Introduction
This literature review will
provide support for the need to investigate the relationship between beginning
secondary mathematics teachers’ beliefs about the nature of mathematics,
mathematics teaching, and mathematics learning and their teaching practices.
This will lead to the identification of factors that affecting this
relationship between teachers’ beliefs and their teaching practice. The
influences of interventions will also be discussed very briefly in this
literature review.
Relationships between beliefs and
practice
Poor results in Penilaian Sekolah Rendah (PSR) and Penilaian Menengah Bawah (PMB) are partly
due to lack of understanding of the mathematical concepts among students. The
lack of understanding of these concepts is partly influenced by the
instructional practice the teachers employed. Explain-practice is a typical
teaching approach in many mathematics lessons whereby a teacher explains a
procedure and then students practice. Thompson (1992) explains that adopting
traditional teaching approach (instruction) not only misrepresents mathematics
to the students but also has an impact on their poor performance in national
and international assessments. In many studies, explain-practice instruction
has been shown to not be an effective or a successful way of teaching
mathematics. This type of practice has failed to produce good results in
mathematics (Stevenson and Stigler, 1992). Teachers view their task as one of
presenting mathematics to their students. Their main concern is to ensure that
their students learned to perform easily the tasks required by their homework
and tests. They teach their students to know how rather than to know why (Kesler, 1985 and McGalliard,
1983).
According to Baroody (1987), all teachers hold beliefs about
mathematics, mathematics teaching, and mathematics learning that influence
their teaching strategies. Brown & Borko, (1992)
suggest that teachers who are initially have nontraditional beliefs about
mathematics teaching, tend to employ traditional teaching when faced with
constraints in their actual teaching. Over the last 15 years, there has been
much research that has been taken in many directions involving the study of the
relationship between mathematics teachers’ belief and practice (Raymond, 1997). Thompson (1992) suggests that research should
more closely examine the relationship between conceptions of mathematics and
instructional practice.
Taken together the
definitions of beliefs used by Cobb (1986), Hart (1989), and Schoenfeld (1985)
and the categorization of mathematics beliefs by Lester, Garofalo,
and Kroll, (1989) and McLeod (1989), mathematics beliefs can be defined as
personal judgements about mathematics evolved from
experiences in mathematics, including beliefs about the nature of mathematics,
learning mathematics, and teaching mathematics.
Hersh (1986) claims that teaching
must be viewed on the basis teachers’ beliefs of the best way to teach and also based on
their understanding of the nature of mathematics. Dongherty (1990), Grant
(1984), Kesler
(1985), Kuhs (1980),
Lerman (1983), Marks (1987), McGalliard (1983), Shroyer (1978), Steinberg, Haymore,
& Marks (1985), and Thompson (1984)
have indicated that teachers’ beliefs play a significant role in shaping the
instructional behavior in class.
According to Garofalo (1989), the development of one’s beliefs about the
nature of mathematics and their practice is important as it influences how one
thinks, approaches, and follows through on mathematical tasks. This development
is also important as it influences how one studies mathematics and how and when
one teaches mathematics.
Teachers’ beliefs
develop from their teaching experience rather than from formal study. Schuck (1996) argues that these experiences are one of the
factors that inhibit the existing beliefs and attitudes to move towards reform
in the way mathematics is taught in schools.
Models of mathematics teaching
Most teachers believe that
understanding plays an important role in enables students to master
mathematical concepts and skills. This in return, will help students to achieve
success in mathematics. Kuhs and Ball (1986)
suggested four models of teaching namely Learner-focused; Content-focused with
an emphasis on conceptual understanding; Content-focused with an emphasis on
performance; and Classroom-focused, to be used in one’s teaching. All the
models are equally important and should be used in teaching, depending on the
needs and topics taught. However, regretfully, such belief is not put into
practice. Content-focused with an emphasis on performance has becoming the most
popular choice in teaching field in
Another model of teachings
presented by Perry, Howard, & Conroy (1996) indicates that teaching can be
categorized as either a transmission or child-centered. In traditional
teacher-centered transmission, the teacher delivers knowledge and skills,
whereas in child-centered learning, children are actively involved in
constructing their own mathematical knowledge.
Constructivist views of
mathematics learning (Cobb & Steffe, 1983; Confrey, 1985; Thompson, 1985; von Glaserfeld,
1987) underlies the learner-focused
view of mathematics teaching whereby teaching emphasizes students’ active
involvement in the mathematics lesson. This model requires students to explore
and formalize ideas. This model is likely to be used by teachers who view
mathematics as a dynamic discipline, dealing with self-generated ideas and
involving methods of inquiry (Ernest, 1988). Teachers in this model act as
facilitators or stimulators of students learning, addressing interesting
questions and providing situations for investigations, challenging students to
think, and assisting them uncover inadequacies in their own thinking (Kuhs & Ball,1986).
Expert-Novice Paradigm
In my study, the
focus will be on beginning secondary teachers. This group of teachers is chosen
because these are the teachers who are believed to be more progressive and
anxious to promote different teaching strategies. Leinhardt
(1989) explains that the expert is different from a novice is that an expert
has the ability to create a lesson which is rich in agendas, spends less time
in transition and gives better explanations of new materials. So it is expected
that teaching done by the expert is better than that done by the novice. However, this is not the case in
Raymond (1997) argued that the relationship
between beginning teachers’ mathematics beliefs and practice has been
overlooked but are important to study because: “beginning teachers reveal much
about their beliefs as they struggle to develop their teaching practice” and
“their beliefs about mathematics and mathematics pedagogy are likely to be
challenged during the first few years of teaching because their pedagogical
ideals are pitted against the realities of teaching”.
Mediating factors affecting the
relationship between beliefs and practice
It has been suggested by many
researchers that the relationship between beliefs and practice is weakened when
teachers work under perceived constraints, (Thompson, 1984). Though teachers
hold very strong beliefs about mathematics teaching and learning, when faced
with a situation which is discouraging such as when pressure is exerted by
principals, by parents, by class size and so on, their beliefs may not be seen
in their practice. However, Kaplan (1991) maintains that “beliefs and practice
can always be shown to be consistent, regardless of other contributing factors,
when one distinguishes between types of beliefs, such as deep versus surface
beliefs, and corresponding practices, such as pervasive versus superficial
practice” (pp. 550).
According to Brown (1986),
Cooney (1985), Thompson (1984, 1992), the relationship between beliefs and
practice is not consistent due to the existence of other mediating factors (Fazio, 1986). Factors like Social teaching norms and the immediate
classroom situation can affect the relationship between beliefs and
practice (Brown & Borko, 1992; Leinhardt, 1989; Thomson & Schuck,
1987; Westerman, 1990). New teachers who are
initially very ambitious to teach, full of creative ideas tend to be put off
when they are not getting full support from students, parents, fellow teachers,
and administrators. The Principal and parents obviously want their children to
achieve success in examinations.
Therefore, they emphasize to teachers the need to cover the syllabus
before the examination.
Some inconsistencies between
teachers’ beliefs and practice may be due to lack of knowledge and skills.
Knowledge and skills play a very important role for a teacher to be able to
teach well. Without a strong knowledge and skills, a teacher will not be able
to convey content effectively. A teacher needs to be good at varying teaching
techniques to provide different alternatives for weak students. Similarly, a
teacher needs to be very well versed in subject matter in order for him or her
to provide challenging questions for very good students. Lack of strong
knowledge and skills tends to limit teachers ability to vary their teaching
strategies. One cannot apply the learner-focused model identified by Kuhs and Ball (1986) if the teacher does not have a strong
and broad knowledge base in mathematics. Furthermore, without strong knowledge
and skills, one will not be able to teach with confidence.
Teachers’
personal beliefs about their capabilities to help students learn, sometimes,
called personal teaching efficacy, have been shown to be an influence in
promoting their classroom performance. Teachers with low levels of teaching
efficacy make less effort in finding materials and planning that can bring
about interesting and challenging lessons for their students. On the other
hand, teachers with high levels of efficacy tend to be more progressive, seek
resources and are able to develop challenging lessons (Bandura,
1997; Pajares, 1996; Tschannen-Moran
et al., 1998).
D’Andrade (1981) suggests that
children’s beliefs develop gradually as they react to the situations in which
they find themselves. They develop beliefs that are consistent with their
experience. This process of developing beliefs is characterized as “guided
discovery”. Definitely cultural factors play an important role in this process
of developing beliefs. Schoenfeld (1989) emphasizes
that in mathematics education most researchers claim that the cultural setting
of the classroom influences development of beliefs about mathematics.
Gender is also an important
factor that influences the relationship between beliefs and practice. Female
teachers tend to show more interest and effort in creating a better learning
environment. They are more creative and always come up with interesting
teaching models or aids. Male teachers, on the other hand, tend to teach
plainly. This scenario is evidently shown in higher institutions which are
dominated by female students. Need to find a literature review.
Language is another factor
which contributes to the inconsistencies. English as a second language in
Conclusion and Implications
Ernest (1988)
suggests that teachers’ attitudes towards the subject have an impact on their
teaching practice, which in turn impacts upon the ethos and culture of the
mathematics classroom. This in turn impacts upon the attitudes of their
students which in turn influences teacher’s attitude. As a result, poor attitudes
and poor achievements levels are passed from one generation to the next
generation. This is very difficult to remove or improve. According to Cooney
(1987), substantive changes in the teaching of mathematics to more
student-centered learning such as suggested by the NCTM Standards (1989) will
be slow in coming and difficult to achieve. This is due to the basic beliefs
teachers hold about the nature of mathematics. Collier (1972) and Shirk (1973)
suggest that prospective teachers’ conceptions are not easy to change; one
should not expect that a single training program can bring about the desired
changes. It is not easy to change beliefs. What needs to be changed are the
mediating factors (encourage nontraditional beliefs about mathematics teaching and
learning) that affect beliefs.
D’Andrade (1981) and Schoenfeld (1988) argue
that there is nothing wrong with the students’ mechanism for developing beliefs
about mathematics. What needs to be changed is the curriculum (and beyond that,
the culture) that encourages such beliefs.
However, there
was sufficient indication in data from many surveys and journals, to show that
intervention played a significant role in bringing about changes in teachers
beliefs. The intervention has to be combined with steps to promote factors that
encourage nontraditional beliefs.
A considerable
effort in teachers professional development has been
made which is intended to help them to be able to conduct lessons that can
develop students understanding in mathematics and thus lead to the production
of fruitful results and achievement in mathematics education. Nevertheless,
this effort will not be meaningful without the cooperation and support of other
related agencies or authorities. Teachers
at the same time must make an effort to use the knowledge they acquire from the
course and must continuously upgrade themselves with up to date teaching
approaches. Most often, it is found that
these teachers who have participated in an in-service program are not
practicing what has been taught to them. To develop the teachers’ development
in mathematical knowledge and a constructivist
pedagogy, they are endouraged to participate relevant
workshops (Simon & Schiffer, 1991; Schiffer & Fosnot, 1993;
Simon, 1995).
From the findings
of this research, it is expected that the influences that effect beliefs and
practices will be studied in greater depth. It is also anticipated that
findings from this research will provide a starting point for discussion of the
influence of interventions on teaching. It is hoping that this study will
encourage teachers to reflect on their experiences as students and to discuss
how their beliefs and practices may better prepare them to use new approaches.
23 Feruary 2005
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