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Assessment of teaching
mathematics as a process involving problem solving, reasoning, and
communication should provide evidence that the teacher-
models and emphasizes aspects of problem solving, including formulating
and posing problems, solving problems using different strategies,
verifying and interpreting results, and generalizing solutions;
demonstrates and emphasizes the role of mathematical reasoning;
models and emphasizes mathematical communication using written,
oral, and visual forms;
engages students in tasks that involve problem solving, reasoning,
and communication;
engages students in mathematical discourse that extends their
understanding of problem solving and their capacity to reason
and communicate mathematically.
Elaboration
Problem solving, reasoning,
and communication are processes that should pervade all mathematics
instruction and should be modeled by teachers. Students should be
engaged in mathematical tasks and discourse that require problem
solving, reasoning, and communication. Consequently, assessing the
teaching of mathematics should determine whether teachers and students
are actively involved in these processes. The acquisition of the
ability to represent mathematics in these ways takes place over
time and hence should be a continuing focus of instruction. It follows
that assessing the existence of these processes in the teaching
of mathematics must similarly take place over time.
Teaching mathematics from
a problem-solving perspective entails more than solving nonroutine
but often isolated problems or typical textbook types of problems.
It involves the notion that the very essence of studying mathematics
is itself an exercise in exploring, conjecturing, examining, and
testing - all aspects of problem solving. Tasks should be created
and presented that are accessible to students and extend their knowledge
of mathematics and problem solving. Students should be given opportunities
to formulate problems from given situations and create new problems
by modifying the conditions of a given problem.
Teachers should engage students
in mathematical discourse about problem solving. This includes discussing
different solutions and solution strategies for a given problem,
how solutions can be extended and generalized, and different kinds
of problems that can be created from a given situation. All students
should be made to feel that they have something to contribute to
the discussion of a problem. Assessment should focus on the notion
of whether mathematics is being taught in such a way as to promote
these aspects of problem solving.
Teaching mathematics as
an exercise in reasoning should also be commonplace in the classroom.
Students should have frequent opportunities to engage in mathematical
discussions in which reasoning is valued. Students should be encouraged
to explain their reasoning process for reaching a given conclusion
or to justify why their particular approach to a problem is appropriate.
The goal of emphasizing reasoning in the teaching of mathematics
is to empower students to reach conclusions and justify statements
on their own rather than to rely solely on the authority of a teacher
or textbook.
Assessment should seek evidence
that students are using inductive reasoning, proportional reasoning,
and spatial reasoning and are constructing arguments. Assessing
whether mathematics is being represented as a process of reasoning
should focus on whether the teacher demonstrates the pervasiveness
of mathematical reasoning throughout all areas of mathematics and
whether the teacher requires students to use various reasoning processes.
Communication is the vehicle
by which teachers and students can appreciate mathematics as the
processes of problem solving and reasoning. But communication is
also important in itself, since students must learn to describe
phenomena through various written, oral, and visual forms. The notion
of communication emphasized in this standard cannot be fully realized
in a lecture-oriented lesson or when students' responses are limited
to short answers to lower-order questions. This standard suggests
that mathematics is learned in a social context, one in which discussing
ideas is valued. Classrooms should be characterized by conversations
about mathematics among students and between students and the teacher.
Mathematical communication
can occur when students work in cooperative groups, when a student
explains an algorithm for solving equations, when a student presents
a unique method for solving a problem, when a student constructs
and explains a graphical representation of real-world phenomena,
or when a student offers a conjecture about geometric figures. A
teacher should monitor students' use of mathematical language to
help develop their ability to communicate mathematics. This could
be done by asking students if they agree with another student's
explanation or by having students provide various representations
of mathematical ideas or real-world phenomena. The emphasis should
be on all students communicating mathematics, not just on the more
vocal students. In order for teachers to maximize communication
with and among students, they should minimize the amount of time
they themselves dominate classroom discussions.
Vignettes |
| The
teacher constructs her lesson on the basis of her students' previous
experiences. In selecting the task, she considers its potential for
fostering mathematical reasoning and communication. |
5.1 Pat Kowalczyk's
kindergarten class enjoys activities involving continuing patterns
that have been started using blocks, beads, themselves, and other
items. Today Mrs. K, as the children call her, plans on having her
class construct patterns using their names. She thinks that this
activity will extend the work she has been doing to encourage them
to reason and communicate about mathematics with one another. She
has prepared a paper with a 5 x 5 grid of 2-centimeter squares for
each student.
At their tables the students
fill out the grid, using one square for each letter of their name.
When they finish writing their names the first time, they start
over and continue until each of the 25 squares contains a letter.
Mrs K: Select your
favorite crayon and color in the squares that contain the first
letter of your name.
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| In
assessing her teaching, the teacher focuses on the students' ability
to rely on their own mathematical reasoning. |
Mrs.
K walks around the room observing and listening to the students as
they work. When Susan wants to know if she should color both the S's
in her name, Mrs. K responds with a question, "Are both the first
letter of your name?" Susan thinks for a moment and then says,
"No, only this one is," and she colors only the first S
in Susan. Mrs. K makes a mental note that Susan seems confident in
her decisions and does not seek additional confirmation from her.
As she continues to walk around, Mrs. K observes that some children
seem to understand the activity and work independently, some are actively
conferring with others, and some are waiting for her to help them.
She muses, not for the first time, about what more she could be doing
to foster greater self-reliance by her students. |
| The
teacher poses questions that engage students in mathematical reasoning
and communication. She then analyzes her own ability to ask questions.
|
When the students complete
their grids, Mrs. K asks the class if they can predict who has the
same patterns of colored-in squares on their grids. She tries to
phrase the question so as to encourage the students to reason and
to communicate their ideas. She notices that she is improving in
her ability to construct good questions on the spot.
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| The
teacher checks the effectiveness of her strategies for involving students
in mathematical discourse. |
The
students quickly guess that the two Jennifers in the class should
have the same pattern. Mrs. K asks several students to explain how
they can be sure of this without even checking the girls' grids. When
she hears Marcus say, "'Cause they have the same name so their
papers gotta be the same too," she is really pleased. Calling
on him more often seems to be paying off. |
| The
teacher continually poses questions to extend her students' capacity
for reasoning. |
Searching for the next good
question, Mrs. K challenges the students to find similar patterns
where the students do not have the same first name. After some checking
around, the students find that Kent's and Kyle's grids have the
same pattern.
Kent: Maybe names
that begin with the same letter look the same.
Mrs. K:
Is there anyone else whose name begins with the letter K?
(Katrina, Kathy, and Kevin all jump up, waving their hands.)
Katrina:
But my grid is different from Kent's and Kyle's.
Kathy:
But mine is the same as Kevin's.
Mrs. K: Does
this fit the rule that the names that begin with the same letter
give the same pattern?
Students (in
unison): No!
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| The
teacher monitors classroom discourse to make sure that all students
are participating. |
Mrs. K looks around, trying
to decide on whom to call and tries to remember who has not spoken
much today. She remembers that Nikki has not said anything today,
although she did complete her grid quickly.
Mrs. K:
Nikki, how can we change our rule so that it works?
Nikki:
Well, I think it will work if they have the same number of letters
and if their name begins with the same letter.
Laura (excitedly):
Mine matches Kathy's, but our first letters are different.
Mrs. K:
Let's check it out. (She holds them up to the window, one on
top of the other.) Hey, it looks like they do match!
|
|
The teacher is
attentive to how the students are communicating their ideas.
|
At
this point, Dave, Jane, and José put their patterns by Kyle's
and Kent's and are surprised that the patterns match. They don't know
how to express their finding. Mrs. K is a little surprised that this
is hard to explain. Judy says that it has something to do with the
length of the name. Short names seem to match short names but not
long names. Finally, Stanley says that the names with the same number
of letters will match. Some of the other students question whether
he is right. After examining many other examples, they conclude that
he is correct. |
|
The teacher reflects
on the lesson and the students' ability to reason and communicate
effectively.
The teacher's
self-assessment focuses on the effectiveness of the task in promoting
reasoning and communication and how the task can be improved.
|
After
school, Mrs. K reflects on the lesson. She writes a few notes in her
journal - about Marcus, Nikki, and several other students. She also
writes down the task so that she can remember it for the future and
indicates that she thinks it could be used profitably again. She is
impressed with the students' ability to reason. She thinks that letting
the students use different-colored crayons to color in the grids may
have distracted them from the lesson's primary objective. She makes
a note to let students pick only one color next time she uses this
activity. Although she thinks she is getting better at formulating
good questions, she also thinks that she needs to find more ways to
encourage students to communicate their ideas with one another and
to build on one another's reasoning. |
|
The principal
encourages collaboration as a professional development activity.
The teacher's
goals for the lesson emphasize communication through written and
visual forms.
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5.2
The principal has given Doug Walker, an experienced seventh-grade
teacher, release time to observe Louise Knight's seventh-grade mathematics
class and to help her develop professionally. Ms. Knight is a young
teacher who demonstrates a considerable amount of energy and creativity
in her teaching. The day before the scheduled visit, Mr. Walker talks
with Ms. Knight about the upcoming lesson, her objectives, activities,
and her expectations for the students. The focus of the lesson is
on the use of geometric terms to describe where objects are located.
Ms. Knight provides Mr. Walker with a list of terms that the students
are to use in writing the directions. This list includes perpendicular
bisector, angle bisector, midpoint, right angle, acute angle,
and obtuse angle. |
|
The teacher demonstrates
sensitivity to students whose native language is not English.
The teacher has
provided an opportunity for written, oral, and visual communication
in the context of a problem-solving activity.
|
The
next day, Ms. Knight starts the lesson by reviewing the geometry terms.
She then organizes the students into pairs. She carefully selects
the pairs so that each student whose first language is not English
will be working with a student whose first language is English. Out
on the school grounds each pair determines where their treasure is
hidden. They write directions for finding the treasure, using the
geometry terms they had reviewed earlier. They check the accuracy
of their directions by following them from the beginning to see if
they arrive at the spot they had identified for the treasure. Then
they draw a map that indicates the location of the treasure. |
|
The colleague
observes that the teacher encourages communication but avoids placing
herself as the center of attention.
The students write
directions, a form of communication.
|
Mr. Walker notes how Ms.
Knight checks the progress of each pair of students, helping them
when necessary but not providing students with correct descriptions.
She also records the specific difficulties of individual students.
The following set of directions
is written by Dave and Julio:
- Start at the intersection
of the sidewalks. Bisect the right angle and proceed 25 yards
along the bisector. Stop.
- Now go to the midpoint
of the line segment between where you are and the flag pole.
- Turn left 90 degrees
and go 10 yards. Stop.
- Go halfway to the cedar
tree. Find the treasure.
Mr. Walker accompanies Dave
and Julio as they check the accuracy of their directions.
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| The
colleague observes students engaged in mathematical discourse - discussing
terms and checking references. |
Mr.
Walker is impressed with the quality of the students' mathematical
communication and with the way Ms. Knight has organized the pairs
of students. Most of the students used all the terms identified by
Ms. Knight. He notices that several students double-check the meaning
of some of the terms in their textbook. He observes that Ms. Knight
spends a considerable amount of time encouraging the students to be
as precise as possible in writing their directions. She usually does
this by asking questions rather than by telling the students what
the directions should be. |
|
The colleague
offers a suggestion to improve the activity.
|
After
the lesson Ms. Knight and Mr. Walker talk about the ability of the
students to write directions. They both think that although the students'
directions were generally clear and reasonably concise, a few were
confusing and did not lead to the treasure. Mr. Walker suggests that
the students should write the directions first and then hide the treasure
accordingly. This might avoid the problem of locating a point and
then trying to write the directions that guides a person to that point.
Ms. Knight likes this suggestion. She agrees that locating the point
first and then writing the directions may have led to some confusion.
|
|
The colleague
offers additional suggestions for improving the lesson.
The colleague
is supportive of the teacher's creative efforts.
|
Mr.
Walker also suggests that when a treasure is not found because of
faulty directions, the "search party" describe their path
to the students who wrote the directions so that the discrepancy can
be discussed. Ms. Knight sees this as an added opportunity for students
to communicate mathematics. She likes having the students validate
the directions rather than just relying on the teacher. Mr. Walker
compliments Ms. Knight on the lesson and her courage for trying an
activity-oriented lesson. He wishes her well on the next day's treasure
hunt. |
|
The teacher reads
professional journals to improve his teaching.
The teacher and
colleague work together.
The colleague
offers specific suggestions.
|
5.3 Art Heyen has
been reading various articles in the Mathematics Teacher
about the importance of emphasizing mathematical processes when
teaching mathematics. He decides to make a concerted effort this
year to incorporate these ideas into his teaching. At the beginning
of the year, he meets with Diane Rowan, an experienced mathematics
teacher, to discuss how his teaching could become more process oriented.
Diane suggests that he start with a few selected topics to "get
the feel of it" and then work from there. Diane offers suggestions
for a lesson on graphing parabolas that Art could use later in the
year. The emphasis is on considering the equation
y
= ax2 + bx + c
and examining the effect
on the graph when different values of a, b, and c
are used.
|
|
The teacher initiates
contact with a colleague to observe his teaching.
The teacher notices
his students' mathematical disposition has improved, perhaps because
of his more process-oriented approach.
In the past, the
teacher used lecture/listening as the primary instructional technique.
|
In November, Art is ready
to teach the lesson on graphing parabolas. He invites Diane to observe
the lesson and make suggestions. He indicates that he has had moderate
success with other lessons in which mathematical processes have
been emphasized. He complains somewhat about it taking so long to
find good materials but notices that the students seem more interested
in mathematics this year than any of the previous two years that
he has been teaching.
Art typically teaches the
lesson on graphing parabolas by modeling several graphs and helping
the students locate the vertex and several other points, which are
then plotted. After several demonstrations, he assigns practice
problems. This year he will teach the topic with a greater emphasis
on conceptual development.
|
| The
teacher incorporates technology into the lesson. He then begins by
determining and building on what the students know. |
Art begins the lesson by
passing out graphing calculators that have just recently been obtained.
He asks the students to write down three statements or words they
associate with the equation
y
= ax2 + bx + c.
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| The
colleague considers how the teacher could have improved a student's
reasoning ability. |
Some
of the phrases are quadratic equation, parabola, and horseshoe
shaped. One student mentions that if x = 0, then y
= c. Art asks what this means, but the student is not sure.
Diane thinks Art might have spent more time helping the student reason
through his conclusion. |
| The
teacher involves students in mathematical exploration by having them
consider special cases using graphing calculators. |
Art asks the students to
use the graphing calculators to graph the cases when a =
0.25, 0.5, 1, 4, and 8 and b and c are
both 0, and then sketch and label the graphs on the same set of
axes on the graph paper he has provided.
|
|
The teacher asks
the students to conjecture about the effects on graphs when changes
in the value of a occur.
The teacher's
questions require the students to engage in written communication.
|
The students use these sketches
to answer the following questions on the worksheet that Art has
passed out.
Use your sketches on sheet
1 to answer the following questions:
- What property is common
to all the graphs?
- Under what condition
does the graph of y = ax2
open upward? Downward?
- As the absolute value
of a increases, what happens to the graph of y = ax2?
|
|
The colleague
encourages students to use mathematical reasoning.
The teacher makes
the decision not to pursue the question of whether the graph of
a quadratic function can be a straight line. He intends to discuss
this later with the students.
|
As the students work on
the questions, Art and Diane walk around the room and check the
students' progress. Diane notices that one student has drawn a horizontal
line for one of the sketches. She asks if it is possible for the
graph of a quadratic function to be a horizontal line. The student
seems puzzled. After rechecking his figures, the student finds out
that he had entered y = 42 on the calculator
rather than y = 4x2.
A student who was listening claims that you could get a horizontal
line. Diane asks her how this could happen. The student argues that
it can happen when the a and b coefficients are zero
because then you have a constant function. Both Art and Diane are
impressed with the student's reasoning, even though they recognize
an error in her thinking.
Art next asks the students
to consider various graphs of equations having the general form
y = ax2 + c.
They are to first use their graphing calculators and then sketch
the graphs on a set of axes for the following equations:
y = 3x2
+ 10
y = 3x2 +
5 |
|
y = 3x2
5
y = 3x2
10 |
|
| The
teacher has the students use graphing calculators as tools in mathematical
discourse. |
|
| The
teacher asks the students to use inductive reasoning to determine
the nature of the graphs. |
When the sketches are completed,
the students are instructed to address the following questions:
a. How does the value
of c affect the graph of y = 3x2 + c?
b. At what point does the graph of y = 3x2 4
intersect the y-axis?
|
| The
teacher gives the students an opportunity to apply their generalizations
in sketching other equations. |
On the basis of the graphs
they have sketched, the students are then asked to consider the
following questions:
Without using the calculator,
indicate whether the graph of each of the following equations will
open upward or downward; whether the graph will be relatively narrow
or wide; and where the graph intersects the y-axis. Then
sketch the graph on a sheet of graph paper.
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| The
teacher provides students with a different means of communicating
mathematics. |
To
conclude the lesson, Art asks the students to write one or more statements
about what they have learned about the graphs of quadratic equations.
He collects these papers and assigns additional equations for the
students to explore and sketch. |
| The
teacher and the colleague reflect on the lesson. |
After school, Art meets
briefly with Diane to discuss the lesson. He mentions that he wanted
to cover more material - particularly the relationship between the
value of the discriminant and the number of x-intercepts.
He realizes that it would be easier to just tell the students what
he wants them to know, but he was very pleased with their ability
to communicate mathematics in their written statements and to use
inductive reasoning to figure out what the sketches of the last
set of graphs will look like.
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| The
colleague suggests how the teacher can enhance his students' ability
to solve problems using inductive reasoning. |
Diane
concurs. She was particularly pleased with Art's repeated efforts
to encourage the students to write statements about what they had
discovered. She suggests that next time he might begin the lesson
by asking the students what the graph of the equation y = 1.5x2 - 3x + 4.2
(that is, an equation that the students wouldn't normally encounter)
would look like. After students' conjectures are recorded, the lesson
could be developed, following which the equation could be revisited
to determine its graph. The students could use their knowledge developed
in the lesson to determine its graph. |
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The colleague
is supportive.
|
Art likes the suggestion.
He sees this fitting in with his intention of helping students to
reason mathematically. Diane thanks Art for inviting her into his
classroom. She compliments him on his efforts to improve his students'
ability to use mathematical processes.
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