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The evaluation of the
teaching of mathematics should be a cyclical process involving
the periodic collection and analysis of information about an individual's
teaching of mathematics;
professional development based on the analysis of teaching;
the improvement of teaching as a consequence of the professional
development.
Elaboration
Evaluation is the vehicle
that connects a teacher's current teaching with the professional
development necessary to enable that teacher to improve the teaching
of mathematics. The evaluation process begins by collecting data
representative of the teacher's current practice. The collected
data is then analyzed with respect to what is valued in the teaching
of mathematics, such as the vision of teaching presented in the
first section of this volume. Aspects of instruction that are deemed
consistent with what is valued should be identified as well as those
needing improvement. Although this analysis may result in a report
for the teacher's personnel file, the more important outcome is
the creation of a plan to help the teacher develop professionally.
This plan should consist of instructional alternatives that have
the potential for improving teaching as well as strategies for implementing
these alternatives. Subsequent lessons are then observed and analyzed
to determine whether improvement has been made; hence, the evaluation
process is cyclical.
The cycle may require only
a few minutes, as would be the case if a teacher thoughtfully reviews
an algebra lesson taught during one period before teaching the lesson
again during a later period, or it may require a year, if college
coursework is recommended as a professional development activity.
In most cases the length of the cycle would be between those two
extremes. For example, a teacher may be trying to increase her repertoire
of assessment techniques and is interested in determining the impact
of the various techniques on student learning and disposition to
do mathematics within a given grading period.
Too often the evaluation
process involves only a supervisor making a single observation during
an academic year. This process is limited in at least three ways.
First, annual observations are much too infrequent to provide the
basis for a comprehensive professional development plan. Second,
evaluations by a single observer are too unreliable and ignore the
wealth of expertise available from the teacher and the teacher's
colleagues (see Standard 2). Third, evaluations based on a single
source, such as a single classroom observation, are similarly unreliable
and ignore other important sources of data that furnish additional
information about teaching that would be useful for planning professional
development (see Standard 3).
Professional development
can take many forms (see the third section of this volume), including
independent study, participation in in-service programs provided
by the school, enrollment in college courses, discussions with colleagues,
observations of colleagues, and attendance at professional meetings.
Evidence of successful professional development should appear in
subsequent teaching and be documented in future assessments.
The major goal for any evaluation
of mathematics teaching should be to improve teaching and enhance
professional growth. This emphasis would be a significant change
from present evaluation practices in many school districts in which
the goal is to provide documentation for personnel decisions or
simply to comply with a requirement that all teachers have an assessment
report added to their file according to some specified schedule.
Although it may not be possible, or even desirable, to eliminate
such reports from the evaluation process, it is critical that the
primary emphasis be placed on the use of evaluation to furnish the
basis for professional development activities aimed at improving
the teaching of mathematics.
Vignettes
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The school prepares
mentor teachers to support the professional growth of young teachers
and provides release time for their work.
The mentor teacher
collects data on her colleague's teaching.
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1.1
Before school begins, Jan Williams, an experienced fourth-grade teacher,
is assigned to work as a mentor with Tom Burton, a first-year fifth-grade
teacher at Valley Elementary School. Their classrooms are across the
hall from each other, so Jan has many opportunities to observe Tom's
class and confer with him about his professional development. On this
day Jan observes Tom handing out worksheets to his class after reviewing
the standard multiplication algorithm. She takes extensive notes that
describe the students as well behaved but often uninvolved and passive.
Tom tends not to ask many questions; Jan notes that of the fifteen
questions she observed, all but one required a response of a number
or a single word. Several students quickly finish their work and set
it aside. |
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The mentor teacher
observes students at work.
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When
Jan talks with the students, she notes that they have made a number
of errors in doing the worksheets. They don't seem very interested
in checking their work, however. |
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The teacher reflects
on the lesson and discusses the lesson with the mentor teacher.
The mentor teacher is helping him develop professionally.
The focus of the
discussion is on the questioning techniques that a teacher can use.
A goal is set
for improving a teaching skill.
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That
afternoon, Jan and Tom meet to discuss and analyze the information
she has collected. She asks Tom to reflect on the lesson and give
his evaluation of it. Tom recalls that several of his questions didn't
seem to spark much discussion; Jan shares her observation that most
of his questions could be answered by a number fact, a simple computation,
or a single-word response. He says that he is so concerned with his
own teaching activities that he neglects to focus on what the students
are doing. He admits that often the students seem uninterested in
the mathematics lessons but that he is hard pressed to figure out
any alternatives. Besides, he explains, this particular lesson was
a review lesson; he queries Jan on whether there are any better ways
of conducting review lessons. Jan offers several suggestions on how
he could rephrase his questions so that students would become more
involved in class discussions and on various types of activities he
could use to review the material more effectively. Tom decides that
he will work on improving his skill in questioning students. |
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the teacher and the mentor identify management strategies to monitor
students' work. |
When
discussing the part of the lesson involving seatwork, Tom recalls
that some students' primary objective appears to be to finish the
work ahead of the other students. Jan and Tom discuss ways of changing
this attitude. Jan explains the differences between monitored practice,
seatwork, and homework to help Tom plan how to structure class time.
She reminds Tom that the teachers have been working with Claude Andrews,
the principal and curriculum coordinator, to encourage the use of
different teaching techniques. |
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principal notes that the teacher has improved his teaching by incorporating
concrete materials and by improving the way he asks questions and
interacts with the students. It is clear to the principal that the
mentor has helped the teacher develop more effective ways of managing
students and improving the students' attitudes toward mathematics
and their willingness to work. |
In November, Claude makes
his second visit to Tom's class after discussing Tom's progress
with Jan. This time the class is studying the area of rectangles.
The students are asked to draw rectangles that would have an area
of 20 square units, each unit being a 1-inch square that the students
have cut out of card stock. The principal observes that Tom monitors
the students' work by looking at their drawings and quietly interacting
with them as they work. This activity is followed by a discussion
with different students sharing their work on the overhead. The
following drawings are put on the overhead.
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The evaluation
cycle continues as a means of staff development.
The principal
offers suggestions for improving the teaching of mathematics.
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During the conference with
Tom, Claude compliments him on implementing Jan's suggestions. Claude
suggests that since the students have just reviewed perimeter, Tom
incorporate problem solving involving both perimeter and area into
the lesson by using questions like the following:
If the perimeter of a
rectangle is 18 centimeters, what are the dimensions of the rectangle
with the smallest area if the lengths of the sides are whole numbers?
If a rectangle has a length
6 times its width and has an area less than 50 square centimeters,
what could be the dimensions of the rectangle, assuming the lengths
of the sides are whole numbers?
The principal suggests that
as a technique to enhance communication and involvement, Tom could
have the students work in pairs. Tom thinks this is a good suggestion.
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References are
provided for professional development.
Additional phases
of the evaluation cycle are scheduled.
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After
discussing how to get students to work in pairs or in small groups,
Claude lends Tom a reference book on cooperative learning and suggests
that he observe Jan's class again, since she often uses cooperative
learning groups when teaching mathematics. He schedules another set
of observations with Tom in February. |
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The teacher assumes
responsibility for periodic review and collecting information about
her teaching.
The teacher senses
that the problem involves the allocation of time within a given
class period.
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1.2
Three weeks before school begins, Mary Fisher examines two
videotapes of her ninth-grade algebra class from last spring. As a
seventeen-year veteran, she does a yearly review of goals and expectations
before the new school year begins. The videotape helps her recall
ending last year feeling vaguely disturbed by her inability to find
classroom time to emphasize more problem-solving activities and discussions.
She recognizes the need to find time to integrate student exploration
and computer-based modeling into the required curriculum. Yet, she
barely has enough time to cover the homework and present the new material.
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The teacher analyzes
her teaching performance using a videotape.
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Self-analysis
of the videotape reveals a startling observation regarding her allocation
of class time. She notices that she spends almost twenty-five minutes,
nearly the first half of the period, covering homework. More important,
she observes that many of the students are off-task and passive while
she does the problems on the board. |
| Professional
development includes reading professional journals and collaborating
with colleagues. |
Mary
calls Delores Laco, a ninth-grade teacher and colleague. She asks
Delores's opinion about changing their homework-review techniques
this fall. Delores tells Mary about a recent journal article that
she has read during the summer. The article offers various suggestions
for reviewing homework Mary says that she will read the article. They
agree to discuss alternatives for covering homework when they meet
during the in-service days the last week in August. |
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The teacher changes
her teaching practice as a consequence of her professional development
activities.
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Subsequently, Mary and Delores
decide to try four different methods of covering homework at various
times during the first quarter:
- Have students keep their
homework in a notebook that will be periodically reviewed.
- Pair students to discuss
their homework briefly at the beginning of the class period.
- Give frequent short
quizzes on the homework.
- Write solutions to selected
problems on a transparency and put these solutions on the overhead
sometime during the class period.
Mary discusses these strategies
with the students during the first week of school. In addition,
she decides to start each class period with a problem-solving activity,
moving homework to later in the period.
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log allows for a periodic collection of information for analysis.
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In an effort to monitor
her current classroom time spent on homework, Mary keeps a daily
log of the amount of time spent going over homework in class. Delores
suggests that they also keep track of the part of the period in
which homework review takes place. In early October, Mary reviews
her second-hour algebra class chart for the previous week.
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The teacher reflects
on her improvement of teaching as a consequence of her professional
development activities.
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Mary
shares the results with Delores. Mary is very pleased that she has
been able to reduce significantly the amount of time she has been
spending on homework. Yet she has not detected any drop in student
performance as a result of this new approach. To the contrary, the
students seem to be more attentive when covering homework. In addition,
she is pleased with her attempts to engage students in more problem-solving
activities and discussions. |
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The teacher notes
student difficulties and discusses them with his mathematics supervisor.
The supervisor
notices a particular problem.
The teacher confers
with the supervisor about the learning problem. The supervisor makes
a suggestion for addressing the problem.
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1.3 Jim Waseskuk
is discussing his seventh-grade mathematics class with his mathematics
supervisor, Ellen Davenport, as part of his quarterly evaluation.
He tells her that the class is studying relationships between parallelograms
and rectangles. Jim indicates that the class can identify properties
of a given figure but that they are having difficulty making comparisons
between parallelograms and rectangles. When Ellen observes the class,
she notes that many students have difficulty with questions such
as "How are the diagonals of a rectangle different from the
diagonals of a parallelogram?"
During a planning period,
Jim and Ellen discuss the problem. Jim expresses frustration at
not being able to get the students to visualize the various properties
of rectangles and parallelograms and, in particular, the diagonals
of the figures. Ellen suggests using cardboard strips with brads
at the corners to form a parallelogram that could be moved to form
a rectangle. Ellen also suggests that elastic thread could be used
to demonstrate how the diagonals change as the parallelogram becomes
a rectangle. Jim likes the idea and thinks that he will have each
of his students make a figure similar to the one Ellen has described.
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teacher follows through by constructing the figure and demonstrating
it to students. |
The next day Jim describes
to the students how to make the figures. He demonstrates how they
should work using one he has made the night before.
The students bring their
constructed figures to class and explore the following motions with
them.
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teacher analyzes the lesson and notices the improvement in students'
learning on the basis of the questions they can now answer correctly.
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Jim
is very pleased that the students are beginning to understand what
happens when a figure with given sides is transformed from a parallelogram
into a rectangle. During their explorations, they conjecture that
the diagonals become congruent when the parallelogram becomes a rectangle
but that other properties remain unchanged for instance, that the
diagonals of both parallelograms and rectangles bisect each other.
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The cycle continues
as the supervisor checks back with the teacher. The teacher has
demonstrated professional growth by indicating how the ideas can
be extended to other lessons.
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Ellen sees Jim at a meeting
the following week and asks how the lesson went. Jim indicates how
pleased he was with the lesson and that next year he plans to do
even more with concrete materials. He also plans on extending the
lesson with figures representing rhombi and squares.
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