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Assessment of the teacher's
ability to create a learning environment that fosters the development
of each students' mathematical power should provide evidence that
the teacher
conveys the notion that mathematics is a subject to be explored
and created both individually and in collaboration with others;
respects students and their ideas and encourages curiosity and
spontaneity;
encourages students to draw and validate their own conclusions;
selects tasks that allow students to construct new meaning by
building on and extending their prior knowledge;
makes appropriate use of available resources;
respects and responds to students' diverse interests and linguistic,
cultural, and socioeconomic backgrounds in designing mathematical
tasks;
affirms and encourages full participation and continued study
of mathematics by all students.
Elaboration
Any assessment of the teaching
of mathematics should consider whether the teacher has established
and maintained an environment that promotes the development of students'
mathematical power. A spirit of inquiry should pervade all mathematics
teaching and learning. In establishing such an environment, the
teacher must be sensitive to students' ideas and encourage mathematical
communication among all students. Technology can be used effectively
in creating such an environment, since it provides a tool for making
mathematical explorations more efficient and accessible.
An inquiry-oriented classroom
can be promoted by engaging students in extensive mathematical discussions
and encouraging them to reason mathematically. The teacher must
respect students' ideas by listening to them and incorporating their
ideas into the class discussion. For their part, students should
demonstrate a willingness to propose hypotheses, to support their
own hypotheses, and support or challenge hypotheses set forth by
the teacher or other students. As the teacher models encouragement
and support for students and respects and accepts their ideas, so
should students learn to support and respect each other and to work
collaboratively and actively to solve problems and to validate proposed
solutions.
Rather than rely solely
on the logical structure of mathematics to determine and sequence
classroom activities, the teacher should use his or her knowledge
of students' understanding as a primary basis for selecting and
sequencing mathematical tasks at an appropriate level of abstraction.
Tasks that promote the active involvement of all students should
be selected. It is imperative that the teacher set high expectations
for every student and work vigorously to see that those expectations
are met.
Ultimately, students must
assume responsibility for their own learning. However, the teacher
has the responsibility to create an environment in which students
are encouraged to accept that responsibility. This standard identifies
indicators for determining whether a teacher is creating such an
environment.
Vignettes |
| The
supervisor and teachers work together to address problems related
to the teaching and learning of percent. |
8.1
A group of middle school teachers and Larry Parker, the district's
mathematics coordinator, have been meeting regularly for the past
semester. The group's goal is to improve the teaching and learning
of various specific topics. They are now focusing on the concept of
percent - a concept that is typically difficult for their students.
Larry schedules his visits to individual classrooms so that he can
observe lessons on percent. On Wednesday afternoon he observes Betty
Mathison, an experienced sixth-grade teacher. |
| The
teacher selects a task that allows students to use what they have
previously learned about fractions. |
Betty has prepared transparencies
of 100-square grids that are partially shaded. Each grid is shown
briefly (a couple of seconds) on the overhead before she asks the
question, "What percent of the grid has been shaded?"
Examples
of transparencies used.
|
|
By having the
students write their estimates, the teacher is involving all the
students in the lesson.
The teacher is
providing the students with an opportunity to share their mathematical
reasoning.
|
Students
write their guesses and their reasoning on a sheet of paper. Betty
walks around the room and observes what they have written after showing
each transparency. Betty then randomly selects students and asks them
to indicate what their guesses are and to share their rationales for
the guesses. The students begin to appreciate the strategies that
their classmates are using.
Students give the following
reasons for their estimates:
- The first one was 36%
because I saw 3 rows of ten and 6 more.
- The first one was more
than 30%. I saw three rows but didn't have time to count the other
ones. It was probably about 35%.
- In the second one, I
think there were 8 rows of ten and then 7 more. That makes 87%.
- I know there was one
row that wasn't shaded so it had to be less than 90%. Maybe it
was 86%.
- The last one was hard.
I think it was about 20%. It seemed like it was one-fourth covered.
- The third one was less
than 50% but I don't know exactly. Maybe it was 10%. I'm just
guessing.
|
| The
teacher encourages students to engage in mathematical reasoning. |
Betty
is supportive of the students' responses; sometimes she asks them
to elaborate on their reasons. |
| The
mathematics supervisor notes that the teacher provides students with
an opportunity to represent different percents. |
Following
this whole-class activity, Betty has the students work in pairs to
represent different percents using cardboard base-ten materials. In
addition to asking students to represent whole number percents less
than 100, Betty asks students how they might represent 150% or 200%
or a percentage less than 1%, like 0.5%. Larry is impressed with the
extension. |
| The
teacher encourages students to make connections with previously learned
material. |
Betty
is hoping the students will see the connection between representing
percents like 36% and 87% to representing percents that are more than
100% or that are not whole number percents. She reminds the students
of the earlier work they had done with decimal representations using
the cardboard base-ten materials. |
|
The supervisor
is very supportive of the tasks the teacher designed.
The supervisor
supports the teacher's creativity in using available resources.
|
Both
Larry and Betty are pleased with the lesson. Although the students
had a great deal of difficulty representing percents greater than
100 and percents that are not whole numbers, they nevertheless felt
that it was a good start in developing a concept of percent. Larry
asks Betty to share the lesson with the other middle school teachers
at next week's meeting. He wants the other teachers to see how Betty
has made effective use of base-ten materials constructed from old
cut-up cardboard boxes to teach percent. Larry indicates that he will
try to secure funds so that the materials can be made from card stock
and thus be more attractive for the students. |
|
University methods
students will confer with the teacher about their reactions to the
lesson.
The mathematical
environment and how it affects learning is recognized as being important.
|
8.2
Pierre Bordeaux, a master teacher, is on the telephone talking
with his friend Sally Witt, a mathematics educator from Southwest
State University. Sally would like her preservice teachers to observe
a teacher who has a reputation for having students explore mathematics.
She would also like her students to have the experience of evaluating
a mathematics lesson. The methods students have developed criteria
they think are important to the teaching of mathematics. In particular,
they think the mathematical environment is important. Mr. Bordeaux
indicates that he would welcome the university students and whatever
comments they might have about his teaching. He suggests that the
students come next Thursday because he is planning to do some explorations
involving the Pythagorean theorem. |
|
The teacher engages
students in mathematical discourse.
The teacher demonstrates
respect for the students and their cultural backgrounds. He also
represents mathematics as a human endeavor.
|
The
next Thursday Dr. Witt's methods class is sitting along the sides
of Mr. Bordeaux's class ready to take notes. Mr. Bordeaux begins the
class by asking his students if they remember anything about the Pythagorean
theorem. The students offer that it has something to do with right
triangles, the Greeks, and that it has a formula. Yvonne claims that
it says a2 + b2 = c2.
Clare says that it works only for right triangles. Ben adds that the
c side must be opposite the right angle. Mr. Bordeaux compliments
them on remembering so much about the theorem and tells them that
the Pythagorean theorem was also known to the ancient Egyptians and
Chinese. He encourages them to check other books in the school library
on the history of the Pythagorean theorem to see what else they can
learn about it. |
|
The teacher asks
the students to engage in mathematical communication.
The teacher helps
the students extend their understanding of the theorem.
The teacher gives
the students an opportunity to engage in mathematical exploration.
The university
students take note of the teacher's questions.
The teacher is
making appropriate use of resources.
|
Mr.
Bordeaux asks the students to provide an interpretation or a
drawing to represent what the theorem says. The students indicate
that they don't know what he means. After some discussion, the
students draw figures similar to the following and interpret
the theorem in terms of the figures. They confess that they
had thought of the theorem only in terms of squaring numbers.
|
|
|
Mr. Bordeaux then
asks them if other figures were drawn on the legs and the
hypotenuse, would the same relationship among the areas hold
true. Jeff asks, "Figures like what?" to which the
teacher responds, "Like equilateral triangles."
Both Mr. Bordeaux's students and Dr. Witt's students are struck
by the question; they had never considered such an extension
of the Pythagorean theorem. Most of the geometry students
volunteer that it won't work; a few think it will. Having
anticipated that there would be some disagreement, Mr. Bordeaux
distributes a worksheet with three right triangles drawn,
construction paper, and scissors. He instructs the students
to work in pairs and construct and cut out equilateral triangles
to fit on the sides of the right triangles.
|
|
|
| The
university students are impressed with how the teacher supports students
in validating a conjecture. |
Mr.
Bordeaux encourages the students to cut the triangles into smaller
pieces in order to determine if the sum of the areas of the two smaller
equilateral triangles is equal to the area of the larger equilateral
triangle. The university students observe the way the teacher engages
the students. |
| The
methods students focus on the exploring nature of the mathematical
environment and on the emphasis given to mathematical communication.
|
The
methods students are quite impressed with the mathematical activities.
They observe that the students are busy exploring mathematics and
that the teacher is supportive of their doing so. They are particularly
impressed when Mr. Bordeaux has the students verbalize what they had
found. This is consistent with the emphasis Dr. Witt had given in
the methods class and with the criteria for effective teaching that
they had established. |
|
The teacher poses
a mathematical question.
The methods students
observe that a different means of validating a conjecture is used.
|
Mr.
Bordeaux then asks the students to consider the case when semicircles
are drawn on the sides of the triangles. The students draw the figure
and determine that the sum of the areas of the semicircles on the
two legs is equal to the area of the semicircle on the hypotenuse
by using the formula for the area of the circle. Mr. Bordeaux points
out that this is a different kind of justification from what they
had previously used with the construction paper and the equilateral
triangle. The methods students take note of this. |
|
The student forms
a conjecture.
The university
students observe that the teacher has asked a student to produce
a counterexample.
|
Nathaniel asks if the Pythagorean
theorem works for any figures placed on the sides of the triangles.
Manuel and David think not. They state that you can't just draw
any rectangles on the sides - they would have to be related in some
way. Mimi agrees. She says, "You couldn't expect one to be
skinny and another one thick and expect it to work." The university
students take note of Mr. Bordeaux's next question, "Can you
draw an example to show one that doesn't work?" Mimi draws
the following picture to illustrate her point:
|
| The
methods students note that the teacher is supportive and observe that
the geometry students take initiative in the discourse. |
Mr.
Bordeaux compliments Mimi for her counterexample. After vigorous discussion,
the students decide to consider the case in which the lengths of the
rectangles are twice as long as the widths. They will study this case
as part of tomorrow's homework. |
|
The university
students inquire about the extent to which the teacher uses instructional
aids in engaging students in exploration activities.
The university
students ask how the teacher is meeting the needs of individual
students by pairing them as he did.
The teacher acknowledges
that he needs to consider more carefully how to organize the students.
|
After the class, Mr. Bordeaux,
Dr. Witt, and the university methods students meet to discuss the
lesson. The methods students ask whether Mr. Bordeaux always uses
materials like he did today to explore mathematics. He replies,
"Not always," but indicates that he likes to use them
as much as possible. He explains that it is very important for students
to see mathematics as a subject to be explored and not just as statements
in a textbook. He emphasizes to the university students that he
likes to have the students work in pairs or in small groups to explore
various mathematical topics. The methods students ask how he decided
to pair the students. Mr. Bordeaux confesses that he hadn't thought
much about it but that he would the next time. One of the university
students recalled that one of the students had a broken hand and
couldn't do much of the physical manipulation during the exploration.
Mr. Bordeaux acknowledges this and indicates that he should have
spent more time helping that student.
Summary: Foci of Evaluation
The five standards in this
section present foci for evaluating the teaching of mathematics.
All these standards, and particularly Standard 6, emphasize the
importance of significant mathematics when evaluating the teaching
of mathematics. It is through encountering significant mathematics
that students develop mathematical power. But attaining mathematical
power requires more. It requires a disposition to do mathematics
and an environment in which the processes of doing mathematics are
continually emphasized.
All this cannot occur without
teachers who present stimulating tasks and create an environment
in which problem solving, reasoning, and communication are valued
and promoted. Further, the message teachers send students should
not be limited to instruction alone; it must also include what and
how mathematical learning is assessed. It is through assessment
that we communicate to our students what mathematical outcomes are
valued.
A consistent message throughout
the standards for the evaluation of teaching is the importance of
teachers' being reflective about their teaching and working with
colleagues and supervisors to improve their teaching. Although the
standards in this section can provide a focus for improvement, such
improvement is more likely to occur when teachers have the support
to engage in professional development. As suggested in vignette
8.2, professional development spans a teacher's professional life.
In the next section standards for the professional development of
teachers are presented and discussed.
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