M. Bressoud June, 2006
C.1: Develop mathematical thinking and communication skills
Courses designed for mathematical sciences majors should
ensure that students
For me, the greatest revelation to come out of the Calculus Reform
movement was the central importance of writing in the effective
teaching of mathematics. When I began to give students challenging
problems that would take several days to solve and then required them
to explain the solution in detail, I discovered for the first time how
very confused their thinking could be, even when they were able to find
the correct answer. When they had to explain themselves in writing, I
began to appreciate where they were having difficulty, which
definitions and concepts were not clear, and how they were thinking
about a problem. To an extent that was totally impossible with short
answer assignments and exams, I began to understand what I needed to
emphasize and clarify. And, as I critiqued early drafts and forced
students to rework their writing, I witnessed real learning take place,
learning that would be reflected in subsequent assignments.
We who teach and who write professionally know that nothing
clarifies our own understanding of an idea or concept so thoroughly as
having to explain it, either in writing or orally, to another person.
This is the gift that we give to our students when we force them to
convey mathematical knowledge through written and oral communication. I
have found that these are the experiences that the students remember
when the course is over, the lessons that stick with them.
I design each course around roughly three major projects, projects
for which I expect clear explanations of how the problem was solved,
how it is justified, what it means. All project reports involve at
least one preliminary draft on which I comment. There is no getting
around the fact that this takes time and effort on my part, but there
is now a lot of help available on how to provide feedback that is most
helpful as well as how to use my time most efficiently. I especially
recommend Steve Maurer’s A Short Guide to Writing Mathematics
 and Annalisa Crannell’s A Guide to Writing
in Mathematics Classes and her Checklist for Writing and
Grading Essays . The MAA has several Notes
volumes with great advice on the use of writing [3,
5, 6]. What I find to be the
most useful general reference on writing across the curriculum with
suggestions for different types of writing assignments, a discussion of
their relative strengths and weaknesses, and a great chapter on how to
read, comment on, and grade student writing is John C. Bean’s Engaging
Few of us have been trained to use writing as a tool for teaching
mathematics. Fear of the cost in time and effort keeps many of us from
attempting it. But there are many levels at which writing can be used
to effectively support our teaching. We owe it to our students to draw
on the expertise that has been developed and is now readily available.
 John C. Bean, Engaging Ideas: The Professor’s
Guide to Integrating Writing, Critical Thinking, and Active Learning in
the Classroom, Jossey-Bass, San Francisco, 1996.
 Annalisa Crannell, A Guide to Writing in
Mathematics Classes and Checklist for Writing and Grading Essays, http://server1.fandm.edu/departments/Mathematics/writing_in_math/writing_index.html
 Bonnie Gold, Sandra Z. Keith, William A.
Marion, eds., Assessment Practices in Undergraduate Mathematics, MAA
Notes #49, Mathematical Association of America, Washington, DC, 1999.
 Stephen Maurer, A Short Guide to Writing
Mathematics, contact information at http://www.swarthmore.edu/NatSci/smaurer1/WriteGuide/
 John Meier and Thomas Rishel, Writing in the
Teaching and Learning of Mathematics, MAA Notes #48, Mathematical
Association of America, Washington, DC, 1998.
 Andrew Sterrett, ed., Using Writing to Teach
Mathematics, MAA Notes #16, Mathematical Association of America,
Washington, DC, 1990.
Do you know of programs, projects, or ideas
that should be included in the CUPM Illustrative Resources?
Submit resources at www.maa.org/cupm/cupm_ir_submit.cfm.
We would appreciate more examples that document experiences with the
use of technology as well as examples of interdisciplinary cooperation.