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The teacher of mathematics
should promote classroom discourse in which students-
listen to, respond to, and question the teacher and one another;
use a variety of tools to reason, make connections, solve problems,
and communicate;
initiate problems and questions;
make conjectures and present solutions;
explore examples and counterexamples to investigate a conjecture;
try to convince themselves and one another of the validity of
particular representations, solutions, conjectures, and answers;
rely on mathematical evidence and argument to determine validity.
Elaboration
The nature of classroom
discourse is a major influence on what students learn about mathematics.
Students should engage in making conjectures, proposing approaches
and solutions to problems, and arguing about the validity of particular
claims. They should learn to verify, revise, and discard claims
on the basis of mathematical evidence and use a variety of mathematical
tools. Whether working in small or large groups, they should be
the audience for one another's comments - that is, they should speak
to one another, aiming to convince or to question their peers. Above
all, the discourse should be focused on making sense of mathematical
ideas, on using mathematical ideas sensibly in setting up and solving
problems.
Vignettes
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| Establishing
norms of discourse such as those described in this section is hard
work, especially with older students who have become accustomed to
a different set of standards for school thinking and talking. |
3.1 It is late September
in a sixth-grade class. Mrs. Fondant wants to engage her students
in a problem that will yield multiple solutions to help break down
their image of mathematics as a domain of single right answers.
One aim she has right now is to establish the norms and routines
of discourse in the class. She knows that much of what she does
is different from what the students have grown accustomed to in
previous grades. Therefore, Mrs. Fondant takes this aspect of her
task at the beginning of the year seriously. She thinks she is finally
making some progress with this group, after a month of concentrating
on this dimension of her work with them.
She writes the following
problem on the board:
The Wolverines scored
30 points in the first half of last night's basketball game. The
unusual thing is that they did it without scoring a single foul
shot. How did they score the 30 points?
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| Students
are eager to share answers. |
Immediately, one student
yells, "That's easy! They scored ten 3-point shots!"
Another quickly interjects,
"There are a lot of possibilities. It could be fifteen 2-point
shots." Mrs. Fondant has one of the students explain scoring
possibilities.
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| Students
are using a variety of tools to work on the problem. In this example,
solution strategies vary, but answers are the same. |
Several students offer possible
answers. The teacher directs them to work alone or with a partner
to figure out as many answers as they can. Walking around, she notices
that some are constructing tables, others are using formulas, and
still others are randomly writing down combinations as they occur
to them.
After a few minutes, Mrs.
Fondant asks if they are ready to talk about what they have found.
They seem to be, so she asks one girl for one of the combinations
that she worked out.
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This student already
assumes that justifying her answer is part of giving it.
Students initiate
strategies.
The teacher expects
students to take a large share in the pursuit of one another's suggestions.
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The girl says, "Two
3-pointers and twelve 2-pointers - 2 x 3 is 6 and 12 x 2 is 24 and
6 + 24 is 30."
Several others add possible
combinations.
Another girl asks, "How
can we keep track of all these? Let's make a table."
Mrs. Fondant looks expectantly
at the group. "Does someone want to make a table on the board?"
One student comes up and
makes a two-column chart:
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3-point
shots
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2-point
shots
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He stands at the board recording
the combinations other suggest.
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3-point
shots
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2-point
shots
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8
10
0
4
2
6
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3
0
15
9
12
6
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| Students
question one another's ideas. |
One
student questions the solution, six 3-point shots and six 2-point
shots. "That makes 33 points." |
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Students support
their ideas and solutions in response to others' challenges or counterarguments.
Students search for patterns and question inconsistencies
that puzzle them. They are key participants in the discourse.
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A girl explains, "6
x 3 is 18 and 6 x 2 is 12. 18 + 12 is 30.
"Oh,
yeah," he says.
Another
student suggests redoing the table so that the combinations are
in order. "Maybe we'll see a combination we missed."
The teacher asks him to come up and do that.
Looking at
the revised table, one student raises her hand. "Why are
there no odd numbers in the 3-point box?"
Several
seconds pass. Everyone seems to be pondering this. A couple of
students whisper to one another.
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The teacher's
question does not endorse or dismiss this idea.
Students puzzle
about the mathematical clues.
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The teacher asks what they
think about this. One girl says that maybe they missed some. A few
students begin searching for other combinations. After a few moments,
a student says she thinks it has something to do with the fact that
an odd number times an odd number equals an odd number, but she's
not sure what that tells her.
In the third
row, a few students are leaning together, talking quietly. One
says, "I think that's important."
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The teacher expects
students to look for mathematical evidence.
Students make
conjectures publicly and try to convince themselves and one another
of their validity.
The teacher does
not directly evaluate the correctness of students' comments. Instead
she interprets the comment as a conjecture and expects the students
to examine its validity. She then extends the problem, trying to
foster students' habits of mathematical inquiry.
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Mrs. Fondant asks why it
would matter.
"Because
for 2-point shots, the total number of points will always be even,"
begins one student and then pauses.
"Oh!
Is it because 30 is even? And you need two even numbers to equal
an even?" bursts in one of the boys who has been talking
in the little group.
"What
do you think?" asks Mrs. Fondant.
Before class
ends, Mrs. Fondant asks the students, "What if they had scored
31 points-would that have changed our table?"
3.2
Ms. Chavez has rolled the math department computer into her class
for the morning and has connected it to her LCD viewer. Her 28
first-year algebra students, seated at round tables in groups
of threes and fours, are working on a warm-up problem. The day
before they had had a test on functions. For the warm-up to today's
class, Ms. Chavez has asked students to set up a table of values
and graph the function y = |x|.
She has
chosen this problem as a way to introduce some ideas for a new
unit on linear, absolute value, and quadratic functions. During
the warm-up, students can be heard talking quietly to one another
about the problem: "Does your graph look like a V-shape?"
"Did you get two intersecting lines?" Walking around
the room, Ms. Chavez listens to these conversations while she
takes attendance. After about five minutes, she signals that it
is time to begin the whole-group discussion.
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| Students
are expected to communicate about mathematics. They seem used to paying
attention to one another's ideas and to reasoning together. They also
accept responsibility for helping others. |
A girl volunteers and carefully
draws her graph on a large wipe-off grid board at the front of the
room. As she does this, most students are watching closely, glancing
down at their own graphs, checking for correspondence. A few students
are seen helping others who had some difficulties producing the
graph.
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| Students
use a variety of tools to reason together about mathematics. They
do not rely on the teacher to initiate all ideas or to certify results. |
Another
student suggests that they enter the function into the computer and
watch it produce the graph. Several other students chime in, "Yeah!"
The first girl does this, and the class watches as the graph appears
on the overhead screen. It matches the graph she sketched, and the
class cheers, "Way to go, Elena!" |
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By directing them
to think through comparisons, the teacher creates a context that
is likely to promote students' reasoning and communication about
these functions and their graphs.
The students communicate
with one another about mathematics without the teacher asking them
questions or directing their comments. They also use mathematical
language that they have developed through the discourse.
The teacher listens
to students carefully.
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Ms. Chavez then asks the
class to sketch the graphs of y = |x| + 1,
y = |x| + 2, and y = |x| - 3
on the same set of axes and write a paragraph that compares and
contrasts the results with the graph of y = |x|.
"Feel free to work alone or with the others in your group,"
she tells them.
After a few
minutes, two students exclaim, "All the graphs have the same
shape!"
A few other students look
up. Another student observes, "They're like angles with different
vertex points." "Then they're really congruent angles,"
adds his partner.
Ms. Chavez
circulates through the class, listening to the students' discussions,
asking questions, and offering suggestions. She notices one group
has produced only one branch of the graphs. "Why don't you
choose a few negative values for x and see what happens?" Another
group asks, "What would happen if we tried |x - 3|?"
"Try it!" urges Ms. Chavez.
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| Students
initiate conjectures publicly. They make connections between this
graphing activity and transformational geometry. |
The
students continue working, and the conversation is lowered to murmurs
once again. Then the members of one group call out, "Hey, we've
got something! All these graphs are just translations of y = |x|,
just like we learned in the unit on geometry." |
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Students present
and explain solutions to the rest of the class.
The teacher does
not press for closure on these ideas simply because the period ends.
These journals
give the teacher insights into students' thinking. They also offer
students the opportunity to reflect on their understandings and
feelings.
The teacher plans
homework to strengthen students' developing ideas from class and
also to extend and stretch their thinking in preparation for the
next class.
Both teachers
and students benefit from collaborating on assessment. Teachers
can gain additional information and insights about students; students
gain additional opportunities to integrate and reflect on their
understanding.
Students are continually
pressed to seek connections.
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Lionel sketches his graph
on the dry-erase board. Elena again enters the equation of the graph
into the computer and the class watches as the graph is produced.
The computer-generated graph verifies Lionel's attempt. Again there
are cheers. Lionel gives a sweeping bow and sits down.
Ms. Chavez
asks the students to write in their journals, focusing on what
they think they understand and what they feel unsure about from
today's lesson. They lean over their notebooks, writing. A few
stare into space before beginning. She gives them about ten minutes
before she begins to return the tests. She will read the journals
before tomorrow's class.
At the end
of the period, she distributes the homework that she has prepared.
The worksheet includes additional practice on the concept of y = |x| ± c
as well as something new, to provoke the next day's discussion:
y = |x ± c|.
Over the
next couple of weeks, students explore linear, quadratic, and
absolute value functions. Nearing the end of this unit, Mrs. Chavez
decides to engage students in reflecting on and assessing how
far they have come.
As she assigns
homework for that evening, she announces, "I'd like each
of you to write two questions that you think are fair and would
demonstrate that you understand the major concepts of this unit.
I'll use several of your ideas to create the test. And here's
a challenge for the last part of your assignment: you just drew
the graph of f(x) = x2 - 2x
as a part of the review. Think about everything we've done so
far this semester, and see if you can remember any ideas that
will help you draw the graph of |f(x)|."
3.3
Ms. Pizzo has been working on fractions for a little over a week
with her thirty-six students, the biggest class she has ever had.
She feels that she is not connecting very well with them - the
group is simply too large. Many students have a conception of
fractions that they picked up last year, which is that a fraction
is a certain size piece of something. For example, "one-fourth"
is this:
"Three-quarters"
looks, as one student said, like "a baby carriage":
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| The
students bring their own ideas, ways of talking, and reasoning. |
Ms. Pizzo is worried, though,
for her students do not seem to understand fractions as numbers,
nor do they see fractions as relational to some referent whole:
for example, the idea of three-fourths of eighteen makes no sense
to them. Three-fourths is the baby carriage shape.
Trying to think
of something that will engage them and get them talking about fractions
in some other ways, Ms. Pizzo decides to give them the following
problem:
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| The
teacher thinks that because they can think about "one-half of"
something in a variety of ways, they may be "launched' into another
way of interpreting and making sense of one-third. |
1/3 of the crayons
in James's box of 12 crayons are broken.
1/3 of the crayons
in Fred's box of 12 crayons are broken.
Who should be sadder and
why?
The students
seem interested in this. They set to work, some drawing pictures,
some getting out real crayons. Ms. Pizzo overhears Hilda tell Robbie,
"Look, one-half is this much-
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| The
students are using a familiar representation of fractions to model
and reason about an unfamiliar situation. |
and one-third
is this much-
so you can just tell that James should be sadder."
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The two children
pursue a conjecture, exploring examples and looking for counterexamples.
They are trying to convince themselves of what they have discovered.
The students realize
that a counterexample would change what they had found.
Listening to students'
conversations can yield valuable information for the teacher. By
attaching the label "conjecture" to their idea, she begins
to help students develop a language for the mathematical thinking
they are doing.
The students are
in the habit of letting mathematical evidence determine the validity
of an idea or an answer.
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Robbie, staring at Hilda's
drawing, exclaims suddenly, "Look! One-third is smaller
than one-half, even though three is more than two!"
"Hey!"
answers Hilda. "Does that work with others?"
Robbie quickly
draws one-fourth. The two children look at each other, excited.
Steven leans
over. "It doesn't work. Lookit." He draws the three-fourths
baby carriage shape. "Three-fourths. It is bigger than one-half,
and four is more than two."
Ms. Pizzo is
enjoying overhearing this interchange. "What was your idea,
your conjecture?" she asks Hilda and Robbie. She hopes that
she can get them to articulate what they were noticing.
By now, several
other children have wandered over to Hilda's desk, having overheard
this excitement.
"I guess
we were saying that, with fractions, if there is a bigger number
on the bottom, the fraction is smaller. It's like opposite,"
answers Robbie.
"But now
with what Steven showed us, we see we were wrong," adds Hilda.
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The teacher does
not press to elaborate or qualify the students' partial understanding
of the meaning of the denominator in relation to the size of the
fraction - for instance, showing them that their conjecture would
work with unit fractions. Instead, she assumes that the class as
a whole can work with and clarify what Hilda, Robbie, and Steven
have been working on.
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"Let's talk about this
in the whole-group discussion," suggests Ms. Pizzo, pleased
both that the students are finding this interesting, that they are
beginning - despite the size of the group - to produce little mathematical
sparks that kindle the rest of the students' interest, and that
they are on to a key concept in fractions. She makes a note to herself
to develop some kind of task that will help students investigate
their ideas about the relationships between the size of denominators
and the size of fractions.
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