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Assessment of a teacher's
fostering of students' mathematical dispositions should provide
evidence that the teacher
models a disposition to do mathematics;
demonstrates the value of mathematics as a way of thinking and
its application in other disciplines and in society;
promotes students' confidence, flexibility, perseverance, curiosity,
and inventiveness in doing mathematics through the use of appropriate
tasks and by engaging students in mathematical discourse.
Elaboration
If students are to develop
a disposition to do mathematics, it is essential that the teacher
communicate a love of mathematics and a spirit of doing mathematics
that captures the notion that mathematics is an invention of the
human mind. Sometimes this entails an exploration of a student's
query or a consideration of multiple ways of solving a problem.
Certainly, it involves a sense of communicating mathematical ideas.
There is little value in telling students how exciting mathematics
is if they are not actively engaged in doing mathematics themselves.
Using mathematics to explore
real-world phenomena is one means of developing mathematical disposition.
For example, students could consider sampling problems and forms
of statistical inference using proportional reasoning as a means
of understanding how mathematics relates to their lives. The notion
of connections is central to this means of promoting mathematical
disposition. Students could explore Euclidean properties on a sphere,
such as the sum of the measures of the angles of a triangle, to
consider the generality of those properties as another means of
developing mathematical disposition.
Assessing the teacher's
fostering of students' mathematical disposition should focus on
whether the teacher facilitates students' flexibility, inventiveness,
and perseverance in engaging mathematical tasks and on whether students
demonstrate confidence in doing mathematics. Verbal cues that encourage
students during instruction and supportive written comments on homework
and test papers are obvious means of promoting a disposition to
do mathematics. Teachers should be nonjudgmental when students give
answers or present solutions to problems; teachers should help students
to correct their mistakes, but mistakes should be recognized as
a natural part of the learning process. Students should be expected
to raise questions and challenge ideas generated by other students
as well as by the teacher. Above all, students should have ample
time to be active participants in doing mathematics.
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| The
teacher is concerned with her students' dispositions to do mathematics
and that they have a narrow perspective on what constitutes mathematics.
The teacher is engaged in self-analysis of her teaching. |
Vignettes
6.1 Stephanie
Douglas, a fourth-grade teacher, feels that her students are not
having enough experiences relating mathematics to other subjects
or other aspects of their lives. She sees mathematics as a subject
that involves many aspects of living and would like her students
to share that view as well. In particular, she is concerned that
her students think mathematics is basically computational in nature
and not very interesting. She would like to change this.
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The teacher provides
an opportunity for students to participate in solving a real-life
problem.
The teacher encourages
the students by building their confidence.
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As
part of a school project on the environment, Ms. Douglas's class decides
to plant a maple tree in the school yard on Arbor Day. The cost of
the tree including planting is $50. The class discusses how they can
raise the money for doing this. After considering several options,
the class decides to collect aluminum cans - from home and ones that
they find lying around in various places. The students feel good about
this idea because they can make a contribution by cleaning up litter
and by planting a tree for the school. Ms. Douglas compliments the
students on their idea for raising money. |
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The teacher engages
the students in mathematical discourse.
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Several
of the students check on how much can be earned by collecting aluminum
cans. They find out that they can earn 40 cents a pound and that it
takes about 30 cans to make a pound. The students ask how long it
will take to earn the money. Ms. Douglas sees this as an excellent
opportunity to become involved in the mathematics of prediction and
estimation. The class determines that it will take 125 pounds of cans,
or about 3750 cans, to earn the $50. Hank, one of the students, interjects
that he can't drink that much soda pop. The other students think they
can't get that many cans. |
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The teacher encourages
students to propose solutions to a logistical problem.
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Ms.
Douglas suggests that students from other classes could bring in cans
and that their class would be in charge of collecting them and taking
them to the recycling company. Eleanor suggests that they put a big
container in the hallway where students from other classes could drop
off their cans. The other students concur with Eleanor's suggestion.
Ms. Douglas says that she will check with the school principal but
she thinks it is a workable idea. |
| The
teacher helps students model the situation using mathematics. |
Ms. Douglas suggests that
the students make a graph to determine how long it will take and
how their collection is proceeding. Since they have three months
before they want to plant the tree, the students decide to consider
six points on the graph - one for about every two weeks. The students
plot their data points and eventually produce a graph. They decide
that they need an additional graph, one that relates the number
of cans to the amount of money they can raise. The students construct
the following graphs:
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| The
teacher provides an opportunity for students to use a graph to represent
real data. |
The students decide that
every two weeks they will mark the graph to indicate how many pounds
they have collected. After several months they construct the following
graph to represent the collection process:
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| The
teacher reflects on the activity. She concludes that she has helped
the students develop a more positive disposition to do mathematics
and that the students see mathematics in a broader context than just
textbook problems. |
Ms.
Douglas is very pleased with the activity. She thinks the students
are beginning to see mathematics as more than just paper-and-pencil
algorithms and that it has helped the students better understand the
project. The students look forward to plotting points on the graph
every two weeks. Some of the students find out that in a town in another
state a store gives 45 cents a pound. They draw a graph to represent
that situation and determine how much more quickly they could earn
the money had they lived in that town. |
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supervisor holds a preobservation conference as a first step in determining
the teacher's goals for the lesson. |
6.2 Mr. White is
in a preobservation conference with his mathematics supervisor,
Mrs. James. Mrs. James will be observing Mr. White's seventh-hour
advanced algebra class tomorrow and is gathering information regarding
the class. She is using the district's preobservation conference
report form.
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Teacher ________________ Date ____________________
Grade/Subject _____________ Observer ___________________
1. Learner Objectives
- Content (What will students be learning?)
- Process (How will students be learning?)
- Rationale (Why are students learning this content?)
2. Assessment
- What processes will be used to check for student understanding
in class?
- What processes will be used to check for student understanding
at the end of the lesson/unit?
3. Instructional Strategies
- What special resources, questioning techniques, or
motivational techniques will be used?
4. Observer Focus
- What is the major focus of data collection?
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The teacher is
concerned about the mathematical disposition of the students.
The supervisor
and teacher work together to make the coming observation as productive
and helpful as possible.
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In discussing his goals
for the lesson on the tangent function, one of the topics in the
chapter on right-triangle trigonometry, Mr. White emphasizes that
he would like his students to be more confident in doing
mathematics than they are
now. He indicates that he is working hard to get the students more
actively involved during the lessons. Mrs. James reviews the form
to make sure she is clear about what his expectations are for the
lesson. She asks if there is anything in particular on which Mr.
White would like her to focus during the observation. He indicates
that he would like her to consider how effective he is in getting
all the students involved.
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| The
supervisor points out a positive aspect of the teacher's performance.
She then gives a factual description of his teaching but avoids using
rating terms such as good, excellent, adequate, or inadequate. |
The next day Mrs. James
observes the class, and she and Mr. White sit down after school
to discuss the lesson. Mrs. James begins the postobservation conference
by complimenting Mr. White on his ability to get all the students
involved. She could tell that he had made an extra effort to involve
all the students. She then provides a factual description of his
teaching: It consists primarily of small steps in which Mr. White
models a concept or procedure and then engages students in practicing
other similar problems. She notes that this occurred five times
during the lesson. In each instance the students used Mr. White's
sixty-second rule, namely, that they work on a problem for sixty
seconds by themselves and then share their solutions and questions
with their neighbor. She reported several instances in which students
had generated their own solution methods - ones different from what
Mr. White had apparently anticipated. She shared with him the following
exchange at one point during the lesson.
Mr. W:
Class, find the value of x in the figure on the board.
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| The
teacher monitors students' work. |
(Mrs. James reports at this
point that Mr. White circled the room, briefly stopping at desks
as students worked on the problem. After two minutes, the following
dialogue continued.)
Mr. W:
How can we find x using the tangent ratio? Cindy?
Cindy: Well,
(pause) I used the Pythagorean theorem to find it.
Mr. W: That's
usually okay, but today we are focusing on the tangent ratio. How
can we find x using the tangent ratio? Marc?
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| The
students present different solution methods. |
Marc: Let
cos 21o = x/51. Then x is approximately
48a little less, actually.
Mr. W: But,
Marc, you used the cosine. How can we solve the problem using the
tangent ratio? Allison?
Allison (excitedly):
Well, I don't know. But I think I got a good way to solve it. Take
the complement of 21 degrees and get 69 degrees; then you have sin
69o = x/51. You get about the same answer as Marc
said.
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| The
teacher is not supportive of students' different solution methods.
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Mr. W (somewhat angry):
Did anyone solve the problem using the tangent method? That's what
we are studying!
Jim: Sure. Tan 21o
= 18/x.
Cindy: But isn't
the Pythagorean theorem easier?
Mr. W: It probably
is for this problem. But I want you to know how to use the tangent
ratio.
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| The
supervisor asks the teacher for his analysis of the data she has collected.
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After
reviewing this exchange, Mrs. James asks Mr. White if he sees any
discrepancy between this exchange and his goals for building students'
confidence in doing mathematics. In particular, she asks Mr. White
to consider the case of Allison, who was obviously excited that she
had discovered a way of doing the problem. Allison is not a strong
student and is often a reluctant participant in class. |
| The
teacher realizes that his actions do not help students develop a disposition
to do mathematics. |
At
first Mr. White is defensive, but then he realizes that his actions
were counterproductive to his intent of helping students develop confidence
in doing mathematics. He hadn't realized that he had neglected to
compliment the students for their discoveries. He wondered what he
could have done differently. He confessed that he was concerned about
several of the students acting up, and therefore he wasn't listening
very carefully to what the other students were saying. Mrs. James
suggested that they consider some alternative strategies. |
| The
teacher reflects on the lesson and considers how he might have taught
the lesson differently. |
Mr. White suggests that
either he should have been more accepting of the students' alternative
solution methods or he should have changed the problem so that the
students would have to use the tangent ratio. For example, he could
have deleted the fact that side AB is 51 units.
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| The
teacher recalls that he wanted to improve his students' disposition
to do mathematics and that his suggestion might not help realize this
objective. |
With
this revised problem, the students would have had to use the equation
tan 21o = 18/x. Mrs. James agrees that this would
be one way to force students to use the tangent ratio. But then she
asks, "What did you consider to be one of your primary objectives?"
Mr. White recalls that he wanted to increase the students' confidence
in doing mathematics. He agrees that narrowing the task would not
necessarily contribute toward realizing this objective. Mrs. James
suggests that he could change the question as follows. |
| The
supervisor provides a specific suggestion that better enables the
teacher to reach one of his objectives. |
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Class, we want to
find x in ABC.
How many different
ways can we find a solution, including the new one we discussed
today?
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| The
importance of considering multiple ways of solving a problem is recognized
as a means of promoting mathematical disposition. |
Mr.
White and Mrs. James discuss how this type of question can help him
achieve his objective of teaching the tangent ratio but also go a
long way toward promoting the students' confidence, flexibility, and
inventiveness in doing mathematics. The question also emphasizes that
problems sometimes have multiple ways of being solved and that students
can take pride in discovering a strategy for solving a problem that
nobody else in class had thought about. Allison may have been a case
in point. |
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The supervisor
is supportive of the young teacher.
The teacher reflects
on his teaching and recognizes areas of needed improvement.
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Before leaving the postobservation
conference, Mrs. James makes it clear that Mr. White is making significant
progress as a second-year teacher. His lessons are organized, and
he is genuinely interested in his students. Before Mrs. James's
next visit, Mr. White agrees to work harder to create better questions
and to be more sensitive to recognizing students' solutions, especially
when those solutions are not the ones he was anticipating.
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