as a Language
As with all language acquisition, we learn by talking, listening, reading,
and writing. In mathematics class, we build the skills that allow us
to communicate our thinking with many others. We make conjectures, try
these out, report on our progress, and refine our thinking. An important
aspect of communication of mathematical ideas is to have a variety of
ways to think about and express those ideas. Students may use sketches,
graphs, tables, symbols, or words to facilitate both thinking and communicating.
The overarching goal is to make sense of and take ownership of mathematical
concepts. This goal is more efficiently reached when students are given
opportunities to discuss their thinking with peers and teachers. The
metacognitive activity of formulating, representing, clarifying, and
communicating ideas leads to an increase in learning. (Lampert and Cobb,
in the CPMP Classroom
An observer would see students poring over their collaborative work,
making suggestions for improvements, and in the process, making mathematical
sense of the ideas being studied. Investigative questions deliberately
call for collaboration. "Effective learning environments are community-centered.
These communities can build a sense of comfort with questioning rather
than knowing answers and can develop a model of creating new ideas that
builds on the contributions of individual members." (Pellegrino, 2000)
Homework questions consciously ask for explanation and reflection, all
with the goal of being able to do new and somewhat different problems.
See CPMP Classrooms.
Parents can also be part of this metacognitive process (Garafolo, 1985)
when they allow students to explain what they have learned and identify
where they still have difficulty. Having your child teach you a concept
learned in class is a powerful way of reinforcing that learning. See Helping
with Homework and Tips for Helping a Student.
as an Assessment Tool
Communication is integral to a CPMP classroom, both to develop understanding
of mathematics and as a means for the teacher to assess what each student
knows. Thus, teachers can be seen monitoring group investigations, leading
class discussions, making informal assessments of individual and whole-group
knowledge, and adjusting their teaching plans as they gather information.
Of course, tests and quizzes and homework also require students to communicate
their thinking clearly, to convince themselves and their teachers that
they really have mastered new ideas.
The emphasis on communication in mathematics is a fairly new development
in this country. Most state mathematics standards now reflect this. In
some other countries, this emphasis on communication is not so new. Watching TIMSS videos
of Japanese classrooms or reading about Chinese classrooms in Liping
Ma's book will reveal careful communication, far more elaborate than
the short answers quickly given that you may recall from your own schooldays.
Research to consider:
- National Research Council. How People Learn: Brain, Mind, Experience,
and School. Committee on Developments in the Science of Learning
and the Committee on Learning Research and Educational Practice.
J. Bransford, A. Brown, R. Cocking, S. Donovan, and J. Pellegrino
(eds.). Washington, DC: National Academy Press 1999.
- Kaput, James J. "Linking Representations in the Symbol Systems of
Algebra." In Research Issues in the Learning and Teaching of Algebra, edited
by Sigrid Wagner and Carolyn Kieran, pp. 167-194. Research Agenda for
Mathematics Educators, vol. 4. Reston, VA: Lawrence Erlbaum Associates
and NCTM 1989.
- Garafolo, Joe and Frank K Lester, Jr. "Metacognition, Cognitive Monitoring,
and Mathematical Performance." Journal for Research in Mathematics
Education 16 (May 1985): 163-76.
- Hiebert, James. "Relationships between Research and the NCTM Standards." Journal
for Research in Mathematics Education 30 (January 1999): 3-19.
- Lampert, Magdalene. "When the Problem is not the Question and the
Solution is Not the Answer: Mathematical Knowing and Teaching." American
Educational Research Journal 27, no. 1 (Spring 1990): 29-63.
- Lampert, Magdalene, and Paul Cobb. "Communication and Language." In Research
Companion to NCTM's Standards, edited by Jeremy Kilpatrick, W.
Gary Martin, and Deborah Schifter. Reston, VA: National Council of
Teachers of Mathematics, 2003.
- Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers'
Understanding of Fundamental Mathematics in China and the United
States. Mahwah, NJ: Lawrence Erlbaum Associates, 1999.
- Silver, Edward A., Jeremy Kilpatrick, and Beth G. Schlesinger. Thinking
Through Mathematics: Fostering Enquiry and Communication in Mathematics
Classrooms. New York, NY: College Entrance Examination Board,
- Silver, Edward A., and Margaret S. Smith. "Implementing Reform in
the Mathematics Classroom: Creating Mathematical Discourse Communities." In Reform
in Math and Science Education: Issues for Teachers. Columbus, OH:
Eisenhower National Clearing House for Mathematics and Science Education,
- Stigler, James W., and James Heibert. The Teaching Gap: Best Ideas
from the World's Teachers for Improving Education in the Classroom. New
York, NY: The Free Press, 1999.