Instructional programs from
prekindergarten through grade 12 should enable all students to
 organize and consolidate their mathematical thinking though
communication;
 communicate their mathematical thinking coherently and clearly to peers,
teachers, and others;
 analyze and evaluate the mathematical thinking and strategies of
others;
 use the language of mathematics to express mathematical ideas
precisely.

As students are asked to communicate about the
mathematics they are studyingto justify their reasoning to a classmate or to
formulate a question about something that is puzzlingthey gain insights into
their thinking. In order to communicate their thinking to others, students
naturally reflect on their learning and organize and consolidate their thinking
about mathematics.
Students should be encouraged to increase their
ability to express themselves clearly and coherently. As they become older,
their styles of argument and dialogue should more closely adhere to established
conventions, and students should become more aware of, and responsive to, their
audience. The ability to write about mathematics should be particularly nurtured
across the grades.
By working on problems with classmates, students
also have opportunities to see the perspectives and methods of others. They can
learn to understand and evaluate the thinking of others and to build on those
ideas. For example, students who try to solve the following problem
algebraically may have difficulty setting up the equations:
There are some rabbits and some hutches. If one rabbit is put in
each hutch, one rabbit will be left without a hutch. If two rabbits are put in
each hutch, one hutch will remain empty. How many rabbits and how many hutches
are there?
They may benefit from the insights of students
who solve the problem using a visual representation. Students need to learn to
weigh the strengths and limitations of different approaches, thus becoming
critical thinkers about mathematics.