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Proceedings of the 28th Conference of the International
Group for the Psychology of Mathematics Education, 2004 Vol 2 pp 111–118
THE IMPACT OF TEACHERS’ PERCEPTIONS OF STUDENT
CHARACTERISTICS ON THE ENACTMENT OF THEIR BELIEFS
Kim Beswick
University of Tasmania, Australia
This paper reports on one aspect of a larger study and comprises an analysis of the
beliefs concerning mathematics, its teaching and its learning, and the classroom
practice of one secondary mathematics teacher. It focuses on the question, “What
specific teacher beliefs about students are relevant to teachers’ classroom practice in
various classroom contexts?” The teacher’s practice was examined in relation to
several of his mathematics classes and significant differences, consistent with the
teacher’s beliefs in regard to the various classes, were found. The findings confirm
the contextual nature of beliefs and highlight the importance to teachers’ practice of
specific teacher beliefs about the various students that they teach.
BACKGROUND AND THEORETICAL FRAMEWORK
A fundamental premise of teacher beliefs research has been that an individual’s
behaviour is ultimately a product of his/her beliefs (Ajzen & Fishbein, 1980; Cooney,
2001). Consequently, any attempt to change the practice of teachers must, of
necessity, involve change in the beliefs of teachers. Teachers’ beliefs have, therefore,
long been regarded as critical to the reform of mathematics education (Cooney &
Shealy, 1997). Despite this there is no agreed definition of the concept of beliefs
(McLeod & McLeod, 2002). It is thus the responsibility of researchers in the area to
make clear the meaning that they attach to the term (Pajares, 1992). In this paper
“beliefs” is used to mean anything that a person regards as true, and is essentially the
meaning assigned to the word by Ajzen and Fishbein (1980). Furthermore, since
beliefs must necessarily be inferred (Pajares, 1992), more certainty can be attached to
the existence of a belief that is evident in both the words and the actions of an
individual. Indeed, the degree to which a subject’s actions and statements in other
contexts are compatible with a given stated belief, the more centrally held (Green,
1971) that belief is likely to be. It is also recognised that individuals may hold beliefs
that they do not articulate for a variety of reasons, including the fact that they may
not be consciously aware of their existence (Buzeika, 1996).
Wilson and Cooney (2002) observed that since the 1980s context has been
increasingly recognized as relevant to studies of teaching and learning and that the
teacher’s beliefs in fact constitute part of the context in which classroom activity
occurs. In their theory of planned behaviour, Ajzen and Fishbein (1980) emphasised
the context specificity of beliefs and Green (1971) also asserted the relevance of
2–112 PME28 – 2004
context to the enactment of beliefs, suggesting that the relative strength with which
various beliefs are held is dependent upon the particular context.
Contextual constraints have also been recognised as exerting significant influence on
the relationship between beliefs and practice (Sullivan & Mousley, 2001) while
Hoyles (1992) described all beliefs as situated as a consequence of their being
constructed as a result of experiences which necessarily occur in contexts. Hoyles
(1992) argued that it is thus meaningless to distinguish between espoused and enacted
beliefs or to examine the transfer of beliefs between contexts since differing contexts
will, by definition, elicit different beliefs. Thus, rather than contextual factors
constraining teachers from implementing certain of their beliefs, such factors in fact
give rise to different sets of beliefs which are indeed enacted. Such a view is
consistent with that of Ajzen and Fishbein (1980). Pajares (1992) also stressed the
contextual nature of beliefs and the implications of their being held, not as isolated
entities, but as part of belief systems as described by Green (1971).
Context is thus relevant to both the development and the enactment of teachers’
beliefs, as well as to the particular beliefs that are relevant in a given situation. Hence
an important challenge for researchers is to identify specific teacher beliefs that
significantly impact their practice and that, while context specific, are relevant across
a sufficiently broad range of contexts to be generally applicable. Hoyles (1992)
described the emergence within PME of research that contributes to this end. In
particular, she cited Romberg (1984) as identifying a relationship between teachers’
beliefs about students’ ability and the nature and difficulty of the tasks that they
assign to them. Hoyles (1992) also called for more attention to be paid to the study of
teachers’ beliefs as they exist in relation to various specific contexts and particularly
in relation to the characteristics of their students.
In spite of this there is still little knowledge regarding specific teacher beliefs in
relation to students that are likely to be helpful or otherwise in the creation of
classrooms that reflect the principles of mathematics education reform. Exceptions
include the finding of Stipek, Givvin, Salmon and MacGyvers (2001) that teachers
who believe that students’ mathematical ability is fixed are more likely to hold
traditional views of mathematics teaching, and Cooney, Shealy and Arvold’s (1998)
findings regarding the beliefs of pre-service secondary mathematics teachers. While
acknowledging the context specificity of beliefs they identified a number of beliefs
that such teachers tend to hold about themselves and their role as a teacher and also
made use of aspects of Green’s (1971) description of belief systems in accounting for
both the varying impacts of these beliefs on teachers’ practice and their susceptibility
to change. This paper reports on an examination of the variations between classes in
the beliefs and practice of an individual experienced mathematics teacher, enabling
insight into the nature and place within the structure of the teacher’s beliefs system,
of his beliefs with respect to students.
PME28 – 2004 2–113
Practice, in the current study, was considered in terms of the extent to which the
teacher’s classroom environments could be characterized as constructivist. This was
done cognisant of the facts that constructivism is not prescriptive in relation to
teaching (Simon, 2000) and that any teaching strategy could be part of a
constructivist learning environment (Pirie & Kieren, 1992). Rather a constructivist
classroom environment was considered to be one in which: students were able to act
autonomously with respect to their own learning; the linking of new knowledge with
existing knowledge was encouraged and facilitated; knowledge was negotiated by
participants in the learning environment; and the classroom was student centred in
that students have opportunities to devise and explore problems that are of relevance
to them personally (Taylor, Fraser & Fisher, 1993). Such elements align well with the
principles and standards promoted by the National Council of Teachers of
Mathematics (NCTM) (2000).
THE STUDY
The subject
Andrew had been teaching secondary mathematics and science for 25 years. He had
studied mathematics for three years at University as part of his B.Sc. and had since
completed an M.Ed. Andrew was currently teaching mathematics to two classes in
grade seven and one in grade ten. Both of the grade seven classes were heterogeneous
while the majority of students in the grade ten class were in the average ability stream
with a few studying a separate mathematics course designed for low ability students.
Instruments
Data concerning Andrew’s beliefs were collected using a survey requiring responses
on a five-point Likert scale, to twenty six items relating to beliefs about mathematics,
its teaching and its learning, and from a semi-structured interview of approximately
one hour’s duration. The survey items were taken from similar instruments devised
by Howard, Perry, and Lindsay (1997) and Van Zoest, Jones, and Thornton (1994)
and were originally part of a forty-item survey that was shortened after use in a pilot
study. The audio-taped interview required Andrew to: reflect upon his own
experiences of learning mathematics; describe an ideal mathematics classroom and
compare this with the reality of his own mathematics classes; respond to 12
statements about the nature of mathematics based upon the findings of Thompson’s
(1984) case studies of secondary mathematics teachers; and respond to a further 12
statements about the teaching and learning of mathematics derived from the same
source. The 12 statements regarding the nature of mathematics consisted of four each
that represented Problem Solving, Platonic, and Instrumentalist views of mathematics
as defined by Ernest (1989), and the 12 statements relating to the teaching and
learning of mathematics were similarly representative of three corresponding views
of mathematics teaching and learning.
2–114 PME28 – 2004
Observations of approximately six lessons with the classes in each grade provided
data on Andrew’s classroom practice as well as further opportunities to gather data
from which his beliefs could be inferred. Data on Andrew’s classroom practice were
also gathered from the interview and from both teacher and student versions of
Constructivist Learning Environment Survey (CLES) described by Taylor et al.
(1993) and requiring respondents to indicate on a five-point Likert scale, their
perceptions of the frequency of various teaching/learning practices in their
mathematics classroom. The CLES measures the four aspects of a classroom
environment described above and respectively named Autonomy, Prior Knowledge,
Negotiation and Student-Centredness (Taylor et al., 1993).
Procedure
Andrew completed the beliefs survey during the first few weeks of the school year.
After a gap of several weeks he was asked to complete the teacher version of the
CLES with respect to at least of two of his mathematics classes (one grade seven and
the grade ten), and then to give the student version of the survey to students in these
classes. Interviews were conducted in early October and observations of Andrew’s
mathematics lessons occurred throughout November and December. Inferences
concerning Andrew’s beliefs were made on the basis of the complete data set. That is,
Andrew’s interview transcript and the detailed notes made during and after each
observation period were examined for evidence supporting, contradicting or
clarifying his belief survey responses and the CLES responses of both Andrew and
his students. A set of five centrally held beliefs that emerged as most relevant to
Andrew’s practice were suggested and put to him, along with details of the data
analysis, for comment and verification.
Results and discussion
Andrew’s belief survey responses indicated that he held a Problem Solving view of
mathematics (Ernest, 1989) and a constructivist view of mathematics learning. This
was exemplified by his agreement or strong agreement with statements such as the
following:
Mathematics is a beautiful, creative and useful human endeavour that is both a way of
knowing and a way of thinking.
Ignoring the mathematical ideas that children generate themselves can seriously limit
their learning.
A vital task of the teacher is motivating children to resolve their own mathematical
problems.
However, Andrew seemed unsure as to the most effective pedagogical approach to
employ in enacting those beliefs. For example, he was undecided about the following
items:
PME28 – 2004 2–115
Mathematical material is best presented in an expository style: demonstrating, explaining
and describing concepts and skills.
Providing children with interesting problems to investigate in small groups is an effective
way to teach mathematics.
Students in one of Andrew’s grade seven classes and his grade ten class completed
student versions of the CLES. Andrew opted to complete just one CLES (teacher
version) survey rather than one for each class. The three sets of responses were
similar for three of the four scales with the exception being the extent to which the
classroom environments were perceived to be Student-centred. Both classes
perceived their classrooms to be less Student-centred than did Andrew, with the
difference being greatest in the case of the grade tens. Individual items that
contributed to these differences are shown in Table 1. While individual differences
are small, the data suggest that in his grade ten class Andrew was more likely than in
his grade seven class to set the tasks and to be the arbiter of correct solutions.
Table 1: Items contributing to differences in Student-Centredness
In this class… Teacher
Grade seven
(av. response)
Grade ten (av.
response)
I/the teacher give the students
problems to investigate
Seldom Sometimes Often
The activities students do are set
by me/the teacher
Often Often Very often
Students learn my/the teacher’s
method for doing investigations
Sometimes Often Often
I/the teacher show(s) the correct
method for solving problems
Often Often Very Often
Andrew’s interview responses confirmed his Problem Solving view of mathematics
and his constructivist view of learning as conveyed in his belief survey responses.
The following quotations are illustrative:
I’ve swung in the past from being a Platonist to being more a social constructivist. So
mathematics is, I don’t think it’s out there, I don’t think there is a number one sitting
somewhere in the ether however effectively it sits there because we as a society have
created it.
Mathematics to me is for exploring, conjecturing…, well, there’s probably no such thing
as right answers to any problem.
His ambivalence regarding teaching approaches was also evident in that he expressed
agreement that one of the ways that students learn is “by attentively watching the
teacher demonstrate procedures and methods for performing mathematical tasks, and
by practising those procedures”, but was careful to stress that this was just one way
2–116 PME28 – 2004
that they learn, and that “there’s got to be a big balance”. Andrew described his own
teaching as follows:
I suppose I’m very teacher directed but at the same time what I like to do is not to give
the kids the answers, but what I try to do is to make them think …
Andrew talked primarily about his grade seven classes and described his grade tens
as “a totally different kettle of fish”. He described them as more difficult to motivate,
and also talked about what he regarded as the inappropriateness of much of the
content of the course designed for grade ten students in the middle ability level,
suggesting that these students needed “survival numeracy skills” and not “space and
all these other things”. In the lessons observed, the grade ten class worked from
worksheets on coordinates and mapping. Apart from brief introductions there was no
whole class teaching. Rather students worked with minimal noise either individually
or in twos or threes as they chose. Andrew rarely directly answered students’
questions about the mathematics, but instead made suggestions. Several of the
students did very little work, and this was largely ignored provided that their
behaviour was not disruptive.
In contrast with his grade ten, Andrew’s grade seven lessons consisted primarily of
whole class discussions facilitated and guided by Andrew. Both of the grade seven
classes seemed accustomed to being asked to explain their answers and comfortable
with going to the board to write their answers. Questions such as, “How did you do
it?”, “Is it right?” and “Will it always work?” were recurrent with the students
sometimes using them too. The students consistently appeared to be engaged in
genuinely grappling with the meaning of the mathematics. In one lesson on
multiplying fractions, Andrew allowed the students to run with a conversation
without comment until a student articulated that the effect of multiplying by a
number greater than one makes it bigger and multiplying by a number smaller than
one makes it smaller. Such conversations remained orderly, with one person speaking
at a time and everyone else listening.
Andrew’s beliefs and practice
Three beliefs emerged as centrally held and relevant to Andrew’s teaching in both the
grade seven and grade ten classes. These were:
1. The teacher has a responsibility to maintain ultimate control of the classroom
discourse.
2. The teacher has a responsibility actively to facilitate and guide students’
construction of mathematical knowledge.
3. The teacher has a responsibility to induct students into widely accepted ways of
thinking and communicating in mathematics.
However, although Andrew’s teaching in both contexts was consistent with
constructivist principles there were clear differences. Andrew agreed that he was less
PME28 – 2004 2–117
inclined than with the grade seven classes to make the effort to maximize the
engagement of all students in the grade ten class, or to establish the social norms
required to make whole class teaching effective for all students. Two further
centrally held beliefs underlay this, namely:
4. Older students of average ability are not interested in mathematics.
and,
5. Mathematics that is suitable for older students of average ability is not interesting.
CONCLUSION
While the teacher in this study held beliefs that were essentially consistent with the
aims of the mathematics education reform movement it is clear that his beliefs in
relation to older students of average ability had a significant impact on his practice in
their lessons and in fact limited the extent to which at least some students this class
were likely to engage in mathematical thinking as embodied in the NCTM’s (2000)
process strands. Thus if our aim is to promote teaching that is consistent with a
constructivist view of learning then it is insufficient to assist teachers to develop
beliefs that are considered helpful to this end without attending to other beliefs that
they may hold in relation to specific contexts.
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