The evaluation of the
teaching of mathematics should provide ongoing opportunities for
analyze their own teaching;
deliberate with colleagues about their teaching;
confer with supervisors about their teaching.
The emphasis in this standard
is on the teacher being a significant participant throughout the
evaluation process. In particular, teachers should be given the
opportunity and encouragement to engage in reflecting on and evaluating
their own teaching and to discuss their teaching with colleagues
who have observed their teaching. When evaluations are conducted
by supervisors, it is imperative that teachers play a central role
in providing information about their own teaching, including their
goals and analysis of teaching. It is crucial that teachers see
the evaluation process as one that contributes to their professional
growth as teachers of mathematics, thereby necessitating their participation
in the evaluation process.
Supervisors should establish
collegial relationships with teachers in ways that foster an atmosphere
for evaluation that is conducive to improving instruction. Growth
is nurtured when all parties are interested in improving the teaching
of mathematics and recognize that improvement is an activity for
all teachers, regardless of their level of professional development.
The notion of "coaching" is relevant here, since continual
feedback should be furnished on progress in mastering a greater
repertoire of instructional strategies. Preparation for peer coaching
is an important aspect of staff development.
When a purpose of evaluation
is to produce a report on a teacher's competence, teachers should
be given opportunities to provide their interpretation of events
and to share their instructional goals and expectations of students.
Further, teachers should have access to any information accumulated
during the evaluation process.
It is important that administrators
create an atmosphere in which teachers are encouraged to solicit
colleagues' help in guiding the improvement of instruction. Teachers
should be given time to collaborate with peers and to observe and
analyze each other's lessons. Even in the context of self-analysis,
teachers should have confidence that the administration is supportive
of change and will provide resources necessary to initiate change.
The teacher attends
a mathematics conference as a professional development activity.
The teacher analyzes
her teaching and recognizes a shortcoming in the tasks she has been
giving her students.
2.1 Pat Schuette,
a second-grade teacher, is attending her first state mathematics
conference, which is being held in a neighboring school fifty miles
away. She is particularly interested in attending sessions on the
teaching of geometry and spatial reasoning. Her students seem to
enjoy these topics, and she enjoys teaching them. As she listens
to one of the presenters, she reflects on the tasks she has given
her students. She was pleased that many of the speaker's tasks using
the geoboard were similar to the ones she used with her students.
She is also impressed with the activities involving geometric solids,
activities she rarely used with her students. As Pat listens to
the rationale for helping children develop spatial perception, she
realizes its importance and that she has restricted her tasks to
ones involving only planar figures.
The teacher confers
with other teachers during a workshop.
Later in the day, Pat attends
a workshop for K3 teachers on making and exploring three-dimensional
figures. She welcomes the opportunity to learn more mathematics
as well as obtain some practical activities she can use with her
students. As the teachers explore the various activities, they engage
in a lively discussion of how their students would react to the
tasks and how they could use the activities in their classrooms.
Many of the activities involve using D-Stix to make cubes, rectangular
solids, and different kinds of pyramids. Pat enjoys making the figures
and discovering Euler's formula for the relationship among the edges,
vertices, and faces of the figures. She thinks her students will
enjoy these activities as well.
development consists of exchanging ideas with other teachers.
the workshop, some of the teachers compare notes and share activities
they feel work successfully with their students.
teacher confers with her principal about obtaining materials for teaching
geometry and providing students with experiences in spatial reasoning.
she returns from the conference, Pat approaches the principal about
obtaining additional materials to help teach spatial reasoning. She
describes her experiences at the state mathematics conference and
shows the principal the emphasis on spatial reasoning in the Curriculum
and Evaluation Standards for School Mathematics. The principal
agrees to provide a modest amount of money to purchase D-Stix and
other materials that Pat and the teachers could use to promote students'
understanding of geometry.
The teacher is aware of the need to incorporate calculators into his
teaching but is reluctant to do so. He seeks specific suggestions
from his mathematics supervisor.
Pete Wilder has been teaching eighth-grade mathematics at Blackhawk
Middle School for the past ten years. Although he has read that calculators
should be emphasized at the middle school level, he has been reluctant
to use them. His supervisor, Tim Jackson, has been urging him to use
calculators whenever possible. Tim has been disappointed that Pete
has not made more progress in using calculators and in moving away
from his teacher-dominated lessons. Pete admits that most of his students
use calculators only for checking computations with whole numbers
The teacher analyzes the lesson and identifies one of the problems
namely, that the calculators do not all work the same way and that
he is not clear on how each of the calculators works.
agrees to make a greater effort to use calculators if Tim will come
to his class and make specific suggestions on how they can be used
in a reasonable way. On the day Tim observes the class, the students
are doing computational problems using the memory key on the calculator.
It becomes clear that the memory functions do not all work the same
way on the different calculators. This causes Pete problems in conducting
the lesson. It is clear that Pete is not happy with the lesson. After
the lesson he expresses his frustration to Tim and indicates that
lessons like this are one of the reasons he hesitates to use calculators
with his students.
supervisor is supportive as he recognizes the need to reduce the teacher's
anxiety. He also wants to help the teacher place less emphasis on
in the day, Pete and Tim meet to discuss the problem. Pete is quite
discouraged, which contributes to his anxiety in using calculators.
Tim has two concerns. First, he wants to help Pete feel more comfortable
and confident in using calculators. Second, he wants Pete to use calculators
to explore more substantial mathematics.
teacher and supervisor work in a collegial manner to address the problem.
Assistance is obtained from another teacher.
and Pete decide that they need more information on what his students
know about using the calculator. Pete remembers that Juanita Criswell,
another eighth-grade teacher, developed an activity sheet that can
be used to assess students' understanding and competence in using
different calculator functions. Pete is sure that Juanita would be
willing to let him use the activity sheet with his students. Pete
and Tim agree that this is a good idea. Tim also agrees to see if
he can obtain district funds to buy calculators.
supervisor suggests specific calculator uses that go beyond checking
Tim and Pete spend the rest
of the time talking about how the calculator could be used to explore
mathematics rather than just use it for checking computational exercises.
Pete indicates that the next unit will be on statistics finding
medians and means. Tim suggests using the calculator to investigate
the following types of problems:
- Suppose ten students
had test scores of 68, 73, 77, 81, 84, 88, 89, 91, 94, and 95.
What is the average score? Suppose the teacher made a mistake
and each student should receive an additional 3 points. What will
the new average be? Make a guess before doing the calculation.
- If five workers each
earn $32 500 a year and one of the workers gets a $5 000
raise, how much will the average salary increase?
- If four out of six workers
earn $26 00, $32 000, $27 000, and $31 000
and the average for the six workers is $30 000, how much do the
fifth and sixth workers each make?
The teacher indicates
that the problems seem reasonable and interesting; he is willing
to try to use calculators in a more substantive way.
Pete comments that these
kinds of questions are quite different from those in the textbook.
He likes them and indicates that he is willing to involve students
in such explorations using calculators once he knows more about
how well the students can use calculators.