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Assessment of the teaching
of mathematical concepts, procedures, and connections should provide
evidence that the teacher
demonstrates a sound knowledge of mathematical concepts and procedures;
represents mathematics as a network of interconnected concepts
and procedures;
emphasizes connections between mathematics and other disciplines
and connections to daily living;
engages students in tasks that promote the understanding of mathematical
concepts, procedures, and connections;
engages students in mathematical discourse that extends their
understanding of mathematical concepts, procedures, and connections.
Elaboration
The primary emphasis in
this standard is on the teaching of mathematical content. The teacher
should demonstrate a deep understanding of mathematical concepts
and principles, connections between concepts and procedures, connections
across mathematical topics (e.g., providing geometric interpretations
of probability concepts or of factoring whole numbers), and connections
between mathematics and other disciplines. A teacher who has a sound
knowledge of mathematics can respond appropriately to students'
questions, can design appropriate learning activities involving
a variety of mathematical representations, and can orchestrate mathematical
discourse in the classroom. Furthermore, the demonstration of asound
knowledge of mathematics includes the orientation that mathematics
is, and continues to be, the result of human endeavor and that the
uses of mathematics permeate modern life. On the contrary, making
frequent mathematical mistakes, using limited or inappropriate representations,
or presenting mathematics as a static subject whose meaning is derived
solely from symbolic representations suggests that the teacher does
not have an acceptable command of mathematics.
The teacher should engage
students in a series of tasks that involve interrelationships among
mathematical concepts and procedures. The acquisition of mathematical
concepts and procedures means little if the content is learned in
an isolated way in which connections among the various mathematical
topics are neglected. Instruction should not be limited to a narrow
range of outcomes, such as memorizing definitions or executing computational
algorithms. Instead, instruction should incorporate a wide range
of objectives as suggested in the Curriculum and Evaluation Standards
for School Mathematics. Further, the teacher should emphasize
mathematical communication with the intent of expanding students'
understanding of mathematical content and connections.
Connections should occur
frequently enough to influence students' beliefs about the value
of mathematics in society and its contributions to other disciplines.
Regardless of what mathematics is being studied, students should
have the opportunity to apply the mathematics they have learned
to real-world situations that go beyond the usual textbook word
problems. Students should see mathematics as something that permeates
society and, indeed, their own lives. This standard implies that
instructional activities aimed at promoting students' appreciation
of mathematical connections should take advantage of students' experiences
and interests.
Vignettes |
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The teacher reflects
on her previous teaching of fractions and decides to create new
tasks involving multiple representations of fractions.
The teacher notes
her students' performance.
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4.1
Sara Rasmussen has been teaching seventh-grade mathematics for several
years. Before that she taught mathematics to fifth and sixth graders.
Although Sara is an excellent teacher, she is continually concerned
about her teaching of common fractions, particularly with her students'
ability to interpret fractions in a variety of contexts and to interpret
various operations with fractions. This year she has made a special
effort to create tasks that require the use of a variety of representations
of common fractions, including the number line, regions, parts of
sets, decimals, and measurement. She feels that the tasks have helped
the students develop a good grasp of translating among representations
- representing 3/4 as a region within a rectangle, as a
point on the number line, and as a decimal, for example. She is also
pleased that her students are proficient in adding fractions. |
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The teacher observes
that students have difficulty making the connection between the
concept of fraction and adding fractions using the number line.
The teacher adjusts
instruction to help students make the connection.
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Sara
decides to see how well the students can make the connection between
the concept of fraction and the addition of fractions. She asks them
to add 3/4 + 1/2 and to provide an
interpretation of finding the sum using the number line. She is surprised
that the students have very little sense of how to interpret the addition
of fractions using the number line. They can mark the points 3/4
and 1/2 and the sum 11/4, but they fail to make
the connection with finding the sum by starting at the point 3/4
and moving 1/2 of a unit to the right to obtain 11/4.
She spends the greater part of one period helping the students understand
this connection. |
| The
teacher discusses the problem with a colleague, who offers a suggestion. |
Later in the day Sara is
talking with another teacher about the problem to see if she has
any suggestions for activities that could extend what she has started.
Her colleague suggests a task that requires students to reason why
certain procedures for adding fractions don't work. Sara decides
she will try the activity.
The next day
Sara writes the following examples on the board:
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| The
teacher provides students with an opportunity to use their mathematical
reasoning in a situation involving concepts and procedures. |
She asks
the students to copy the examples into their journals and to write
a brief explanation on whether they think the examples are correct
or not.
A few days
later, Sara collects the journals and observes the following entries:
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It couldn't
be right because 1/2 plus 1/2 must be
more than 1/2.
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You
need like terms - like 1 centimeter and 2 millimeters is not
3 centimeters. They need to be all in centimeters or all in
millimeters.
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It is
a good way to add fractions because it is easy.
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If you
take one half a pie and one half a pie you get a whole pie,
not part of a pie.
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If you
start off at 1/5 and don't add anything you wouldn't
go back to 1/9. You would stay put at 1/5.
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I used
my calculator and used decimals. It gave me 0.7 for 0.2 + 0.5.
That method can't be right.
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| The
teacher analyzes the students' explanations. |
She
is impressed with the depth of some of the students' understanding.
She is pleased that they have considered the examples in light of
the various representations of fractions they have been studying.
Most of the students used the representations appropriately to show
why the procedure of adding numerators and denominators doesn't work.
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The teacher provides
opportunities for students to discuss the connections they have
made.
The teacher observes
that one student uses multiple representations.
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The
next day Sara divides the class into small groups and hands out the
journal entries above after the students agree that it is okay to
share them. She asks the students to indicate whether they agree or
disagree with each of the statements and why they agree or disagree.
The students discuss their reactions within their group and then share
their reactions with the class. Sara feels that most of the students
are making excellent contributions in analyzing the statements. One
of the students illustrates the problem with the first example using
both the number line and regions of a circle. Sara is impressed. |
| The
teacher observes that the students may be making connections with
other lessons. |
Several
days later Sara listens to a student explain that you couldn't add
1/3 and 5/6 unless you found a common unit for
thirds and sixths. She conjectures that these verbalizations may have
been stimulated by the previous discussions on the journal entries.
She feels good about this. |
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The teacher has
identified a problem and seeks help from his mathematics supervisor.
The supervisor
suggests connecting the topic to a real-world situation and offers
suggestions for increasing student involvement.
The teacher begins
the lesson with a real-world example. By having the students guess,
he increases student involvement.
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4.2
Steve Cooper has taught at North High School for ten years. He generally
uses his seventh-hour planning period to write his lesson plans
for the next day, but today he is meeting with the mathematics supervisor,
Mr. Johnson, in preparation for a scheduled observation. Mr. Cooper's
concern about tomorrow's algebra class dominates the discussion.
The topic is writing the equation of a line given the coordinates
of two points on the line. Typically he has not been able to make
the topic interesting to the students, and, perhaps as a consequence,
the students find it difficult. Mr. Johnson suggests connecting
the lesson to an activity involving statistics and using a computer
to graph a data set and determine a line of best fit. Together they
plan the lesson and discuss some questions Mr. Cooper might use
during the lesson.
The next day
Mr. Cooper begins class by asking students to guess which is longer,
their foot or the inside of their arm from the wrist to elbow. The
students measure and record the lengths to the nearest centimeter.
Mr. Johnson notes the improvement in student involvement and interest.
Data for the
entire class, including Mr. Cooper, are recorded on the chalkboard
in tabular form. Mr. Cooper then asks the students to graph a scatter
plot of the data with arm length on the x-axis and foot length
on the y-axis. Rob and Karen enter the data into a computer
at the front of the room, displaying the data and the scatter plot
on the overhead screen. The other students compare their graphs
to the graph on the overhead screen. Mazie notes that points that
appear more than once on the table appear only once on the screen.
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The supervisor
observes that the students are using the data in different formats,
illustrating mathematics as a network of interconnected concepts
and procedures. A computerized representation facilitates discourse.
The supervisor
sees the effects of their previous planning in the use of stimulating
questions connecting mathematics to the real world.
The teacher connects
writing the equation of a line to real data.
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| Mr. Cooper asks the students whether they could
make a prediction about the length of a foot of a person with
a 35-centimeter-long forearm. In a spirited discussion the students
agree that, in general, people with longer forearms have longer
feet, but it is difficult to make a numerical estimate from
these data. Tim proposes that since ten of the people have the
same arm and foot lengths, 35 centimeters would be a reasonable
guess. Mr. Cooper uses his pen to highlight the two points (22,
22) and (25, 25) on the projector and then lays his pencil across
those two points. The students notice that most of the points
are on or close to that line. After noting that the equation
of this line is y = x because every
point has equal x and y coordinates, Mr. Cooper
helps the students understand how to find the equation of the
line through (23, 23) and (26, 25). He then asks the students
to pick two points on their scatter plots and follow the same
technique to produce the equation of their line. The students
compare their equations. To conclude the lesson, Mr. Cooper
has the |
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computer generate the equation
of the "best fitting" line, y = 0.655424x + 7.9987,
and graph the line on the screen. The bell rings before Mr. Cooper
can have the students calculate the predicted length of a foot of
a person with a 35-centimeter forearm using the computer-generated
equation.
Later that day, during his
seventh-hour planning period, Mr. Cooper and Mr. Johnson discuss
the lesson.
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| The
supervisor and teacher confer on the lesson. The teacher notes that
students have a more positive attitude toward the topic but need more
practice. |
Mr. Cooper: This
seemed to go better than the textbook approach I have used in the
past.
Mr. Johnson: The
students were very engaged.
Mr. Cooper: Yes,
but they will still need lots of practice to get the procedure down
pat.
Mr. Johnson: Some
of that practice should include situations like you used today.
The students learned much more than how to write the equation of
a line.
Mr. Cooper: It takes
so long to do a lesson like this. We didn't get finished in a class
period.
Mr. Johnson: Student
interaction in inquiry lessons like this one take more time, but
the students really seemed to understand what was happening. I think
they will remember this much longer than a chalkboard explanation.
Mr. Cooper: I can
probably do a better job of managing such a lesson next time.
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The supervisor
comments on the strengths of the lesson and offers encouragement.
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Mr. Johnson:
You'll find that things get easier with experience. Keep in
mind the high level of student interest and the mathematical potential
of the lesson. It was an excellent first lesson with this new material
and approach.
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