Fall 1997 - Volume I, Issue
Since the publication of Everybody Counts: A Report to
the Nation on the Future of Mathematics Education (National Research Council,
1989) and the Curriculum and Evaluation Standards for School Mathematics
(National Council of Teachers of Mathematics, 1989), teachers at all levels as
well as high school mathematics teachers have been discussing and debating the
vision of the Standards documents and endeavoring to make reforms or changes in
their mathematics teaching practices. These reforms include teachers helping
students develop mathematical power, that is, the ability to explore,
conjecture, reason logically, to solve nonroutine problems, to communicate
about and through mathematics, and to connect ideas within mathematics and
other intellectual activity (National Council of Teachers of Mathematics,
1991). A question that arises from this debate is "what are teachers doing to
make changes in their mathematics instruction since the publication of these
In order to address that question, I designed a small
research project that centered around the question "What do you believe is
effective mathematics teaching?" In asking this question, I wished to gather
information that would contribute to knowledge about the mathematics education
reform movement in the eyes of middle school mathematics teachers. I chose
semi-structured interviews and classroom observations as the vehicles for
inquiry. I wanted to learn how teachers talked about their teaching and also
how they were translating their thoughts to working with students.
study builds on the literature on teacher thinking and knowledge literature of
Shulman (1987) which includes the thinking about pedagogical content knowledge.
This study also looks at the development of a possible framework in looking at
teacher change in mathematics. This framework comes from the "major shifts" in
mathematics teaching practices as written in the Professional Standards for
Teaching Mathematics (National Council of Teachers of Mathematics, 1991).
In these Teaching Standards, "major shifts" are presented that must occur in
the practice of teaching mathematics so that teaching for the empowerment of
students can occur. To teach mathematics effectively, classrooms must become
mathematical communities, not collections of individuals. Teachers must
encourage students to use logic and mathematical evidence as verification,
reason mathematically, conjecture, invent, and engage in problem solving, and
connect mathematics with its ideas and applications. These ideas present a
shift from students viewing the teacher as the sole authority for correct
answers, that students memorize procedures, that students find answers in a
mechanistic way, and that students view mathematics as a body of isolated
concepts and procedures.
I pursued this research because I want to
learn about how instruction is changing for our future undergraduate college
students in which there is a group who will study to become teachers. The
phenomenon of "teachers teach the way they are taught" has the potential to
change as students are taught differently in school. This research also
contributes to how professional development programs can produce shifts in
teaching practices. Teacher education can be examined in a way that provides
more information about how teachers and students can be better prepared to
become effective teachers.
In an effort to
understand the shifts that are occurring in middle school teachers' practices,
during the Fall of 1993, I interviewed three middle school teachers about what
they believed was effective mathematics teaching. As a follow up to the
interviews, I observed a mathematics class taught by each teacher. Each
interview was audio taped and field notes were recorded for each classroom
observation. This data helped me answer the research question.
the teachers were from the same urban school district. Two of the teachers were
sixth grade teachers and one was an eighth grade teacher. All three of the
teachers were teaching in regular education classrooms. These teachers were
chosen because of their interest in the improvement of mathematics education in
the district. This interest is known because of their participation in a
project for the improvement of teaching mathematics offered by a mathematics
education professor from a local university.
The research question,
"What do teachers in an urban setting believe is effective mathematics
teaching?" guided me through my interviews with teachers and observations of
each of the teachers' classroom lessons. My interview questions often lead to
asking the teachers to clarify their answers by giving explicit examples from
their teaching experiences.
The "major shifts" outlined in the
Professional Standards for Teaching Mathematics (National Council of Teachers
of Mathematics, 1991) served as a framework to guide the design of the data
collection. I relied primarily on the semi-structured interviews and
observations as data collection strategies. All interviews were conversational
in style with the purpose of allowing teachers to express their own personal
views on effective mathematics teaching. I audio taped the interviews at each
teachers' own school site and transcribed the three interviews. The classroom
observations followed within a week after each interview. The interviews were
used to inform my observations to observe how each teacher's beliefs affect
The main research question was posed to learn about the
"shifts" that are taking place in the teaching of middle school mathematics.
With such a small sample of middle school teachers, it will be hard to make
general statements about what is happening in middle school classrooms as a
whole, but this study is one of many that is a start at looking for general
trends in the practice of middle school teachers involved in the mathematics
education reform movement.
Findings from this study
indicate that there are, in fact, "shifts" in practice as outlined in the
Teaching Standards. It was found that responses to the main research question
reflect the shifts away from mathematics classrooms as collections of
individuals, and mathematics as isolated concepts and procedures, to
mathematics classrooms as mathematical communities and mathematics as
connecting ideas and applications. The shift of teacher as sole authority and
student reliance on memorization of procedures toward logic and mathematical
evidence as verification and mathematical reasoning was also observed. The
teachers in this study also spoke about the shift in assessment activity from
paper and pencil assessment to assessment aligned with instruction. These
findings have the potential to have important implications in how teacher
change can be examined and reported. Below are snapshots of findings from
interviews and classroom observations.
In reviewing the
data, it was found that there was a shift from mathematics classrooms as
collections of individuals to mathematics classrooms as communities of
students. The teachers in this study reported that they felt that cooperative
learning is a major factor in effective mathematics teaching. These teachers
emphasized that during math time, "it's okay to share, it's okay to ask other
students in your group to share their process in problem solving." In a
classroom observation of students working on fraction computation problems,
students worked in cooperative groups and shared their strategies and processes
with their group and with the whole class. The class worked together at
resolving any difficulties that students were having with the problems. The
teacher asked group leaders to report group results and students had the
responsibility of listening to each other and working as a community rather
than a set of individuals.
A major theme that arose in teachers interview
responses was making mathematics real for their students, in other words, to
connect mathematical ideas and applications to a student's life. From the
classroom observations, one of the sixth grade teachers designed lessons around
an activity about aquariums. In this activity, students were given a 30 gallon
aquarium and $25 to spend on fish. Students brainstormed about their
experiences with aquariums and what materials they would need to keep an
aquarium. In solving the problem about what kind of fish to buy for the
aquarium, students came up with questions about the types of food that fish
eat, and what kinds of fish get along in the same aquarium. When asked what
this activity has to do with mathematics, students spoke about measurements
(gallons, cubic centimeters or inches, feet, teaspoon, pinch, degrees of
temperature, water air pressure). The students also spoke at length about the
type of problem solving this type of project needs. This discussion about
problem solving showed the power the students had in "thinking about the
thinking" or metacognitive skills needed to solve a problem they may have a
chance to tackle in their lives. Students exclaimed that they would need a much
larger budget so that they could feed the fish and decorate and clean the tank.
Students were able to extend their thinking on the original problem of how much
fish to buy to many other issues they felt should be under consideration.
The teachers spoke at
length about needing to know what students were thinking about while engaged in
mathematical investigations and activities. This allowed the teachers to gauge
a student's mathematical authority. To do this, teachers designed lesson
activities that allowed students to investigate mathematical situations and to
talk about mathematics, thus allowing the teachers to assess student knowledge
about their mathematical problem solving skills as well as how students verify
their results and gain mathematical authority. During one lesson observation,
students were using fraction factory pieces to show equivalence of fractions.
While setting up examples of equivalent fractions with the pieces, students
were able to verify on their own whether fractions were equivalent or not. When
students had trouble with this activity, the teacher asked the student to show
her how two fractions are equivalent using the fraction factory pieces. It was
through this student investigation guided by the teacher that the student was
able to gain authority through verification of a solution to a problem. In
response to many of the students questions about the activity, the teacher
often asked other students or groups of students to give examples of equivalent
fractions, allowing the students to see examples made by other students. The
teacher transferred authority of correctness of answers and verification of
results to the students, not entirely to herself.
I observed students in the
classroom who were actively engaged in investigating mathematical problems and
showing their mathematical power. Teachers in this study were aware of the
potential of "learning while learning about what students were learning."
Teachers in this project were in the stage of learning about how to assess
student knowledge that is aligned with instruction. Instead of reliance on
paper and pencil quizzes and tests, the teachers in this study were working on
incorporating ways of examining student progress while students are engaged in
mathematical activities. One of the teachers observed that I've learned much
more through seeing students work together on mathematical tasks than I have
seen on a conventional quiz or test. There is a dynamic there that I see much
better when the students are working together on a task rather than when they
are working alone in isolation.
As seen in the above examples, there are
indeed shifts occurring in the practices in the teaching of middle school
The Professional Standards for
Teaching Mathematics (National Council of Teachers of Mathematics, 1991) has
provided a good framework to gather information about teaching practices. The
interview and classroom observation data was easily coded along the lines of
the shifts in practice. There is potential for this framework to be implemented
in the observation and supervision of prospective teachers. The findings
of this study show that the mathematics reform movement is alive and well. The
sixth grade students in the classrooms of the teachers interviewed in this
study are now sophomores in high school. This means that students are coming to
the undergraduate mathematics coursework prepared differently than those
'traditionally prepared.' Given this information, there needs to be more
dialogue about what types of knowledge and dispositions students are bringing
with them to the college experience. A new question arises about how to prepare
teachers that have reformed experiences of how to teach mathematics.
National Council of Teachers of Mathematics. (1989).
Curriculum and Evaluation Standards for School Mathematics. Reston, VA:
National Council of Teachers of Mathematics.
National Council of Teachers
of Mathematics. (1991). Professional Standards for Teaching Mathematics.
Reston, VA: National Council of Teachers of Mathematics.
Research Council. (1989). Everybody Counts: A Report to the Nation on the
Future of Mathematics Education. Washington, DC: National Academy Press.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform.
Harvard Educational Review, 57(1), 1-22. 1 Return to the Journal of
Pedagogy, Pluralism & Practice Main Page
Journal of Pedagogy, Pluralism and Practice Main Page
Fall 1997 Issue Main Page
Preface - A Dedication to Paulo Freire
Solange Lira and Bill Stokes - Toward Pedagogies of Freedom
Maria de Lourdes B. Serpa and Caetano Valadão Serpa - Freire Pedagogia da Autonomia Book Review
Sandras Barnes - Connecting Theory to Professional Growth and Pedagogical Practices in a Multicultural Setting
Peggy McIntosh and Emily Style - The National SEED Project on Inclusive Curriculum
Patricia L. Jerabek - Creating A Multicultural Learning Environment at Lesley College
Walter E. Stone, Jr - Making the Shifts
Barry Sugarman - Learning,Working, Managing, Sharing
Sebastian Lockwood - Density of Coincidence
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