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Classroom Discourse
(Mathematics)
Teresa Maguire and Alex Neill (2006)
Mathematical
classroom discourse is about whole-class discussions
in which students talk about mathematics in such a way
that they reveal their understanding of concepts. Students
also learn to engage in mathematical reasoning and debate.
Discourse
involves asking strategic questions that elicit from
students both how a problem was solved and why a particular
method was chosen. Students learn to critique their
own and others' ideas and seek out efficient mathematical
solutions.
When
to use
Discourse
can be used at any time during a unit of work.
It
can:
-
be used to determine what students are thinking and
understanding in order to build bridges between what
they already know and what there is to learn;
-
offer opportunities to develop agreed-upon mathematical
meanings or definitions and explore conjectures.
The
theory
While
classroom discussions are nothing new, the theory behind
classroom discourse stems from constructivist views
of learning where knowledge is created internally through
interaction with the environment. It also fits in with
socio-cultural views on learning where students working
together are able to reach new understandings that could
not be achieved if they were working alone.
Underlying
the use of discourse in the mathematics classroom is
the idea that mathematics is primarily about reasoning
not memorization. Mathematics is not about remembering
and applying a set of procedures but about developing
understanding and explaining the processes used to arrive
at solutions.
How
the strategy works
Well-designed
distractors provide alternatives that identify particular
misconceptions. Providing a number of alternatives may
scaffold the students' thinking.
When
a class of students offers a range of responses and
strategies to solve a problem, discussion arises over
the validity of each response. This gives the class
the opportunity to explore and reach a common agreement
on which method(s) would be successful and/or most efficient.
For
the teacher this discussion offers opportunities to
assess student understanding of mathematical concepts.
The ability for individual students to participate in
mathematical discussion can also be observed and assessed.
What
to do
In
order for discussion to take place, classroom (sociomathematical)
norms need to be firmly established so students feel
comfortable explaining and justifying their responses.
Establishing
this classroom culture can be done by:
-
expecting students to explain and justify their answers,
whether they are correct or not;
-
emphasising the importance of contributing to the
discussion by explaining their strategy rather than
producing correct answers;
-
expecting students to listen to and attempt to understand
others' explanations;
-
commenting on or redescribing students' contributions
while notating the reasoning for the class on the
board;
-
having other students pose clarifying questions to
the student explaining the problem;
-
expecting students to explain why they did not accept
explanations that they considered invalid;
-
using students' names to label agreed-upon conjectures,
e.g., "Natasha's rule"
Paul
Cobb (2006) states that there are two parts to a mathematical
explanation. The calculational explanation involves
explaining how an answer or result
was arrived at – the process that was used. A
conceptual explanation involves explaining why
that process was selected – what are the reasons
for choosing a particular way. In this way students
have to be able to not only perform a mathematical procedure
but justify why they have used that particular procedure
for a given problem.
Ways
to encourage calculational explanations:
-
Pose a problem and expect students to find their own
way to a solution.
-
Ask questions that are designed to keep students puzzling
like "How are we going to figure this out?"
"What should we do?" "Who has an idea?"
"I don't understand. How will that work?"
-
Listen and watch rather than indicate whether responses
are right or wrong.
- Use
True/False or open number sentences or statements
to generate a range of answers that require individuals
to justify them. Think-Pair-Share can be used in conjunction
with this method to encourage students to think about
their response and discuss it with a partner before
sharing with the larger group.
-
Use questions that encourage a range
of responses. For example:
- "Who
has a different way to solve the problem?"
- "What
is a different way to do this?"
- "Who
has another way to think about this?"
- "Would
someone like to add to that idea?"
- "Can
you explain what John just said in your own words?"
followed by "John, does that describe your
idea?"
Ways
to encourage conceptual explanations:
- Use
questions that require students to justify their own
and other's answers
For example:
- "Why
do you agree with that?"
- "Why
do you disagree?"
- "Is
this true for all numbers?"
- "How
can we know for sure?" (these question are
precursors to mathematical proof)
Limitations
-
Students may not arrive at an agreed-upon answer during
their discussion. The teacher has to decide when to
step in and provide an explanation, when to model,
and when to ask pointed questions that can shape the
direction of the discourse. One way to overcome this
is to ask "If someone from the classroom next
door said '…..' what would you say?"
-
The teacher needs to be able to anticipate responses
and respond spontaneously to students.
-
The teacher needs to develop a deep knowledge of mathematics
concepts and principles in order to understand the
reasons behind students' errors. A teacher needs to
have one eye on the underlying mathematical concepts
while the other eye is focused on the current understandings
of the students.
-
Some students may have difficulty explaining their
reasoning.
Adapting
the strategy
Classroom
discourse has been used in research projects that have
led to ARB resources. In Neill's 2005 set article on
estimation, refer to the 'Method' section and Figure
2 which describe an extended process that includes discussion.
This
strategy has similarities to other strategies where
students are required to explain and justify a position
or point of view. Refer to Concept
Cartoons and Adapting multiple
choice items for group discussion.
Examples
of ARB resources that can be used for classroom discourse
Maths
AL7106
and AL7109
use True/False number sentences to explore the additive
identity and the concept of equality. These can easily
be used as whole-class discussion starters.
NM1200,
NM1202,
NM1203,
NM1204,
NM1206,
NM1207,
and NM1208
get students to discuss and compare the estimation strategies
they use on a problem, and use this to help introduce
new methods of estimation to students.
NM1199,
NM1201,
and NM1221
ask students to identify which cartoon characters are
estimating and which are not. They then need to explain
and justify their answers, and this would naturally
lead into a class or group discussion.
Other examples of ARB resources that can be used for classroom discourse
Resources that use
classroom discourse.
Selected
references
Burns, M. (2005). Looking at How Students Reason. Educational
Leadership, 63 (3), pp. 26-31.
Chapin,
O'Connor & Anderson. Classroom Discussions:
Using Math Talk to Help Students Learn. Retrieved
May 31, 2006, from http://www.k12.wa.us/conferences/JanConf2006/materials/11/Davison_L2.doc
(link has since changed).
Cobb,
P. (2006) Supporting Productive Whole Class Discussions.
Retrieved May 31, 2006, from
http://www.nzmaths.co.nz/sites/default/files/Numeracy/References/PaulCobb.ppt
Neill,
A. (2005) Estimation exposed. set:
Research information for teachers, 1, 48-53.
Schifter,
D. (1996). A Constructivist Perspective on Teaching
and Learning Mathematics. Phi Delta Kappan, 77
(7), 492-499.
Assessment
strategies | ARB
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