TEACHING PERSPECTIVES

Why do I enjoy teaching mathematics? Creating a lesson involves problem solving. I enjoy the creative process of identifying learning objectives, creating activities that require students to think, anticipating students’ responses to those activities, and visualizing a learning trajectory. Teaching a lesson is exciting for me. I enjoy the challenge of having to respond to unanticipated situations during a lesson and having to make decisions within a set of constraints such as time constraints, the need for closure, and the need to preserve students’ autonomy in solving the problem. Metaphorically, planning and teaching a math lesson is like strategizing and playing a sport; a sport in which my students and I are on one side and the problem to be solved on the other. Winning occurs when my students engage in the problem, solve the problem independently, learn the underlying mathematical ideas, and enjoy the process. Winning is rewarding, but winning does not always occur. With practice and reflection, the chance of winning increases over time.

What is my view on learning and teaching? I consider learning to be the process by which students construct new knowledge from their existing knowledge as they interpret and make sense of a situation. Teaching is the process of creating and managing situations conducive for desirable learning to occur.

What is my role as a teacher? While students are responsible for their own learning, I am responsible for creating the opportunities for them to learn. With target learning goals in mind, I select and design meaningful problems that can provoke students’ intellectual need for those goals. For example, the task of finding the speed at a specific time, say t₁, on a distance-time graph that is created by a motion detector will potentially necessitate students’ attending to the meaning of speed. If t₁ is within the at-rest interval of a walk-stop-walk motion, then students who use the formula speed = distance / time to obtain a value, say d₁/t₁, will be disconcerted. I am also responsible for creating an environment conducive for students to explore ideas, to express their ideas, to challenge each other’s ideas, and to learn from mistakes. Like a coach I will facilitate the learning process by intervening discriminately, by challenging them to communicate, formulate and justify their ideas in ways that are amenable to the mathematics community, by orchestrating whole class discussions, and by assuming a figure of authority in situations in which I need to communicate standard mathematical conventions or to clarify misinterpretations. To help students retain what they have learned in class, I will assign homework problems that require them to repeat productive reasoning rather than merely to practice procedures. Students may resist this mode of learning initially but their resistance is likely to diminish as they see how their struggle with mathematics has helped them deepen their mathematical understanding and develop confidence in their problem-solving abilities.

What do I want my students to learn? I once asked why we make students learn algebra, calculus and higher mathematics when most students do not use these topics in adult life. I then realized it is mathematical ways of thinking that our students should be developing instead of a collection of rules, procedures, theorems, and proofs. Examples of mathematical ways of thinking include making and testing conjectures, justifying one’s claim, finding the underlying cause for a mathematical phenomenon, using heuristics to solve a problem, making connections, and seeking efficiency. Some of these ways of thinking can be useful in solving non-mathematical problems such as identifying the cause of a production breakdown or defending one’s business proposal. In teaching a topic, I try to identify two complementary sets of learning goals:(a) key concepts and (b) the associated ways of thinking. In addition, I try to interact with students in a manner that fosters desirable habits of mind such as taking ownership in one’s learning, practicing patience with oneself and with others, accepting and learning from mistakes, welcoming feedback, and being reflective.

What qualities do I value? Based on my teaching experience, I think students respect teachers who are principled yet flexible, confident yet humble, and challenging yet encouraging. A teacher should strive to teach according to a set of teaching principles while accommodating students’ needs and the university’s requirements. A teacher should be competent and efficacious, but should respect students regardless of their mathematical competence. Students should be challenged to meet high expectations, but at the same time the teacher must identify with their struggle and provide support and encouragement. One of the most important attributes is that a teacher should be passionate about teaching and should care deeply about the students’ learning. In summary, teaching is exciting and rewarding for me; it is something that I can continually improve by being reflective and staying current in the field.

Updated on September 1, 2006

Reality is the same for all, yet it is perceived differently by each.