Board work Students come to the board (individually
or in small groups) to write up solutions for class discussion. Sometimes,
several groups will work at the board on the same problem in order for me to
demonstrate that there is no one right answer or approach to solving a problem.
Students learn from each other's mistakes and learn presentation skills as
well. At first, I do not require that students remain at the board to defend
their solution but, as the classroom becomes more supportive, students gain
confidence in their ability to discuss mathematical ideas.
Brainstorming I give students about two to five
minutes in pairs to write down everything they know about a given topic, after
which I call on them randomly or in turn and generate a list on the blackboard.
Discussion centers around evaluating and categorizing the items gathered. I use
this technique most often for review or to begin an investigative class (see
below).
The purpose of assignments is to allow students
to practice with concepts and processes without being penalized. 

Problemposing Students generate their own questions
in a particular area. These are examined and the students are then asked to
commit themselves to a particular conjecture, which then is added to a class
list. Proving this conjecture becomes the focus of attention in the ensuing
weeks.
Investigative class The class examines patterns in
concrete examples to uncover algebraic structure. It then makes
generalizations, describes them in the form of a conjecture, and develops
theory to prove or disprove the conjectures. For example, we generated all
standard structure theorems for finite groups in this way.
Small group work Students work together in small
groups and I circulate. This provides me with information on student
understanding and enables students to intervene in the pace and the development
of the course. Even in one large class, where I could see only the work of the
students at the ends of the rows, I gained immediate feedback on the students'
understanding of the concepts involved. Students in that class accused me of
checking up on them. I asked whether they would prefer I waited for the test
before I found out how much they understood and pointed out that a winning
strategy would be always to sit near the aisle. This produced a very curious
seating arrangement for the remainder of the course: the centre of the room was
empty and all students were concentrated at the ends of the rows.
Proof generation The forwardbackward proof technique
described by Solow in his book, How to Read and Do Proofs, is a very
empowering strategy to teach students (4). The technique takes its name from
the way mathematicians typically organize their thoughts when constructing a
proof of a statement for the first time. It often eliminates students'
complaints that they "don't know how to begin!"
Some students have difficulty adjusting to my teaching
approaches and for this reason, ongoing classroom research is essential. For
example, one student complained that my questioning techniques and practice of
asking her to come to the board were intimidating and humiliating. On the other
hand, she appreciated the opportunity to learn from other students. Through
negotiation, we were able to arrive at a compromise which made her, and other
students in the class, more comfortable about participating.
Evaluation of student
learning
The studentcentered techniques I employ provide immediate,
frequent and regular feedback on students' understanding of the course content
and processes. Following is a brief description of some of the more formal
evaluation methods I employ.
Assignments I assign homework regularly throughout
the course and space the assignments so that, when combined with quizzes and
examinations, students receive regular feedback on written work. I encourage
students to collaborate on assignments but will accept only independent
writeups. I then comment on the homework but do not grade it. The purpose of
these assignments is to allow students to practice with the concepts and
processes of the course without being penalized for doing so.
Participation Students earn participation credit in
a variety of ways, allowing for their individual learning preferences. Some of
these ways include: preparing for class by doing assigned reading and working
problems from the text; participating during the class in assigned exercises;
sharing ideas by coming to the blackboard to present a proof by asking
questions, by offering explanations or by joining in discussions; visiting me
for an office consultation. 