Running
Head: BELIEFS AND PROFESSIONAL DEVELOPMENT
Teachers Beliefs and Professional
Development on Reform Mathematics:
How Beliefs Help, Hinder, and
Change
Pamela R. Hudson Bailey
George Mason University
EDCI 857: Preparation and
Professional Development of Mathematics Teachers
Dr. Jennifer Suh
Teachers Beliefs and Professional
Development on Reform Mathematics:
How Beliefs Help, Hinder, and
Change
What
comes first? Professional
development? Changing teacher
beliefs? Student success with a new
method? All of these are questions that
are a part of cyclic models and very complex and interrelated. Deciding what should be included or put in to
the creation of teacher education or professional development that will affect
teacher beliefs and student out comes has been approached by many researchers
(Boaler & Staples, 2008; Cross, 2009; Cwikla, 2002; Derry, Wilsman &
Hackbarth, 2007; Harel & Lim, 2004; Lappan, 2000; Lin, 2004; Nathan &
Koedinger, 2000a, 2000b; Olson & Kirtley, 2005; Orrill, 2006; Petty, 2007;
Sztajn, 2003; Uworwabayeho, 2009; Weidemann & Humphrey, 2002; Weiss &
Pasley, 2006). However, teacher’s
beliefs about subject matter have not been a focus prior to 1989 (Peterson,
Fennema, Carpenter & Loef, 1989).
The beliefs held by teachers, their knowledge (content and pedagogical),
and decisions made regarding curriculum, content, and student-teacher
interactions effect the way teacher’s teach and student’s learn (Peterson,
Fennema, Carpenter & Loef). This
paper will look at effective professional development, the concerns,
expectations, support, and outcomes, and then proceed to delve into teacher
beliefs, what they are, what influences beliefs, the effects and changes of
beliefs and the different belief systems.
At the center of professional development and teacher beliefs are the
students and their needs with the reform movement as the backbone of teaching
and learning.
Professional
Development
Expectations of professional development
sessions.
Every professional development
should be based on a set of goals (Weidemann & Humphrey, 2002). Cwikla (2002) states that setting small and
large goals will help to acknowledge success with small incremental changes
that lead to instructional improvement.
As professional development leaders, we need to understand why and how
the small changes lead to positive results and influence teachers to continue
on the path of improving their classroom practices (Cwikla). Participants need to be told what the goals
are, the implementation details, the focus of each session, and the role that
they will be playing for the professional development (Cross, 2009; Guskey,
1986). Weiss and Pasley (2006) believe
that the overall goal should be to increase teacher knowledge and pedagogical
content while establishing a vision for change. They go on to state the goals should be
specific, be reachable by all participants, and have high expectations. Goals are also a way to hold student and
teacher learning accountable (Loucks-Horsley, Stiles, Mundry, Love, and Hewson,
2010). First and foremost, professional
development sessions should be focused on the students (Lappan, 2000).
Harel and Lim (2004) state that teachers
should understand that the focus is on how the students are learning the
concepts and how the mathematics teaching affects their learning. This focus on students was regarding
observations that are a part of a professional development workshop. Observations are a good way of creating
teacher awareness to the implementation and results of using reform methods,
more precisely open-ended tasks (Lin, 2004). After the observations, group discussions
should be encouraged to reflect on student actions and interactions with the
concepts. Using more open-ended tasks
may lead to students using manipulatives.
Elementary school teachers are well versed on how to use manipulatives
correctly however high school teachers, who tend to be more procedural, do not
take advantage of manipulatives (Olson & Kirtley, 2005). Professional development sessions should help
teachers become familiar with manipulatives, how and when to use them as well
as student expectations. In addition to
manipulatives, technology, hardware and software, need to be taught and applied
to lessons appropriately (Uworwabayeho, 2009).
Uworwabayeho mentioned Geometer’s SketchPad with students being able to
visualize the concepts being taught.
Observations, manipulatives, and technology are all good ways to
implement inquiry or investigative approaches to teaching and learning.
Investigative/inquiry
approaches and
using manipulatives.
Teacher’s methods of instruction
range from direct instruction to deductive approaches and there appears to be a
disconnect between practice and what the teachers believe as the best way for
students to learn mathematics (White-Clark, DiCarlo, & Gilchriest,
2008). Cwikla (2002) posits that
environments that encourage inquiry are more successful in changing to reform
movement practices and that it will lead to improvement in instructional
practices. Therefore, professional
development needs to focus on mathematical reform that includes investigative
or inquiry approaches (Lappan, 2000).
This leads to the objective of being exposed to new methods in
professional development then is a necessary component and should include
manipulatives and calculators to encourage teacher and student thinking
(White-Clark, DiCarlo, & Gilchriest).
In order for sessions to be productive, teachers should help to
determine what they need to learn and should be involved in the development of
the learning experiences (Orrill, 2006). Investigative or inquiry activities,
including open-ended problems, are difficult for teachers to develop according
to Lin (2004). Professional development
needs to consider this difficulty as part of the development as well as to have
teachers think about the social aspect of teaching and learning. Lin states
that math talk should be encouraged and part of the activity so that students
can explain what they are doing and what it means. Discussions allow students the opportunity to
delve into the conceptual understanding of the concepts they are learning.
Conceptual understanding and expectations.
Teachers need to understand what is
meant by conceptual understanding (Olson & Kirtley, 2005). During a professional development teachers
may be given a unit of study or a big idea and asked to determine the
conceptual understanding involved (Lappan, 2000). By acknowledging the difference between
conceptual understanding and procedural knowledge, teachers will be able to bring
a combination of both in to their instruction (Derry, Wilsman, & Hackbarth,
2007). Gaining insight into the way
students understand mathematics will enable teachers to better meet the
student’s needs (Lin, 2004; Nathan & Koedinger, 2000a, Nathan &
Koedinger, 2000b). By implementing
mathematical reform that includes math talk teachers will know if students
understand the concept instead of just applying procedures rotely. This means that teacher thinking and learning
needs to be focused on the students’ thinking and learning (Cwikla, 2002). Nathan and Koedinger (2000a; 200b) extend
this thought by saying that teacher thinking and learning should also correlate
with the students. What teachers know
and understand may be obvious, not so for students (Uworwabayeho, 2009). Loucks-Horsley, et al. (2010) posits that
teacher knowledge of the “what” and “how” is not enough. They need to understand the “why” so they
will not be hampered in their ability to adapt to different situations. Lappan states several additional aspects that
need to be integrated into professional development.
Professional development is composed of
several aspects.
Content
should be one of the major focuses for a professional development and should be
implemented using mathematical reform ideas such as problem solving (Lappan,
2000; Weiss & Pasley, 2006).
Challenging students to learn the material to the best of their ability
and striving to engage them in problem solving activities will enable a deeper
conceptual understanding of the content to be developed (Cwikla, 2002). By doing so, students will also be encouraged
and given the opportunity to make connections with the content and within their
lives (Lappan). Knowing the content is
important but its implementation is more so according to Lappan.
Pedagogical content knowledge allows the
teacher to have the expertise to present concepts that are centered on what the
student needs to know and how the student need to learn the material (Orrill,
2006). A professional development
session should promote teachers learning of pedagogical content knowledge that
they will be expected to use in their classrooms (Orrill). Modeling the approaches and presenting
lessons in a professional development, discussing explicitly the instructional
strategies, should all be backed by research based teaching and learning
methods (Weiss & Pasley, 2006).
Having the opportunity to understand why they need to learn the theory
behind how to teach using student-centered approaches will help the teachers
when they do implement the strategies with their students (Orrill). Teachers need to be engaged in problem
solving activities so that they can also experience what their students will
during the activity (Derry, Wilsman & Hackbarth, 2007). They need to see what they need to do in
order for the student to gain knowledge (Lin, 2004). Creating disequilibrium or having teachers
acknowledge the positive aspects of the approach will encourage them to become
change agents (Lin; Loucks-Horsley, et al., 2010). Professional development sessions should
consider implementing the creation of activities that challenge teachers,
causing them to doubt evidence they hold that supports their beliefs and then
encourage them to reflect (Cooney, Shealy, & Arvold, 1998). Activities need to correlate with the curriculum
and encourage collaboration as concepts and the connections between them play
an important role in professional development.
Curriculum based professional
development sessions have more of a lasting effect on teachers than
professional development that is general in nature (Weiss & Pasley,
2006). This means that teachers need
time to align the curriculum with state standards, national standards, and
assessments. Along with aligning the
documents curriculum developers need to keep student learning and achievement
at the forefront when planning and facilitating instruction (Cwikla,
2002). The flow of the curriculum needs
to make sense to teachers and students, allowing for connections to be brought
forth and investigated (Cwikla; Lappan, 2000).
Lastly, the curriculum should be centered on the reform movement to
encourage teachers to become change agents and to increase student achievement
(Cooney, Shealy, & Arvold, 1998; Cwikla).
This leads to collaborative settings, calling on the expertise from all
of the teachers involved to discuss, create, and analyze lessons and
assessments (Lappan; Olson & Kirtley, 2005). Being involved in collaborative sessions
leads to instructional improvement and teachers having more confidence to learn
and to share with each other (Cwikla). Given
the opportunity to collaborate, teachers will garner support from each other as
they grow and change and at the same time they will be learning from their
students on how to meet their needs (Guskey, 1986; Lin, 2004). Collaboration should occur with peers within
their building and between schools according to Weiss and Pasley. Lin stresses that a safe environment is
needed in order for teachers to feel free to be creative and solve
problems. This is good practice for how
we expect students to collaborate in the classroom. Along with collaboration teachers need to
talk and reflect about the mathematics they are teaching.
Discussions between teachers provide
support for the questions that they have about a lesson, about a student’s
response, or about an assessment (Guskey, 1986; Lin, 2004). Interactions between students might reveal
some information that the teacher may need to investigate or address in order
for deeper understanding to be gained.
Encouraging participant discourse on how to engage students in thinking
algebraically may also be used as a guide to reflective analysis (Derry,
Wilsman, & Hackbarth, 2007).
Uworwabayeho (2009) states that teacher self-reflection will help to
improve classroom interactions and instruction with Orrill (2006) stating that
the reflection should focus on the mathematics and how it is being learned by
the students. Maintaining a journal
throughout a professional development is an idea given by Lin. He suggests that the journal include
information about strategies attempted, how the students conceptualized the
activity, and student thinking processes.
The journaling should also be about what the teacher is learning in
mathematics and how he or she is encouraging the students to do the same. As a continual learner, teacher’s usage of
journals will help them to see and understand where they are now and where they
want to go in their careers.
Teachers
as continual learners.
There are three types of knowledge that
teachers need to continually assess and strive to stay current: the knowledge
of content, the knowledge of student epistemology, and the knowledge of
pedagogy (Harel & Lim, 2004). The
knowledge of content will affect how a teacher presents material and what they
teach. Harel and Lim discuss the “Ways
of Understanding” or WoU and the “Ways of Thinking” or WoT that a teacher may
consider as they stay current in the types of knowledge. Teachers need to know WoU and WoT for
themselves but also for their students. WoU
is when students develop meaning and interpretation, solve problems, and have
the ability to justify and reason through a problem whereas WoT is the way a
student understands a situation. These
thoughts are extended to approaches one might use to problem solve and to
proving. Problem solving approaches
include solving simpler problems, similar problem analysis, and drawing or
illustrating the problem. The two
approaches to proving are ascertaining and persuading. Ascertaining is an internal persuasion of
truth whereas persuading is an external persuasion toward others. How one thinks and understands will help in
problem solving and proving.
Knowledge of student epistemology is how
a student builds or constructs knowledge (Harel & Lim, 2004). The “psychological principles of learning” is
a term that Harel and Lim use when referring to student epistemology. In order for the students to build their
knowledge teachers must know how to teach the concepts and how to assess for
understanding (Harel & Lim).
Promoting WoU and WoT, therefore keeping both ideas at the forefront of
planning, will enable teachers to determine and guide the pedagogical decisions
that is needed for student achievement.
Weiss and Pasley (2006) maintain that a
common principle of professional development is to provide experiences based on
pedagogical content knowledge and materials.
In order to maximize effectiveness, professional development needs to
focus on all aspects of knowledge (Harel & Lim). Learning new material and being open to
listen and try new ideas may be trying for some teachers but transforming
teacher’s instructional beliefs and abilities takes time (Weiss &
Pasley). New teachers are not done with
learning after student teaching. The study
conducted by Boaler and Staples (2008) discusses that new teachers, in a school
that is successfully implementing mathematical reform, stay one day behind
experienced teachers and are given the time to observe that teacher so they
might learn. Professional development models have been developed that relate
the changes teachers may go through in the process to become more
student-centered.
Professional
development models.
Guskey (1986) shares a model that has
the following steps: professional development, changes in teachers’ classroom
practices, changes in student learning outcomes, and changes in teacher beliefs
and attitudes. In this flow student
learning comes only after changes in classroom practices have occurred. Following student learning, where positive
results that have been experienced by the teachers, will come a change in
teacher beliefs (Guskey; Loucks-Horsley, 2010).
This model does not take into account all the variables that effect
changes in teacher beliefs and actions but is a basic flow. The basis is that teachers are not truly
committed to an approach till they have seen the results. Data collection, before and after
implementation of a new approach, will help teachers to acknowledge student
growth. Therefore using Guskey’s cyclic model,
teacher beliefs can be viewed as a result instead of a cause of a change.
Loucks-Horsley, et al. (2010) revealed
another model for professional development that includes: commitment to vision
and standards, analyze student learning and other data, set goals, plan, do,
and evaluate results. This model is also
cyclic and inputs knowledge and beliefs, context, critical issues, and
strategies at various points on the cycle.
Loucks-Horsley, et al. view their model as what goes into creating a
productive professional development whereas Guskey’s (1986) model reveals when
teacher’s beliefs will be affected with a possible change along with the
changes in the classroom. Beliefs and
knowledge are addressed in the Loucks-Horsley, et al. model but only with
respect to professional development and what is needed to create change. They posit that modeling habits of mind and
activities so that teachers can experience the mathematics, on their level will
enable teachers to relate to their students as an outcome of their
participation in professional development.
Along with models of professional development come the misgivings and
concerns.
Concerns
with professional development.
Teachers
bring access baggage when participating in professional development
sessions. Their goals may not match with
those set for the professional development if the objective is for teachers to
learn, understand mathematics, and the development of the mathematics through
student-centered learning when the teacher wants activities that they can take
back to the classroom and immediately use (Orrill, 2006). When given a choice of activities Orrill
found that teachers only wanted to do those that they thought the students
would be able to do instead of grappling with material to increase their own
knowledge. Teacher attitudes also play a
part in the successful implementation of the goals. Optimistic attitudes of those involved toward
reform goals were shown to positively correlate with the length of time the
participant was involved in the professional development (Weiss & Pasley,
2006). Besides promoting better
attitudes, there was also a willingness to collaborate with peers, an awareness
of how students learn, and a willingness to be a change agent. Weiss and Pasley also revealed that those who
were opposed to change did affect the other participants in the professional
development. The participant’s knowledge
also played a part in the sessions.
Orrill (2006) described teachers in her study
that did not strive to expand their content knowledge and understanding and
focused on learning software instead of the mathematics and making
connections. Only a small group of
teachers in the study viewed the professional development as intended, centered
on increasing content knowledge and conceptual understanding while making
connections. Her reasoning behind the
lack of fulfilling the expectations of the activities was that teachers
possessed low mathematical content knowledge.
Weiss and Pasley (2006) posited that teacher weaknesses in content knowledge
and pedagogical practices are partly to blame for poor student performance in
mathematics. It seemed that these
teachers were more concerned about materials they might need or use and
housekeeping needs. Self-confidence may
also play a role in a teacher’s subject matter knowledge (Cross, 2009). In the study conducted by Orrill teachers
wanted the process that needed to be used when solving the problems, the way
the instructor would like to see it done.
The teachers did not want to rely on their collaborative peers and
instead relied on the teacher to respond to all of their questions and
concerns. Orrill views an over reliance
on the instructor during professional development a major concern when the
participants did not want to even take the explanation or advice of the
graduate student that was assisting.
Just like students, teachers needed the affirmation that they were on
the right track procedurally as well as with their mathematical knowledge. This was noted with the large amount of
procedural questions that were asked. Lacking
perseverance as they struggled with the mathematics, the teachers were not
willing to try new things.
Self-confidence also came into the picture when teachers were expected
to use equipment, materials, and apply strategies (Weiss & Pasley).
Support
needed for effective implementation.
Weidemann and Humphrey (2002) discuss
networking among university staff responsible for teacher education. A
successful network is similar to professional development. A network of teachers should also set
goal(s), establish time lines, and provide time for those involved to communicate
their concerns and share ideas. Timelines
for professional development should also include support through follow up
sessions. The professional development
sessions should be sustained over time which will allow for internal support to
be established and strengthened (Orrill, 2006).
Derry, Wilsman and Hackbarth (2007) promoted presenting professional
development in two parts, a summer workshop and school year follow ups. The
school year portion allowed the participants to implement knowledge and experiences
gained during the summer workshop thereby allowing the participants to continue
learning and adapting their instructions from the students work. Classrooms would then become the site of
ongoing inquiry into students learning conceptually and owning their
knowledge. Follow up sessions supported
the participants and the additional content focus helped to insure positive
results (Guskey, 1986; Weiss & Pasley, 2006).
Schools, administration and peers need
to support those attending professional development (Orrill, 2006). Sztajn (2003) stated that teachers are not
receiving support and assistance to meet the expectations of reform in
mathematics which in turn hampered the effectiveness of the program. They may even feel threatened and anxious as
the change is taking place (Guskey, 1986).
Being able to offer follow ups for ongoing prolonged support will help
and encourage change with systemic support resulting in a more effective implementation
(Guskey, 1986; Weiss & Pasley, 2006).
Petty’s (2007) analysis of teachers input on a questionnaire revealed
that they wanted support in the form of materials and supplies, increase in
salary, smaller classes, support with student discipline, a safe environment,
and quality class time.
In addition to the support listed above,
teachers would like to select the topic of the professional development that
would relate to their needs (Petty, 2007).
As well as selecting the topic teachers would like to also determine the
time when they attend. Many of the
teachers requested training on technology usage and implementation with opportunities
to attend conferences and workshops.
Professional
development outcomes.
Orrill (2006) revealed several outcomes
of effective professional development that included teachers owning their own
learning, and being better able to understand student’s feelings, fears, and
concerns. Teachers that were able to act
as students and investigate, problem solve, and be involved in mathematical discourse
improved their chance of using the material and strategies correctly in the
classroom (Weiss & Pasley, 2006).
Weiss and Pasley revealed that teachers that were actively involved in
professional development for sixty or more hours were more likely to implement
strategies such as hands on activities, investigative lessons, and creating
simulations. However there was some
improvement shown after thirty hours and up to eighty hours. Loucks-Horsley, et
al. (2010) stretch the number of hours to 100 but adds a narrow time frame for
the hours to be obtained from 6-12 months.
Improvement was shown in lesson quality as teachers learned to modify
instruction to an inquiry based, inductive approach. Teacher attitudes toward inductive teaching
methods improved as the time spent in the professional development
increased. A culture of learning may be established
as time on task increased leading to greater communication (Weiss & Pasley).
Increased communication was enhanced
when student generated items were brought into the discussions, widening the
case possibilities (Derry, Wilsman & Hackbarth, 2007). Teachers compared the “cases” permitting them
to analyze the student’s ability to think algebraically, their conceptual
understanding of the material, and their knowledge of the required
procedures. During sessions teachers may
also be assessed for their own content knowledge, pedagogical content knowledge,
and their ability to analyze student thinking (Derry, Wilsman & Hackbarth,
2007). In addition to analyzing case
studies the teachers were also given the chance to analyze videos of student
actions during a problem solving activity.
Discussions and collaboration of what they viewed could be done
immediately instead of waiting to experiment and analyze with their own
students. This led to the teachers
showing an increase when given a problem solving task to compare the multiple
representations and to interpret the solutions, improving their ability to
identify and explain student’s algebraic thinking. Guskey (1986) summed up results of
professional development changes to include teacher instructional practices in
the classroom, learning expectations of students, and the teacher’s beliefs and
way of behaving.
Peterson, Fennema, Carpenter and Loef
(1989) discussed changing teacher’s beliefs with respect to students instead of
the application of new concepts. The
research team stated that teacher’s beliefs were either a result of having
students capable of higher order thinking or the higher order thinkers may be a
result of teacher’s beliefs. But what
causes or affects teacher’s beliefs? A
teacher’s beliefs, as previously stated by Weiss and Pasley (2006), will take
time to change. If their beliefs do not
correlate with the goals of the professional development then the
implementation will be ineffective.
Content knowledge, pedagogical knowledge, and student epistemology are
all governed by an individual’s beliefs.
Guskey (1986) views change as taking time due to the cyclic nature of
change as teachers apply a new concept and note the change in student results
which leads to the change in beliefs. A
lot more goes into changing a teacher’s beliefs besides attending a professional
development. This leads to a discussion on what are the teacher’s beliefs, what
do their beliefs effect, how to change beliefs, what effects beliefs,
contrasting beliefs, and belief systems.
A teacher’s beliefs predict how they
will practice instruction in their classrooms (Cross, 2009; Harel & Lim,
2004; Nathan & Koedinger, 2000a; Nathan & Koedinger, 2000b; Sztajn,
2003). Guskey (1986) posits that
classroom instruction effects student learning then results in changes in
teacher beliefs. However it is cyclic. Some
of the teachers may take new ideas of reform mathematics and adjust it to fit
their own beliefs (Cross; Sztajn).
Others state that “…this is the way I’ve learned it…so I am teaching the
way I’ve learned it (Harel & Lim, p. 28).”
But how can teachers practice something they have never seen or
experienced was a question posed by Orrill (2006). Sztajn disclosed that the current
mathematical reform movement was expected to alter a teachers’ ideological vision
or beliefs. It is the reverse. The teachers’ ideological beliefs influence
the understanding and execution of the mathematics. Many of the teachers felt
that they implemented the standards of reform but they were done according to
their interpretations. Reform documents
are interpreted by each teacher according to their beliefs about teaching and
mathematics.
Zollman and Mason (1992) constructed an
instrument that measured teacher beliefs with respect to the NCTM Standards (1990). It may be used as a reflective tool to make
decisions about one’s own teaching as well as to evaluate the perspectives a
teacher holds toward the current vision of the reform movement based on the
NCTM Standards (2000). By using this
instrument, it was shown that teacher beliefs can be modified if they are given
the opportunities to experience and obtain knowledge to support the change.
Also revealed in Sztajn’s (2003) case
studies was the influence of the socioeconomic level of the students. Those from lower income families do not
receive the mathematics instruction that the students from higher income
families receive. As the child’s income
level decreases instruction changes from classrooms based on problem solving
and creative activities to drill and practice.
The mathematics becomes less demanding and rigorous. When planning instruction the major factor is
the students’ ability (Nathan & Koedinger, 2000a; Nathan & Koedinger,
2000b). If the students are doing well
then many teachers may feel that there is no need to change their beliefs of
teaching and learning mathematics (Cross, 2009). Teacher’s beliefs need to correlate with
their expectations with regard to the investigative/inquiry approach, the
reform movement, so that they will be able to implement it successfully and for
there to be improvement in mathematics (Cross; Lappan, 2000; Nathan &
Koedinger). In order for students to
become problem solvers and critical thinkers, teachers need to believe and
practice mathematical reform, inquiry, and student-centered approaches (Cross;
Olson & Kirtley, 2005). Those who do
believe appreciate and rate professional development activities higher than
those who do not (Cwikla, 2002). Being
able to assess student difficulties while they are struggling through a task is
an aspect that can be brought forth, acknowledged, and practiced during the
teacher activities as they model lessons (Cwikla; Nathan, 2004). Awareness of problem difficulties will affect a
teacher’s beliefs in how students learn and thereby effecting instruction
(Nathan). Nathan’s study of five high
school teachers in a one day professional development session revealed that
most teachers felt that symbolic manipulation was easier for students instead
of the arithmetic or algebraic verbal problems.
Actual student work was revealed and analyzed showing the opposite. Teacher’s lack of awareness of this
difficulty hampers the learning of mathematics.
If the mode of instruction was based on teacher evidence then becoming
aware of the data may create some disequilibrium resulting in a change in
teacher beliefs.
Uworwabayeho (2009) disclosed that
teachers need to accept student autonomy and be comfortable dealing with
situations as they arise. A very
different interaction between teachers and students evolves as the teacher
releases some control of the class. This
requires pedagogical content knowledge, one of the types of knowledge that should
be addressed during professional development.
The three types of knowledge possessed
by the teacher (content, pedagogical, and student epistemology) will effect
what is being taught as well as how the concepts are taught (Cross, 2009;
Cwikla, 2002). How the teacher
conceptualized the mathematics they learned and the courses they took, their
mathematical background, effects their beliefs (Cross; Cwikla). Textbooks are another large influence on teacher’s
beliefs (Nathan & Koedinger, 2000a; Nathan & Koedinger, 2000b). Nathan and Koedinger conducted a study that
assessed teacher beliefs on the difficulties of various types of mathematical
problems and the results correlated with how textbooks flow. Teachers believed that verbal problems, with
or without context, were the most difficult for students to complete
successfully and that the abstract symbolic problems would be the easiest
(Nathan, 2000; Nathan & Koedinger, 2000a, 2000b). As also previously expressed, the results
showed the opposite; success was more pronounced with the verbal problems and
the symbolic problems were the most difficult.
The study did not show much difference between the two types of verbal
problems. This correlates with how
textbooks are set up using the symbol precedence model where symbolic reasoning
comes before algebraic verbal reasoning.
Textbooks influence and appear to support teachers in their beliefs but the
number of years of experience a teacher has does so as well.
Cwikla (2002) found in her study that
teachers with less than two years experience were influenced by professional
development and showed growth in how they perceived student thinking. Teachers having seven to twelve years of
experience showed a desire to try different activities and had the confidence
to do so. This group also rated
experimentation and inquiry activities higher than teachers of other years of
experience groups. Levels of education
influence teacher’s beliefs by making them more aware of research. Cwikla stated that teachers holding advanced degrees
did not appreciate or feel that professional development activities were
beneficial, content and strategies were not important. She used the term that teachers had a “false
ceiling effect” (p. 11) as they knew the math.
Those with weaker mathematical backgrounds might feel threatened,
confused, and as if they did not have the support needed to be successful
during a professional development (Cwikla).
The
effect of teacher’s beliefs.
What
a teacher believes will effect and influence the attention given to student
thinking (Cwikla, 2002; Boaler & Staples, 2008). Cwikla’s investigation of teachers involved
in professional development revealed that they were less willing to participate
in activities that promoted student thinking if it did not correlate with their
beliefs. The teachers in the study that
viewed instructing students as constructing knowledge placed more attention on
how and what student thought. If the
teachers believed that their instruction and responsibility was to transmit
knowledge to others then they didn’t put much weight on student thinking. Harel and Lim (2004) made known that a “way
of thinking” can affect how one interprets and understands a problem situation
and the approach he or she will apply.
How the teacher believes that students learn, symbolic to verbal or
verbal to symbolic, according to Nathan and Koedinger (2000a; 2000b) will affect
how they first plan and approach a concept.
Teachers need to be aware of alternate strategies used to solve problems
to encourage and accept flexibility and efficiency. All participants and stakeholders need to
recognize that changing one’s beliefs take time as there is no quick fix
(Cross, 2009).
Change needs to begin, according to
Cross (2009), with teacher education.
Undergraduates need to be taught mathematics content in a student or
learner-centered environment. Weidemann
and Humphrey (2002) agree with Cross stating that teachers need good
instructional models to imitate and should begin in their teacher education
programs. Teachers in the classroom need
to be exposed to classroom environments that foster inquiry and mathematical
discourse (Cross). Through professional
development teachers can become involved in problem solving activities that
will challenge their belief systems producing disequilibrium which can result
in teachers reflecting on their own actions (Cross; Harel & Lim,
2004). Derry, Wilsman and Hackbarth (2007)
advocate having teachers wrestle with problems prior to being allowed to
discuss the problem with others. This
would allow them to experience problem solving tasks with the same
disequilibrium as felt by students (Lappan, 2000; Olson & Kirtley, 2005). Harel and Lim also stated that teachers
reflecting on their own learning will lead to a better appreciation and
understanding of “…epistemological and pedagogical issues” (p. 32). The sustained, continuous professional
development should encourage awareness of mathematical inquiry and
experimentation, discourse, justification, reasoning, sense-making, and
reflection (Cross). Teachers need to
recognize that change does take time, that it does not come easily, and that it
may occur in steps (Guskey, 1986; Nathan and Koedinger, 2000a; 2000b). One question that may be asked is if all
beliefs can be changed.
Cooney, Shealy & Arvold (1998)
discuss that some beliefs may be changeable while others may not. They based their study of preservice
mathematics teachers on Green’s (1971) research where a difference is made
between beliefs that are or are not held based on evidence. If the belief is not based on evidence then
providing evidence will not influence change.
In contrast if the belief is based on evidence then teachers can
challenge the evidence and modify the belief based on the outcomes. The researchers also compared the teacher
beliefs, feelings, and attitudes with Belenky, Clinchy, Goldberger, and Tarule
(1986) and Perry’s (1970) growth and changes in the lives of women and men
respectively from those that listen to others to determine their path or
actions to one of listening to themselves and listening to others critically,
but making their own decisions. Four of fifteen
preservice teachers were involved in five interviews and all took a survey on
beliefs. Four teachers were selected to
provide a range of initial views about mathematics and teaching
mathematics. The study was to interpret
how their beliefs were held, evidentially or nonevidentially and how they
changed through their education program.
Cooney, Shealy and Arvold surmised that the teacher’s inner voice
effects teacher change and aids in reflecting critically on experiences. Analyzing the four teachers with respect to
where they were on the continuum based on growth by Belenky, et al. and Perry
suggested some modifications. The
modified levels of growth or change were naďve idealist, isolationist, and
connectionist. Their suggestion was that
teacher education programs should be developed around encouraging students to
become reflective connectionists as they would incorporate all the different
voices, look at each critically, and then assimilate all views and information
to change beliefs and become committed to that belief evidentially. Teachers wrestle with contrasting beliefs,
whether they are or are not based on evidence.
Contrasting
mathematical beliefs.
Cross (2009) discussed three different
contrasting views that she uncovered in her study of high school teachers,
their beliefs, and how it correlated with their practices. The first contrasting method is computations
(traditional view) versus a way of thinking (reform view). Expounding on the traditional view, Cross
stated that teachers viewed mathematics as the basic operations with the
teacher being the holder of all knowledge.
Teachers asked the questions, students responded with little or no
mathematical discourse. Instruction was
more focused on the mathematical topic from a computational point of view
versus what it means in real world situations (Harel & Lim, 2004). Mathematical content was considered to be
never changing and absolute with students memorizing facts and procedures
(Harel & Lim). Nothing more was
needed other than concept definitions and rules (Harel & Lim). The reform movement took a different
approach. Instead of a focus on
computations, teachers viewed mathematics as problem solving and students’
thinking processes (Cross). Classes may
be conducted using an investigative/inquiry approach regardless of student
ability levels or the content being addressed (Cross). Even though some teachers practiced
mathematical reform they might still consider the content when determining
whether to use the investigative/inquiry approach (Cross).
The second contrasting belief discussed
by Cross (2009) is demonstration versus guidance. Traditionally teachers felt that students
would watch a demonstration and passively learn the material (Cross). When information was given out and the
students took notes, they should be able to retain it in their memory and were
considered to be engaged in the lesson (Cross).
Low achieving students were believed to not be capable of
student-centered learning (Cross). They
would not be interested in mathematics therefore direct instruction would be
better for these students. Practice
equates to understanding for those who believed in the demonstration,
traditional view. Harel & Lim (2004)
state that this view is more of a lecture format that is a teacher driven
instructional method. In contrast, the
reform view that Cross termed as guidance was from the standpoint that students
were capable of making sense of the mathematics and could piece together their
own ideas. Teachers would be guiding or
coaching their students instead of lecturing.
The teacher’s role as a coach is to develop activities so that students
could learn the material and apply reasoning skills that would support the
concepts that are to be learned.
White-Clark, DiCarlo and Gilchriest (2008) termed this role as the guide
on the side.
Lastly, Cross (2009) discusses the third
contrasting belief, practice versus understanding. Traditional teachers would view practice as
the way a student becomes an expert (Cross).
Practice is the only way that the students would be able to gain
understanding of the concepts (Cross; Sztajn, 2003). This implies that the correct answer is of
the utmost importance which means a focus on more practice and less on problem
solving (Cross). “Remember” is the key
word and stressed with students of lower socioeconomic status (Sztajn). The students need to practice at school due
to lack of parental help and encouragement at home. In order for this group of students to be
successful they must practice and remember the steps thereby requiring
organization skills. Learning is viewed as order with a focus on the “stepping
stones” of what the students need to learn and the many rules that they must
know (Sztajn, p.61). This leads to
teachers organizing material systematically into topics for the students (Harel
& Lim, 2004). Listening to the
students was stressed by several researchers (Harel & Lim; Lappan, 2000;
Uworwabayeho, 2009). Harel & Lim
posit that practice was necessary because teachers lacked listening skills,
truly listening to what the students asked and stated. The teacher focused on what he or she thought
was the best way to teach, or learn, the concepts. Uworwabayeho stated that the teachers needed
to listen to the student’s thoughts and justifications in order for them to
learn the material. Listening is very
important for teachers implementing mathematical reform.
Students construct their knowledge and
engage in mathematical discourse when involved with instruction based on the
reform movement (Cross, 2009). This
means that the teachers need to truly listen to what the student is
saying. Constructing their knowledge by
developing meanings and knowledge to obtain understanding encourages the
mathematical discourse where the students analyze and adjust their thoughts
(Cross). Teachers are better able to
spend time using manipulatives and doing projects due to the support the
students receive at home (Sztajn).
Sztajn’s (2003) case study revealed that the key word for this approach
would be enjoy; students need to enjoy the mathematics. Those students with parents having a college
degree were found to be better able to be successful with the reform approaches
(Sztajn). Even though teachers may have
contrasting beliefs they can be grouped into belief systems.
Cross (2009) discussed three belief
systems that made up the teachers view of mathematics, mathematics expertise,
teaching, and learning. The first belief
system is that mathematics is believed to be basic operations and
formulas. Mathematical expertise for
this system is viewed as expert usage and knowledge of rules, skills, and
facts. Teaching would be focused on
exposing students to the rules and formulas in order to show how to use them
correctly. Knowing how and when to use
the formulas correctly to obtain the correct answer leads to students
understanding and learning the material.
The second belief system is a way of thinking. Mathematical expertise is viewed as problem
solving and critical thinking. Teaching would be focused on being competent at
designing activities and creating lessons so that students could build their
knowledge. Learning is accomplished when
the students take ownership of the material and the process of coming to know
the mathematics. Collaboration, as well
as individual student work, would be assessed so that the teacher might gain
information regarding student knowledge.
The last belief system is solving complex problems as a non-cohesive
domain of knowledge. Teachers view
mathematics as students obtaining a meaningful appreciation and understanding
of the mathematical concepts. Cross also
states that teachers would know when and how to use proper procedures that may
be different depending on the subject.
Teaching would be helping students learn to be independent thinkers and
engaged in the development of learning, taking ownership of their knowledge in
this belief system as well.
Lappan (2000) stresses that teachers
should get to know their students. Being
aware of students’ culture will enable the teacher to determine how to
encourage student success (Boaler & Staples, 2008). Knowing the student as an individual and
their culture is important but teachers need to delve into how the student
learns mathematics and makes sense of concepts (Lappan; Nathan & Koedinger,
2000a; Nathan & Koedinger, 2000b). A
teacher’s beliefs about a student’s culture and ability to learn with out
knowing the student can effect student achievement. Having a clear understanding of expectations
and where they are academically will enable students to become more productive
and successful (Boaler & Staples).
Teachers need to listen to their students as stated previously and
encourage and plan for math talk (Lappan).
Allowing students to communicate their thoughts may be time consuming
but in the long run beneficial for student knowledge gain (Uworwabayeho,
2009). Not only does it take time in the
classroom but teachers need to also realize that it takes time to assimilate
discourse in to their classrooms and to become comfortable doing so
(Uworwabayeho). Anticipating what student responses may be will help in
planning but teachers also need to believe that students can perform, can
respond with thinking skills, and are willing and wanting to learn the
material. Teachers need to employ better
listening skills but students also need to put problems, concepts, and
approaches in their own words, ask good questions, use manipulatives
responsibly, and justify their statements (Boaler & Staples). Having an open mind and a willingness to
change one’s beliefs will enable a teacher to learn from their students.
Participant
Concerns of Student-Centered Learning
Time is a concern not only with
encouraging student discourse but also with developing and implementing student-centered
instruction (Cross, 2009). Teachers are
concerned that they will fall behind the expected benchmarks on the curriculum
and that the students might have to be re-taught if they did not grasp the
material with the new method. Beliefs
may get in the way and the students would then not have the opportunity to try
problem solving, critical thinking, and activities that apply the
concepts. Uworwabayeho (2009) also
elaborates on the concept of time with respect to classroom instruction stating
that habits of asking student to explain their thinking do not happen
suddenly. Cross mentioned, in addition to
time, the concern of institutional factors, assessments, school culture, class
sizes and discipline. Institutional
factors include the administration providing support for change and curriculum
maps being conducive to the time expectations required for student-centered
learning and to the connections of the content.
Assessments and benchmarks will provide a wealth of information to aid
planning, instruction, and classroom assessment but time needs to be provided
for teachers to analyze the data from these assessments and to discuss best
practices. Teachers need to believe in
their administration but the administration must also believe in the teachers
and the students. Materials and equipment are a concern with
the lack of and/or knowledge of how to implement them appropriately and efficiently
(Uworwabayeho). Teachers may need to
share materials so that will effect planning and increase time issues. Attention to how the students use the
materials or equipment places additional concerns on teachers. Internet usage needs to be monitored and
equipment safety issues need to be addressed.
Once concerns are addressed and plans of actions in place to deal with
each then success may result.
Boaler
and Staples (2008) conducted a case study of reform movement vs. traditional
and problem solving approaches on three schools that they called Greendale,
Hilltop, and Railside. The focus was on
Railside as they implemented the reform movement approach. The other two schools were split between
traditional and the problem solving approaches.
Railside teacher’s focused on beginning algebra courses with a
curriculum oriented to mathematical reform involving student’s conceptual
understanding, collaborative work, multiple representations, math talk, and
making connections between algebra and geometry. The teachers acknowledged different
approaches and solution routes as they employed open task problems that
permitted entry levels for different student abilities. All the students at Railside took Algebra I
in their first year of high school and teachers held high expectations for
all.
Students
at Railside came from diverse backgrounds, had the largest English Language
Learners of all three high schools, more students qualified for free/reduced
lunch, lowest percentage of parents with college degrees, and the lowest
student achievement. At the beginning of the year Railside staff tested the
incoming students on middle school skills and then a post test at the end of
the year. Results showed that Railside
students were approaching the levels of achievement of the other two high
schools. Improvement continued through
year two but not as notable in year 3 as that course was not developed with the
same vision and the teachers had less experience. Differences between Black, White, and Latino
students at Railside began to vanish while the achievement gaps between
ethnicities remained the same at Greendale and Hilltop High School’s. Railside teachers believed that their
students could achieve and had the support form their administration. They were given the opportunity to step out
and try new methods, saw the positive affects on the students attitudes and
growth, which in turn helped to change their beliefs.
Deciding when to implement
professional development is a concern that many deliverers of sessions
struggle. Do you only implement if
teacher beliefs correlate with the new approach and goals or do you jump in,
educate and support the teachers so they might see a change in their students
so their beliefs will change and become stronger. Loucks-Horsley, et al. (2010) states that “It
is important not to wait to provide professional development until the entire
school community is united around a common vision (p. 33).” As seen above, teacher beliefs play an
important role with planning and instruction so must be considered, along with
students, when developing sessions. At
the same time developers need to change as the teachers change in order to
continue to meet their needs. Adapting
to their changing needs will reinforce the concept of supporting the teachers
through all of their concerns, the influences that effect their beliefs, and
the effects of beliefs on instruction and learning teachers have that lead to
the changes initiated through professional development.
References
Belenky,
M. F., Clinchy, B. M., Goldberger, N. R., & Tarule, J. M. (1986).
Women’s ways of knowing: The development of self, voice, and mind. New York: Basic Books.
Boaler,
J., & Staples, M. (2008). Creating mathematical futures through an
equitable teaching approach: The case of Railside School. Teachers
College Record, 110(3), 608-645.
Cooney,
T. J., Shealy, B. E., & Arvold, B. (1998).
Conceptualizing belief structures of preservice secondary mathematics
teachers. Journal for Research in Mathematics Education, 29(3), 306-334.
Cross,
D. I. (2009). Alignment, cohesion, and change: Examining
mathematics teachers’ belief structures and their influence on instructional
practices. Journal of Mathematics Teacher Education, 12, 325-346.
Cwikla,
J. (2002). An
interview analysis of teachers' reactions to mathematics reform professional
development. Presented at the Annual
Meeting of the American Educational Research Association, New Orleans, LA.
Derry,
S., Wilsman, M., & Hackbarth, A. (2007). Using contrasting case activities to deepen
teacher understanding of algebraic thinking and teaching. Mathematical Thinking and Learning, 9(3), 305-329.
Guskey,
T. R. (1986). Staff development and the process of teacher
change. Educational Researcher, 15(5).
Harel,
G., & Lim, K. H. (2004). Mathematics teachers’ knowledge base:
Preliminary results. Proceedings of the
28th Conference of the International Group for the Psychology of
Mathematics Education, Vol. 3, 25-32. Norway: PME.
Lappan,
G. (2000). A vision of learning to teach for the 21st
century. School Science and
Mathematics, 100(6),
319-326.
Lin,
P. (2004). Supporting teachers on designing
problem-posing tasks as a tool of assessment to understand students’
mathematical learning. Proceedings of the 28th
Conference of the International Group for the Psychology of Mathematics
Education, Vol. 3, 257-264. Norway:
PME.
Loucks-Horsley,
S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010).
Designing professional development for teachers of science and
mathematics (3rd ed.). CA: Corwin.
Nathan,
M. (2004). Confronting
teachers’ beliefs about students’ algebra development: An approach for
professional development. Psychology
of Mathematics & Education of North America, Toronto, CA.
Nathan,
M. J., & Koedinger, K. R. (2000a). An investigation of teachers’ beliefs of
students’ algebra development. Cognition and Instruction, 18(2),
209-237.
Nathan,
M. J., & Koedinger, K. R. (2000b). Teachers’ and researchers’ beliefs about the
development of algebraic reasoning. Journal for Research in Mathematics
Education, 31(2), 168-191.
National
Council of Teachers of Mathematics. (1990).
Professional
standards for teaching
mathematics.
Reston, VA: Author.
National
Council of Teachers of Mathematics. (2000).
Principles
and standards for school
mathematics.
Reston, VA: Author.
Olson,
J. C., & Kirtley, K. (2005). The transition of a secondary mathematics
teacher: From a reform listener to a believer. In Chick, H.L. & Vincent, J. L. (Eds.), Proceedings of the 29th Conference of the
International Group for the Psychology of Mathematics Education, Vol. 4.
Melbourne: PME.
Orrill,
C. H. (2006). What learner-centered professional development
looks like: The pilot studies of the InterMath Professional Development
Project. The Mathematics Educator, 16(1), 4-13.
Perry, W. G. (1970).
Forms of intellectual and ethical
development in the college years.
New
York:
Holt, Rinehart, & Winston.
Peterson, P. L., Fennema, E.,
Carpenter, T. P., & Loef, M.
(1989). Teachers’ pedagogical
content
beliefs in mathematics. Cognition and Instruction, 6(1), 1-40.
Petty, T. (2007). “Empowering teachers: They have told us what
they want and need to be
successful,”
The Delta Kappa Gamma Bulletin 73(2),
25-28.
Sztajn,
P. (2003). Adapting reform ideas in different mathematics
classrooms: Beliefs beyond mathematics. Journal of Mathematics Teacher Education,
6, 53-75.
Uworwabayeho,
A. (2009). Teachers’ innovative change with countywide
reform: A case study in Rwanda. Journal of Mathematics Teacher Education,
12, 315-324.
Weidemann,
W., & Humphrey, M. B. (2002). Building a network to empower teachers for
school reform. School Science and Mathematics, 102(2), 88-93.
Weiss, I. R., & Pasley, J. D. (2006). Scaling up instructional improvement through teacher professional development: Insights from the local systemic change initiative [Policy Briefs]. Consortium for Policy Research in Education, 44.
White-Clark,
R., DiCarlo, M., & Gilchriest, N. (2008). “Guide on the side”: An instructional
approach to meet mathematics
standards. The High School Journal, 91(4), 40-44.
Zollman,
A. & Mason, E. (1992). The Standards’ beliefs instrument (SBI):
Teachers’ belief
about the NCTM Standards. School Science
and Mathematics, 92(7), 359-362.