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).

Problem-posing
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 forward-backward 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 student-centered 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 write-ups. 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.