The teacher of mathematics,
in order to enhance discourse, should encourage and accept the use
computers, calculators, and other technology,
concrete materials used as models;
pictures, diagrams, tables, and graphs;
invented and conventional terms and symbols;
metaphors, analogies, and stories;
written hypotheses, explanations, and arguments;
oral presentations and dramatizations.
In order to establish a
discourse that is focused on exploring mathematical ideas, not just
on reporting correct answers, the means of mathematical communication
and approaches to mathematical reasoning must be broad and varied.
Teachers must value and encourage the use of a variety of tools
rather than placing excessive emphasis on conventional mathematical
symbols. Various means for communicating about mathematics should
be accepted, including drawings, diagrams, invented symbols, and
analogies. The teacher should introduce conventional notation at
points when doing so can further the work or the discourse at hand.
Teachers should also help students learn to use calculators, computers,
and other technological devices as tools for mathematical discourse.
Given the range of mathematical tools available, teachers should
often allow and encourage students to select the means they find
most useful for working on or discussing a particular mathematical
problem. At other times, in order to develop students' repertoire
of mathematical tools, teachers may specify the means students are
teacher expects not just answers, but also reasons.
Mr. Johnson has presented his first-grade class with several
pairs of numbers and asked them to decide which number is greater
and to justify their responses. He has also been encouraging them
to find ways to write these comparisons.
child uses objects to communicate and justify his answer.
I think 5 is greater than 3 because (he walks to the board and
sticks five magnets up and then carefully sticks three magnets in
teacher encourages students to use symbols to represent and communicate
Mr. Johnson asks whether
that makes sense to other people. The children nod. He asks if anyone
wants to show how they would write this down.
Kevin, up at the board,
Next, Betsy writes:
Mr. Johnson: Can
you explain what you were thinking? Kevin?
The teacher accepts
more than one way of representing the idea with symbols; both are
nonstandard but sensible.
The teacher poses
a challenge that requires students to invent a means of recording
Kevin explains that his
arrow shows that 5 is more than 3 because the bigger number "can
point at" the smaller one.
Mr. Johnson asks Betsy to
explain hers, and she says that she thinks you should just circle
the smaller one.
Mr. Johnson: What
if the two numbers you were comparing were 6 and 6? What would you
do? How would you write that?
The teacher gives
students time to think before responding. He doesnât repeat the
question or call on children; he is silent.
The teacher connects
the studentsâ approaches and reasoning to the conventional notation.
Because the students have thought about what it means for two numbers
to be equal, they are ready to learn how that is conventionally
represented. In this case, the notation follows the development
of the concept in a meaningful context.
Several seconds pass. Ruth
shoots her hand in the air. Several others also have their hands
Ruth: You could draw
an arrow to both of them.
Annie: You could
circle both of them because they are the same.
Jimmy: You shouldn't
mark either one, either way. They are not greater or less.
They are the same.
Mr. Johnson nods at their
suggestions. He writes an equals sign (=) on the board and explains
that this is a symbol that people have invented for the ideas the
children have been talking about.
Rashida: That's like
what Ruth said.
Ruth beams and Annie calls
out: It's like mine too.
teachers have posed as task that requires students to communicate
about mathematics-in pairs as well as in the whole group-and that
lends itself to a variety of tools.
Mrs. Martinez and Mr. Golden, who have teamed up to teach
eighth grade this year, have divided their students into groups of
four. The teachers have challenged them to show why the text says
that division by zero is "undefined." The teachers want their students
to know why "you can't divide by zero." Usually that is all that students
have learned. Once the students figure out why division by zero is
undefined, they are to prepare something that they could use to justify
their explanation to the rest of the class.
The teachers suggest
particular tools in order to stimulate students to make choices
about what might help them work on the problem.
The teacher suggests
another tool-a graph-that might help the students make mathematical
connections, examine the pattern they are seeing, and present their
work to the rest of the group.
Mrs. Martinez suggests that
the calculator may be a useful tool for this problem. "Making up
some kind of story problem for a situation that involves division
might be helpful for others," adds Mr. Golden. The two teachers
have arranged their large classroom so that calculators, graph paper,
Unifix cubes and base ten blocks, felt-tip markers and blank overhead
transparencies, rulers, and other materials are out where students
can freely use them. This facilitates the use of alternative tools.
Students are encouraged and expected to make decisions about which
tool to use. Several students are, in fact, preparing overheads
to display their conclusions about division by zero. Others are
excitedly punching calculator buttons.
"The answer keeps getting
larger and larger!" exclaim a pair of girls as they watch the results
obtained by successively dividing 4 by smaller and smaller divisors
with the calculator. "Why is that important?" asks Mrs. Martinez
as she watches over one girl's shoulder. "Well, because each of
the numbers we are dividing by is getting closer and closer to zero
but isn't zero." "Maybe you could make a graph to show what you
are finding," suggests Mrs. Martinez.
In this situation,
the teacher chooses to point the students directly to a means of
working on and discussing the problem that plays off of important
this idea as a tool for working on the problem, the teacher leaves
the students to use it on their own.
Mr. Golden finds two students
slouching sullenly in their chairs behind the room divider. "We
don't understand what to do," grumbles one. Sitting down next to
them, Mr. Golden begins, "Let's see if I can help. You are trying
to figure out what the special problem is in trying to divide by
zero. Maybe you can use some things you already know about division.
How do you know that 82
is 4? How could you prove that if someone challenged your answer?"
The students look at him disbelievingly. He waits. Then one says,
"Well, I'd just say that 4 times 2 is 8 so 8 divided by 2 has
to be 4. " "Can that help you at all with this problem?" asks
Mr. Golden. He stands up. The two students look at one another and
then, sitting up a bit, begin talking. "Well, that doesn't work
if you take 80,
" Mr. Golden hears one say as he walks away.
Because the discourse of
the mathematics class reflects messages about what it means to know
mathematics, what makes something true or reasonable, and what doing
mathematics entails, it is central to both what students
learn about mathematics as well as how they learn it. Therefore,
the discourse of the mathematics class should be founded on mathematical
ways of knowing and ways of communicating. The nature of the activity
and talk in the classroom shapes each student's opportunities to
learn about particular topics as well as to develop their abilities
to reason and communicate about those topics. Students' dispositions
toward mathematics are also fundamentally influenced by the experiences
they have with mathematical activity. Although teachers may seem
quieter at times, the teacher is nevertheless central in fostering
worthwhile mathematical discourse within the classroom community.
Teachers' skills in developing and integrating the tasks and discourse
in ways that promote students' learning depend on the construction
and maintenance of a learning environment that supports and
grows out of these kinds of thinking and activity. It is to this
issue that we now turn.