professional development for mathematics teachers: a team approach
TRANSCRIPT
Professional Development for Mathematics Teachers: A Team ApproachAuthor(s): P. Mark Taylor, Brandy Ferrell and Theresa M. HopkinsSource: The Mathematics Teacher, Vol. 100, No. 5 (DECEMBER 2006/JANUARY 2007), pp. 362-366Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27972249 .
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Professional Development
for Mathematics Teachers:
A Team
Approach P. Mark Taylor, Brandy Ferrell, and Theresa M. Hopkins
While most teachers are aware that
professional development can be much more than one-shot meet
ings, many still struggle to find the time to do something different.
What does "different" look like, and how does one find the time to do it? How much time and effort will be required beyond our regular workday? We found deep learning on the part of both teacher and student was not as difficult to achieve as we had
anticipated when we teamed up with fellow mathe matics teachers and followed some basic principles.
This story of learning and growth comes from our work as members of a professional develop ment team (PDT), an outgrowth of the Appala chian Collaborative Center for Learning, Assess
ment, and Instruction in Mathematics (ACCLAIM). PDTs are formed as a merger between mathematics teachers within a school or district, one or more
preservice mathematics teachers from a local col
lege, and the preservice teacher's university super visor. We operated under the assumption that there is no dichotomy between preservice teachers and in-service teachers but that preservice and in-service teachers reside at different places on one professional development continuum. In this
framework, the opinions and experiences of the
preservice teacher are valued as much as those of the in-service teachers. The preservice teacher, in
turn, must recognize the richness of the experience of the in-service teacher. The focus of the PDT is the preservice and in-service teachers' mutual goals
362 MATHEMATICS TEACHER | Vol. 100, No. 5 ? December 2006/January 2007
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of (1) having students master mathematics and
(2) moving along the professional development con tinuum by working and learning together.
Telling the entire story of our work on the
professional development team would take a book rather than an article, so we offer it here as three PDT Principles. The first two principles are related to the activities of our PDT over the first two years: (1) Only work on a goal that all of your PDT mem bers see as central to their work and (2) Smaller
goals can lead to greater gains. The third principle, Real change occurs one lesson at a time, is con
nected to the story of the PDT's third year and to the experience of a lesson study action research
plan initiated by our preservice teacher.
PDT PRINCIPLE tt1: Only work on a goal that all of your PDT members see as central to their work as mathematics teachers
Principle #1 can help solve one of the perceived difficulties of a PDT: time constraints. Are students
struggling with solving systems of linear equations? Are they struggling to be motivated in all math ematics courses? Is our aim to have more students enrolled in four years of high school mathematics? The first step in the work of a professional develop ment team is establishing a common goal.
Whatever the goal, it must be focused on the stu dents. In our case, the PDT instantly agreed on what was needed: a shared vision of curriculum. In fact, teachers were not always sure what was happen
ing in the room next door, even if the same course was being taught. We assumed that we knew, but
every once in a while we would hear about a lesson or activity that made us wonder. Hence, the team decided to learn about curriculum and to engage in
creating and teaching such a curriculum. How did this effort address the perceived time
constraint difficulty associated with PDTs? As a result of our common goal, we realized we did not need to seek much additional time because the goal was something we were already going to pursue. What changed was that we would coordinate our efforts and work together more frequently, thus
streamlining the process. In year one, the team began to meet once a
month to discuss curriculum issues. Highlights from that first year included a teacher's trying out a unit from the Integrated Mathematics Program (IMP), a curriculum based on the NCTM Stan
dards, which incorporated "The Pit and the Pen dulum" by Edgar Allan Poe. We chose this unit to
sample an integrated curriculum because it is simi lar in structure and method to Connected Math
ematics, which had been used for a few years in the middle school. This unit was quite different from
the textbooks used in the past. It made connec
tions, not just between mathematics concepts, but also between mathematics and literature. Twice
monthly meetings focused on the impact the story had on students and teacher in terms of instruction and assessment. Students were challenged to think for themselves, and the teacher, as well as the stu
dents, had to make adjustments. The teacher had to allow students to struggle and had to encourage them to answer their own questions whenever pos
sible. The teacher also had to allow more time for in-class work as well as for discussion?time that had previously been devoted to lectures or working examples. Teaching this unit prepared us for the idea that there was a lot to learn when implement ing an integrated curriculum.
The following summer we met on four days in
June and July, and we also met at the beginning of the school year. The idea was to produce a rough draft of a curriculum that was both integrated, emphasizing connections within mathematics, and
coordinated, with one course/year leading to the next with little overlap. We had decided not to be
guided by a textbook, because no available inte
grated textbooks were written specifically for the Tennessee state standards. We began by separating all of the standards from sixth grade through alge bra 2 into content strands.
The standards for each grade/course had five con tent areas: number and operations, algebra, geome
try, measurement, and data analysis and probability. We examined each strand separately at first, looking for when a concept/procedure was introduced, when it was extended, and when it was repeated. We made decisions within each course as to whether the stan dard should be a complete lesson or a secondary les
son, or whether it should just be brought up again as homework to retain student skills.
After each content strand was completed, we
pulled the course standard back together. The next
project was organizing the standards within the course. We read the standards aloud to each other and decided which ones fit naturally together. We discovered that content strands are not the most natural organizing feature. Our end products were more integrated, although most units had a predomi nant content strand. We set up spreadsheets for each course in order to track what had been taught and
what texts or materials had been used and to note
changes suggested for next time. The idea was to connect across content standards by integrating as
much as possible. As figure 1 indicates, the spread sheets began as a list of content objectives from the state standards, grouped on the basis of natural con nections we had identified. The spreadsheet format allowed for ongoing changes as the units were imple mented, as decisions were made about what to group
Vol. 100, No. 5 ? December 2006/January 2007 | MATHEMATICS TEACHER 363
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Algebraic Thinking Unit Objectives TN NCTM Instruction Notes Assessment Notes
IInformally describe and model the concept of inverse (e.g., opposites, reciprocals, and squares and square roots); 1.4 1.3.1
[Describe, model, and apply inverse operations;_ jUse real numbers to represent real-world applications (e.g., rate of change, probability, and proportionality);
3 1.6
Jse mathematical notations appropriately
1.9 2.1.3; 2.4.1
Perform operations on simple algebraic expressions, and
informally justify the procedures chosen;
?2.2.3
2.3 ?2.1.4 Apply measurement concepts and relationships in
algebraic and geometric problem-solving situations; 2.5 4.2.2
IUse estimation to make predictions and determine reasonableness of results; 2.6 1.3.2;
Demonstrate an understanding of rates and other derived and indirect measurements (e.g., velocity, miles per hr, rpm, cost per unit). 2.8
Analyze mathematical patterns related to algebra and
geometry in real-world problem solving;_
2.4.1; 4.2.4
3.2 2.1.1
Solve problems in number theory, geometry, probability and statistics, and measurement and estimation using algebraic thinking and symbolism (attention given to solving linear equations); 3.3
Communicate the meaning of variables in algebraic expressions, equations, and
inequalities;_ Interpret the results of algebraic procedures;
1.2.1;
3.4.4; 4.2.4
2.2.2; 3.4
3.5 2.3.3
Apply the concept of variable in simplifying algebraic expressions, solving equations, and solving inequalities; 3.6 2.2.1; 2.2.2
[interpret graphs that depict real-world phenomena; 3.7 ?2.4.1; 3.4.5
iModel real-world phenomena using graphs.
?Apply the Pythagorean Theorem in problem solving;
3.8 3.4.3; 3.4.4
5 5.6
Flg. 1 First draft of a unit from Foundations 1, a basic mathematics course for technical track students
on a particular day, what activities to use, and what
assessment tools to include, enabling us to shift com
ponents among units more easily. The second year, teachers set out to imple
ment this new curriculum and quickly found it to
be an overwhelming task. The curriculum, as we
arranged it, had no set text. In fact, most teach ers were using more than one text. Poor student
skills were our greatest challenge. One of the basic
ideas we had agreed upon was to avoid unplanned
overlapping of courses/years, but we found that we
continued to feel the need to review material from
previous years. We knew some content would need
additional work, especially in the first year of the
project, but we did not anticipate how much time
would be required. This threw our timing off quite a bit. As a result, teachers mainly fell back to a
position of doing what they had always done while
trying out a few more integrated activities. The following summer, we took a fresh look at
what we had planned and decided that we had bit ten off more than we could chew. The result had
been more of the same kind of curricular inconsis tencies and redundancies. The difference was that
everyone knew better and was dissatisfied. Upon reflection at that point, we had learned (a) the
mathematics that everyone else thought was impor tant, within the team and within our state, (b) how
the curriculum that we had been teaching matched
up with the Tennessee standards, and (c) just how
much work it is to coordinate a curriculum over
several years/courses. The greatest benefit was the
increased communication among the mathematics
teachers on our team. The time we were spending on this project had increased only a little, but our
ongoing conversations were now more focused.
This led us to Principle #2.
PDT PRINCIPLE ?2: Smaller goals can lead to greater gains In an effort to keep our goal manageable, the deci
sion was made that we would focus only on alge bra 1. This course is seen as a flagship course; all
students are required to pass an end-of-year state
assessment on this subject. Algebra 1 was also a
good choice because each teacher, no matter course
or grade level, taught objectives related to those
of algebra 1. This decision allowed us to zoom in,
364 MATHEMATICS TEACHER | Vol. 100, No. 5 ? December 2006/January 2007
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focusing on how to develop a unit on the go. That summer we took a fresh look at the algebra stan dards and how to group them into naturally con nected clusters that would become units.
The following school year, our monthly meet
ings focused on implementation issues around the
algebra 1 units and on developing anchor activities. An anchor activity is a lesson or project that will tie most of the objectives for the unit into one nice
package. One anchor activity we used was to have students measure the steepness of ramps in and around the school building. Students walked up and down the various ramps to feel their steepness, measured their rise and run, and sketched them on a graph to compare. These activities formed a
foundation for the remainder of the unit on linear
functions, providing not only graphs but also experi ences to use as benchmarks to understand slope and
intercept. The need to develop an anchor activity for each unit fit well with the university's teacher
preparation program, which requires each preservice teacher to complete an action research project in a lesson study format. Principle #3 is an outgrowth of the experiences of our preservice teacher, Brandy Ferrell, as she completed her action research project.
PDT PRINCIPLE ?3: Real change occurs one lesson at a time The research conducted during the third year of this program involved the concept of lesson study
within the PDT. In this lesson study, four mem
bers of the PDT collaborated to research, compose, teach, reflect upon, and revise a single lesson as determined by the needs of our students. When we
decided to implement the lesson study, we chose a
mathematics topic that was common to each teach er's curriculum and a concept in which students had encountered difficulty in previous years. By focusing on an introductory lesson on functions, we
hoped to provide students with a better understand
ing of why we study algebra. We were interested in
providing a means of motivation for the algebraic manipulation and graphing with which students had previously struggled. Because our focus as a PDT had been algebra, performing a lesson study on
this topic fit well with our PDT work and fulfilled our goal to create an anchor lesson for a specific topic in algebra. The role of the preservice member of the PDT included the facilitation and implemen tation of the lesson study as well as the analysis of the outcome of using the lesson study process.
After reviewing research on curriculum, instruc
tion, and assessment issues involving the function
concept, we developed a lesson for implementation in one seventh-grade prealgebra class and two high school Foundations II classes. The key ideas we
addressed, as determined by the literature review,
included studying data relevant to student interests, connecting various representations of functions,
illustrating with metaphors, and focusing on the
concept rather than the algorithm. To make the les son relevant to student lives, we allowed them to find examples of functions in the newspaper. The students pursued this activity with interest and
enthusiasm, and it provided a method for assessing their understanding of the function concept. Only after developing students' concept definitions and
introducing a variety of representations of functions did we introduce the symbolic form of this concept. After each stage of the lesson study, we met to evalu
ate, reflect, and revise based upon the results of the lesson. Because of the success of each lesson attempt,
only minor changes were made to the lesson between each stage of the lesson study. Assessment data indi cated that the students understood the concept much better with this approach than they had with lessons used in previous years, and student performance on
algebraic manipulation and graphing improved after the introduction of functions in this class. This indi cated the importance of researching teaching strate
gies for certain concepts and planning lessons as a team prior to implementing lessons in the classroom.
By using the lesson study process, we have dis covered some effective teaching methods that have had a positive impact on our team. Lesson study not only has helped us grow as individual teachers but has also assisted us in uniting as a team and
aligning our algebra curriculum across our middle and high school courses. Because of the extent of research and collaboration required in a lesson
study, this project has been an incredible instru ment for pulling together the coursework for the
preservice teacher and aligning it with her role as a preservice member of the PDT. The lesson study has also provided a resource for future teachers to use when teaching the function concept, and it has
given them a true team experience to duplicate as
they join other professional teams in the future. The
preservice mathematics teacher member of the PDT would encourage all other teaching professionals to participate in lesson study, for educators should
constantly strive to improve teaching strategies, and lesson study is an invaluable tool for doing so.
The PDT's impact on the teacher preparation program at the University of Tennessee is both
personal and practical. Since one of the major focus areas in a teacher education program is curriculum,
working within the ACCLAIM PDT on curriculum
alignment complemented the preservice teachers' coursework and better prepared them for a career as a teacher. Brandy Ferrell, the UT preservice mem
ber in the final year, found that her involvement in the professional development team allowed her to
experience firsthand the act of determining where
Vol. 100, No. 5 ? December 2006/January 2007 | MATHEMATICS TEACHER 365
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and how to implement algebra 1 stan
dards best. According to Ferrell,
UT's program introduced me to
important issues in education and the
many theories about how students
learn, but I found that I was lacking in actual "real" applications related to the classroom, such as the design
ing of curriculum. As we examined
the standards of an entire course and
found how they related to each other, I realized how those standards could
be aligned to provide a relevant con
text for students to learn mathemat ics. This introduction into curriculum
matters provided a framework for the
research I conducted while complet ing my teacher preparation program.
CONCLUSION Our professional development team
started with the common goal of a
cohesive, coordinated mathematics cur
riculum (Principle #1). We began to
find tangible success when we narrowed our focus to a single course, developing
units and anchor activities (Principle #2). Significant change in practice came
when we shifted our focus to a single les son (Principle #3). This placed our daily
planning in the larger context of a coor
dinated curriculum, which enabled us to
learn lessons about mathematics teaching in general from studying a specific lesson.
Following the basic principles pre sented here can take many different
forms. The professional development team merely offers one variation of the
kind of structure that can lead to success
ful professional development for math ematics teachers. PDTs offer practicing teachers the help of a local professor and the energy of a preservice teacher. The
preservice teacher benefits by observing that what the professor has been sharing
truly can be applied in the classroom, and the professor can use these experi ences to coach future students better as
they learn how to teach. The mutual ben efits to the various educators involved in
the PDT allow us to improve upon the most important aspect of our careers: the
education of our students, oo
P. MARK TAYLOR, pmark@ utk.edu, is an assistant
professor of mathemat ics education in the de
partment of theory and
practice in teacher educa tion at the University of
Tennessee, Knoxville, TN 37996. He is interested in
mathematics curriculum and mathematics teacher education and professional
development. BRANDY FERRELL,
[email protected], is a high school mathematics teacher in Knoxville. She is interested in de
veloping integrated mathematics and science curriculum with real-world
applications for the preparation of
future engineering students. THERESA M. HOPKINS, thopkins? utk.edu, teaches mathematics educa tion at the University of Tennessee at
Knoxville. She thanks her coauthors for the opportunity to participate in
this endeavor
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