It is widely acknowledged that what teachers believe influences
their teaching, yet the focus of much professional learning remains on
influencing the specific practices and tools that teachers employ in
their classrooms. In this article it is argued that a greater and more
explicit focus on teachers' beliefs would be beneficial. To this
end an overview of aspects of our understandings of the nature of
beliefs is presented followed by findings from a recent study that
examined mathematics teachers' beliefs and their impact on
classroom practice. Finally, implications for mathematics teachers and
those involved in designing and implementing professional learning for
both teachers and pre-service teachers are suggested.
Belief systems
The idea of belief systems recognises that beliefs are not held in
isolation from one another but are in fact inter-related in complex
ways. Green (1971) provided a description of belief systems that is
still very useful. He described several dimensions of beliefs systems,
three of which are of relevance here. The first is the idea of
centrality. The centrality of a belief is a function of the strength and
number of its connections with other beliefs. Other beliefs may be held
because they are consequences of a central belief and any change in a
central belief would have important ramifications for the
individual's belief system and could be experienced as quite
unsettling. Centrally-held beliefs are thus relatively difficult to
change.
A second aspect of Green's description of belief systems is
the phenomenon of clustering. This means that beliefs with a system can
be held in groups that are isolated from other beliefs. A consequence of
this is that a person may hold beliefs that contradict one another
without being aware of the contradiction. According to Green (1971) such
clusters are likely to develop when beliefs are formed in disparate
contexts. An example might be a student's belief that he is a poor
mathematics student, formed perhaps on the basis of negative experiences
of school mathematics, held at the same time as a belief in himself as
mathematically competent formed as a result of experiences of part time
work in a retail context. The student may not be consciously aware of
one or other or both of these beliefs and may continue to believe both
in the absence of any experience that makes them explicit and stimulates
reflection on their contradictory elements.
The third aspect of beliefs relates to the basis on which they
held. The basis of a belief may be evidence, in which case the belief is
said to be evidentially held, or it may be held for other reasons such
as the perceived authority of its source, or because it is regarded as a
consequence of a another belief which may or may not be evidentially
held. Evidentially held beliefs are by definition susceptible to change
on the basis of evidence to the contrary, while non-evidentially held
beliefs are impervious to evidence and hence very resistant to change.
Implicit in both the centrality and clustering of beliefs is the
importance of context. The relative centrality of beliefs varies
according to the context. For example, in the context of a professional
learning session, a teacher might express a belief in the importance of
providing students with ready access to manipulatives as they engage
with mathematics, but in the context of his grade 8 classroom his belief
that the teacher must maintain control of classroom activity and the
related belief that this particular class would not use manipulatives in
the intended way could be more central. The result might be that
manipulatives are nowhere to be seen in that classroom. It is important
to recognise that this would in no way mean that there was any lack of
sincerity associated with the teacher's statement during the
professional learning session.
The notion of clustering provides an alternative explanation for
apparent contradictions between stated beliefs and practices like that
described above. It allows the possibility that a teacher might
simultaneously hold contradictory beliefs that have developed in
different contexts. Beliefs formed as result of his/her own experiences
of learning mathematics, those formed during teacher education, and
others that have developed as result of classroom experience may contain
contradictory elements that the teacher is unaware of.
The study
The study aimed to examine the connection between secondary
mathematics teachers' beliefs and their mathematics classroom
environments and was described in detail by Beswick (2005, in press). It
involved surveys of 25 teachers and 39 of their mathematics classes as
well as interviews with eight of the teachers and observations of
approximately 12 lessons taught by each of six of the interviewed
teachers. The first teacher survey asked them to indicate the extent of
their agreement with 26 statements about the nature of mathematics,
mathematics teaching and mathematics learning. A second survey, the
Classroom Learning Environment Survey (CLES) (Taylor, Fraser &
Fisher, 1993) sought their perceptions of their classroom environments
and asked them to rate the frequency of occurrence of various classroom
events. The teachers were asked to complete this survey twice with a
particular mathematics class in mind on each occasion. Several of the
teachers completed just one survey either because they believed that the
classroom environments of both classes were the same or because of time
constraints. The teachers administered a student version of the CLES to
the students in the two classes with respect to which they had completed
the teacher version.
The vast majority ([greater than or equal to] 88%) of the teachers
agreed or strongly agreed with statements such as the following:
1. A vital task for the teacher is motivating children to solve
their own mathematical problems.
2. Ignoring the mathematical ideas that children generate
themselves can seriously limit their learning.
3. It is important for children to be given opportunities to
reflect on and evaluate their own mathematical understanding.
4. It is important for teachers to understand the structured way in
which mathematics concepts and skills relate to each other.
5. Effective mathematics teachers enjoy learning and
"doing" mathematics themselves.
6. Knowing how to solve a mathematics problem is as important as
getting the correct solution.
7. Teachers of mathematics should be fascinated with how children
think and intrigued by alternative ideas.
8. Providing children with interesting problems to investigate in
small groups is an effective way to teach mathematics
It is important to note that, with the exception of number 8, none
of these statements prescribe any particular teaching strategy or
classroom arrangement. The teachers were less inclined to agree with
statements that did. For example, less than two-thirds of the teachers
agreed or strongly agreed with the following items:
9. It is the teacher's responsibility to provide children with
clear and concise solution methods for mathematical problems.
10. There is an established amount of mathematical content that
should be covered at each grade level.
11. It is important that mathematics content be presented to
children in the correct sequence.
12. Mathematical material is best presented in an expository style:
demonstrating, explaining and describing concepts and skills.
Number 12 more evenly divided the teachers than any other (32%
agreed or strongly agreed, 40 % undecided, 28% disagreed or strongly
disagreed) indicating a diversity of opinion, as well as considerable
uncertainty, regarding how beliefs such as those expressed in statements
1-8 should be enacted.
Cluster analysis (Hair, Anderson, Tatham & Black, 1998) was
used to group the 25 into three clusters according to their responses to
the beliefs survey and also to group the students according to their
responses to the CLES (student version) (Beswick, 2005). The beliefs
survey resulted in three clusters of teachers. These were:
1. Content and clarity
These teachers believed that they had a responsibility clearly to
explain mathematical content and that it may be necessary to tell
students the answers. They believed that they must cover the prescribed
content in the correct sequence. They also regarded computation is a
major part of mathematics and believed that effective mathematics
teachers enjoyed the discipline.
2. Relaxed problem solvers
Teachers in this cluster viewed mathematics as more than
computation and were the least inclined to believe that it was their
role to provide answers or even clear solution methods. They were also
less concerned than other teachers about either content coverage or
sequencing.
3. Content and understanding
These teachers could be described as the most concerned about the
coverage and sequencing of the content, but the least likely to seek
guidance regarding sequencing from a textbook. They were focussed on
students' understanding of the content, but not comfortable with
students suggesting alternative solutions.
The CLES (student version) resulted in five clusters of classes
based on the classes' average perceptions of the extent to which
they were responsible for their learning and were engaged with the
mathematics and connecting their learning with their existing knowledge
(Beswick, 2005). The more these elements were in evidence the more
consistent with constructivist principles the classrooms were deemed to
be.
Subtle but important relationships were found between the
teacher's beliefs and their students' average perceptions of
their classroom environments (Beswick, 2005). Classes in clusters
characterised by classroom environments most consistent with
constructivist principles were more likely than others to be taught by
teachers whose belief survey responses placed them in the Relaxed
Problem Solvers cluster. It is important to remember, however, that
teachers in this cluster (and each of the others) did not achieve these
classroom environments by implementing identical, or even superficially
similar, practices, but in spite of the variety of ways in which they
were implemented, their beliefs impacted their classrooms in ways that
their students could discern. This fact is illustrated by two of the
teachers in the Relaxed problem solvers cluster, Jim and Andrew
(pseudonyms), who were also interviewed and observed in their
classrooms. Both of these teachers had classes in the cluster
characterised by classroom environments most consistent with
constructivist principles.
The following quotations, some of which also appear in Beswick (in
press), are taken from the interviews with Jim and Andrew and provide an
indication of their beliefs about the discipline of mathematics and
mathematics teaching and learning.
Observations of Jim's classes (grades 9 and 10) and
Andrew's grade 7 classes confirmed their interview responses and
revealed very different teaching approaches at least superficially. For
example, Jim's students almost always worked in small groups and
his interactions with them were primarily at the individual or small
group level. In contrast with this, Andrew's students sat in rows
of twos or threes facing the front of the room and most of the
interactions were at the level of whole class discussions facilitated by
Andrew. Nevertheless, the students perceptions of their classroom
environments indicated that there were similarities in the extent to
which they were responsible for their learning and were engaged with the
mathematics and connecting their learning with their existing knowledge.
The beliefs that emerged as underpinning the practice of Jim and
Andrew related to the nature of mathematics, their students and their
capabilities, the teacher's role in the classroom and professional
learning. Beliefs about mathematics, students, and the importance of
professional learning were most central in Jim's case, whereas
beliefs about the teacher's role were most central for Andrew. The
particular beliefs that emerged as most central to one or other of Jim
and Andrew were:
1. Mathematics is about connecting ideas and sense-making.
2. Mathematics is fun (in the sense of playful confidence with and
enjoyment of mathematics).
3. Students' learning is unpredictable.
4. All students can learn mathematics.
5. The teacher has a responsibility to maintain ultimate control of
the classroom discourse.
6. The teacher has a responsibility actively to facilitate and
guide students' construction of mathematical knowledge.
7. The teacher has a responsibility to induct students into widely
accepted ways of thinking and communicating in mathematics.
8. The teacher is the authority with respect to the social norms
that operate in the classroom.
9. Teachers have a professional responsibility to engage in ongoing
learning.
Beswick (in press) argued that this set of beliefs seems to be
related to teachers' ability to create classroom environments that
can be described as constructivist and that it is such beliefs, rather
than particular teaching methods or materials, that matter in terms of
students' perceptions of their classroom environments. This is
consistent with the findings of Watson and De Geest (2005) and Askew,
Brown, Rhodes, Johnson and Wiliam (1997) concerning the importance of
teachers' beliefs in shaping their practices.
Implications
The literature on teacher change is replete with evidence that real
and lasting change is achieved only if teachers' belief systems
support the underlying premises of the changes they are asked to
implement (e.g., Chapman, 2002). Little is achieved by getting teachers
(or students) to mouth "suitable" views or perform certain
actions if they are not convinced of their value. It is, therefore, not
enough to provide teachers with resources, curriculum materials and
ideas without attending to their relevant beliefs. The point here is
analogous to the more widely espoused view that it is not enough to get
students to recite facts or perform procedures if they are not
meaningful to them--i.e., if they do not really believe the procedures
or their results.
Findings concerning the importance of teachers' beliefs to the
kinds of classrooms that they create highlight the importance of
individual mathematics teachers, and providers of professional learning
or pre-service teacher education related to mathematics, reflecting
carefully on the beliefs that they hold about the nature of mathematics
and about mathematics teaching and learning. The following is a list of
questions that may be helpful in stimulating such reflection:
With respect to each of the nine beliefs listed above:
1. To what extent do I hold this belief?
2. Why do I believe this? What hard evidence underpins my belief?
Is this evidence more than anecdote?
3. How/in what way(s) does this belief shape my practice?
4. How would my practice be different if I believed this?
5. Would an observer in my class (including my students) be
surprised if I told them I believed this? Why?
6. What other beliefs about mathematics or mathematics teaching and
learning, influence my practice? Why do I believe these things? Is there
hard evidence for their veracity?
When considering new practices, ideas, or materials:
7. What beliefs about mathematics and about mathematics teaching
and learning does the author/creator of these materials hold?
8. What does this professional learning provider believe about the
nature of mathematics and mathematics learning and teaching?
9. To what extent do I share these beliefs? Why?
10. What beliefs underpin my negative/positive reaction to this
idea? Are these beliefs reasonable?
In relation to students' perceptions of your beliefs:
11. What might my students think I believe about:
a. their capacity to learn mathematics?
b. how they learn mathematics?
c. what it means to "do mathematics"?
d. my role as a mathematics teacher?
12. How might these perceptions vary from student to student or
from class to class? Would there be differences according to
mathematical ability or grade level? How might students notice these
differences?
Opportunities to talk with trusted colleagues about responses to
these questions would likely be helpful. It would also seem sensible for
professional learning providers to be explicit about their own beliefs
and those that underpin their own practices and recommendations.
Providing time, opportunities and stimuli for teachers' reflection
on their beliefs is also important and certainly consistent with a
social constructivist view of learning that recognises that teacher
change is learning. Similarly teachers should make their own relevant
beliefs explicit for their students. Perhaps teachers and teacher
educators alike could benefit from asking their "students"
what they think their "teachers" believe. All of this has the
potential to be quite confronting and uncomfortable but I believe that
such unsettling is fundamental to learning.
References
Askew, M., Brown, M., Rhodes, V., Johnson, D. & Wiliam, D.
(1997). Effective Teachers of Numeracy. London: School of Education,
King's College.
Beswick, K. (2005). The beliefs/practice connection in broadly
defined contexts. Mathematics Education Research Journal, 17(2), 39-68.
Beswick, K. (in press). Teachers' beliefs that matter in
secondary mathematics classrooms. (accepted February 2006 for
publication in Educational Studies in Mathematics).
Chapman, O. (2002). Beliefs structure and inservice high school
mathematics teacher growth. In G. C. Leder, E. Pehkonen & G. Torner
(Eds), Beliefs: A Hidden Variable in Mathematics Education? (pp.
177-193). Dordrecht: Kluwer.
Green, T. F. (1971). The Activities of Teaching. New York:
McGraw-Hill.
Hair, J. F., Anderson, R. E., Tatham, R. L. & Black, W. C.
(1998). Multivariate Data Analysis (5th ed.). Upper Saddle River, NJ:
Prentice-Hall.
Taylor, P., Fraser, B. J. & Fisher, D. L. (1993, April).
Monitoring the development of constructivist learning environments.
Paper presented at the Annual Convention of the National Science
Teachers Association, Kansas City.
Watson, A., & De Geest, E. (2005). Principled teaching for deep
progress: Improving mathematical learning beyond methods and materials.
Educational Studies in Mathematics, 58(2), 209-234.
Kim Beswick
University of Tasmania
kim.beswick@utas.edu.au
Jim
I read about it, and I enjoy it and I sit here
folding bits of paper in times when I could be
doing something adults think might be more
important ... and I'm constantly excited by it,
and I do a fair bit of personal professional
development and every time I go somewhere,
I find extra little things ... (Beswick, in press)
[I]f we're investigating some aspect of that,
and the kids come up with a "what if" idea or,
"I wonder what happens if we do this", then
I'd absolutely grab it all the time ... If you
think you've planned this lesson, and it's
beautiful and linear and, it's going to work ...
but I think the kids sometimes, won't believe
they've got anything to offer ... and if we're
going to keep inventing this stuff called
mathematics, or discovering it, or making
sense of it, we've got to believe some of our
kids are going to go a lot further than we did,
and if they don't think they can actually offer
anything they won't.
Sometimes when kids have suggestions,
they're incomprehensible, if you just listen to
their words because they haven't got the
language and they haven't got the background ...
and it's easy to dismiss stuff as
being ludicrous, but if you then have got a
culture where they can sit and try and tease
it out and explain it, often they come up with
amazing sorts of things ...
Andrew
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 ... Getting them to come up and
put on the board their ideas, what they think
might be, what they should be doing or their
way of doing something is a struggle ...
Yes, they should guess. They should conjecture,
but at some stage the teacher's going to
have to call a halt and just say well, what
about trying this? ... You're not just a
supporting role, you are a facilitator, but
you're also more than that. You're someone
who hopefully understands the clear path
that might be needed and can also see
different paths to get to the end point and
send the kids off on appropriate paths, not
just let them wander through the minefield.