledge.
If one starts to justify beliefs then, according to Plato, the result will be the
knowledge. ‘Knowledge is justified true belief’ (Mc Dowell, 1987, p. 94, and, in
Furinghetti & Pehkonen, 2002, p. 42).
Trygve Breiteig, Barbro Grevholm, Kirsti Kislenko
Introduction
Beliefs play great role in mathematics learning and teaching. The learning
outcomes of students are strongly related to their beliefs and attitudes about
mathematics (Furinghetti & Pehkonen, 2000). Thus assessing or evaluating of
students’ mathematical knowledge must be made in awareness of their beliefs.
During the last twenty years the research area about beliefs and attitudes has
grown considerably, and many different countries have been included in the
research - the work of Erkki Pehkonen who has made several investigations in
Finland; Peter Kloosterman from United States of America, Günter Törner from
Germany, Gilah C. Leder from Australia, and many others. They have been
looking for answers to several different questions. For example, Kloosterman
asked (2002): “What do students think mathematics is and how does one learn
mathematics?”; Pehkonen & Törner (2004) asked: “How well does information
from different methodological sources and using different methodological tools
to investigate teachers’ beliefs of mathematics fit together?” and “Which method
is best suited to which aspect?”.
Before introducing our study we would like to emphasise the work of Alba G.
Thompson who has been investigating teachers’ beliefs extensively. She has
looked into teachers’ conceptions of mathematics and their relationship to
instructional practice. The issue of changing teachers’ beliefs, comments on
methodology, and descriptions of theoretical frameworks, are included in her
work (Thompson, 1992).
In our project the issue about students’ beliefs are of interest because we
want to find answers to questions about what Norwegian and Estonian students
think about mathematics as a school subject, about the way one learns
mathematics, and what is important in the study of mathematics. In this paper
we report from a small pilot study carried out in Norway and an earlier study in
Norway, which it is based upon.
Since 1995 the project KIM has collected national data on students’
understanding of key concepts in the national mathematics curriculum.
Students’ performances related to one particular area of mathematics, named
Measurements and units, were linked to their attitudes. 105 grade six classes
(2106 students) and 90 grade nine classes (2150 students) took part. Amongst
those students approximately 900 were selected according to their birthdays.
The study is based on data from 891 grade 6 students and 893 students in
grade 9. The same schools were asked to participate in the attitude study later
in that school year, which made it possible to compare the mathematics test to
the questions about their thoughts of mathematics and the teaching and learning
mathematics. The conclusions of this project were following: those pupils who
state a positive interest towards mathematics, on average, performed better on
the mathematics test than their fellow students. There was a strong significant
connection between the performance of the test and the self-confidence in both
grades (Streitlien, Wiik & Brekke, 2001).
Definition of beliefs
Leder and Forgasz claimed that
In everyday language, the term “belief” is often used loosely and synonymously
with terms such as attitude, disposition, opinion, perception, philosophy, and
value. Because these various concepts are not directly observable and have to
be inferred, and because of their overlapping nature, it is not easy to produce a
precise definition of beliefs. (Leder & Forgasz, 2002, p. 96).
Different researchers associate belief with motivation and conception.
Kloosterman (2002) sees the direct connection between belief and effort.
‘Student’s belief is something the student knows or feels that affects effort – in
this case effort to learn mathematics’ (p. 248). Moreover, Kloosterman (2002)
argues that student’s choices are on one hand based on beliefs and on the
other hand on personal goals. Thus, there is a close connection between beliefs
and choices. But sometimes the personal goals and the beliefs are at variance.
One major example is the learning of mathematics. Many students believe that
mathematics is boring, and strong effort is needed to learn it, but still find it
important for life. This is a paradox. The reason for seeing mathematics as
important can be practical – needs for a better profession and to some degree
for a better life. ‘Most youngsters know, as an empirical and sociological fact,
that mathematical competence – even if for unclear reasons – is a key to
attractive education and job opportunities’ (Niss, 1994, p. 377). Jens Højgaard
Jensen has marvellously expressed this idea in one sentence ‘Mathematics is
useless to me, but at the same time I know that I am useless without
mathematics’ (Niss, 1994, p. 377).
There have been two different notions about beliefs and conceptions in
literature. In one case the beliefs are understood as a subclass of conceptions
(Hart, 1989; Thompson, 1992) and on the other hand the conceptions are a
subset of beliefs (Pehkonen, 1994). One can explain the concept of
“conceptions” as an originations, comprehensions, ideas, rules, images etc.
The easiest to understand and thus the widest, is the definition given by
Rokeach (1972). He says: ‘a belief is any simple proposition, conscious or
unconscious, inferred from what a person says or does, capable of being
preceded by the phrase ‘I believe that…’’ (p. 113).
Rejecting the McLeod (1989) idea to watch the person’s affective domain as
an aggregate of beliefs, attitudes and emotions, there can be found the ideas
which claim that belief is only one part of attitude in different researches (Aiken,
1980; Rokeach, 1972). More or less they see three aspects within an attitude: a
cognitive component (beliefs and knowledge), an affective component
(emotions, motivation, feelings) and a performance/behavioural component
(actions). Here, the emotions are one component of attitude, and beliefs with
knowledge are seen as a cognitive component of attitude.
As long as there are different people there will be dissimilar views about
belief, attitude, emotions, meanings, mental images, concepts and so on. The
definition does not play a major part in research, and thus every scientist will
ascribe the importance of different aspects related to particular investigations. It
means that the definition is affected by the questions and the motive of the
research. Hence one cannot say that some definition is wrong and the other is
right, they can be considered to be more or less suitable.
Relationship between beliefs and knowledge
The two parts of the individual – the affective domain and the cognitive domain -
are inseparable and in complex connection. ‘The main difficulty has been the
inability to distinguish beliefs from knowledge, and the question is still
unclarified’ (Pehkonen, 1994, p. 27).
Thompson (1992) points out two ways to distinguish knowledge from belief –
‘degree of conviction’ (p. 129) and consensus. Firstly, beliefs can be held weakly
or strongly. One can claim: “the new mathematics teacher is nice but she has
not assessed us still so it can be changed” or “I know that the test in
mathematics will be hard”. Beliefs can compartmentalize as something uncertain
or certain, important or not so important. But one cannot say that one knows the
fact weakly or strongly. Water starts boiling at a hundred degrees Celsius at sea
level, and that’s it! Secondly, it is possible to believe something despite the
awareness that the others do not agree with it and think about it differently. For
example, “I believe gold can be found in North Pole”.
Underhill (1991) uses the word ‘knowing’ and takes the position that ‘knowing
is believing’ (p. 20). Whatever one knows or does not know is simply the same
that one believes and does not believe.
A. whenever we say we know something, we are simply asserting that we
believe something, whether it is about quality called “red” or the being
called “God” or the relationship “3+4=7”;
B. whatever we do not know, we may not know passively (we have no
belief; we have never been exposed to it!), or negatively (we believe it’s
opposite or some anti-belief or substitute belief; we believe something
other than that);
C. all that we know reflects our beliefs based on empirical data or reason or
faith; these might be thought to exist continuum.
Saying that “the delta parameter of the option price shows how quickly the
option price changes when the asset price is changing” only means that there is
general belief in that. It can be because some trustable person (expert) has said
so or because it has worked very well before, and why should it not work now
(importance of the long experiences), or one does not believe that there is
something else that will work better. Most scientific discoveries have started
from the point that somebody believes in its validity and universality. If one does
not believe in what one wants to prove then it is impossible to do that. We think
the belief in rightness of doing research is one of the basic components of being
scientist. To constructivists ‘knowledge without belief is contradictory’ (Confrey,
1990, p. 111).
One does not have to evaluate or justify the beliefs, it is something which
belongs to the person. But there is definitely a need to explain the ideas related
to knowledge because without justification it will not be accepted as knowledge.
If one starts to justify beliefs then, according to Plato, the result will be the
knowledge. ‘Knowledge is justified true belief’ (Mc Dowell, 1987, p. 94, and, in
Furinghetti & Pehkonen, 2002, p. 42).
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