J Prasad Senesi | jsenesi@uottawa.ca |

Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time.

-Freeman Dyson

While this dichotomy is elegant and descriptive of many mathematicians, it is not one into which an effective teacher can afford to fit. I don't think Dr. Dyson suggests that any one's nature is exclusively that of a bird or of a frog, but only that there are some whose approaches are often dominated by one of these points of view - and that both are necessary in the practice of mathematics. But an educator of math must represent himself equally as both a bird and as a frog, delivering both inspiration and steadfast attention to detail to his students.

In the course of teaching these various courses in different roles, I have learned the obvious lesson that there is no single strategy or 'philosophy' of teaching that can be cut and pasted into any classroom. At the same time, there are certain styles of presentation and interaction that have proven themselves more successful over the years. I will begin by pointing out some of the differences I have encountered in different teaching situations and describing effective approaches, and then I will generalize by describing some ideas and approaches that we can take with us from high school to college and graduate school.

Any lesson plan or lecture is only as good as the instructor's ability to capture and hold the attention of the students. I always begin with a clean board and use an outline style with headings and sub-headings, as much as possible, beginning with a large title for the discussion (What Are The Groups of Order 4?). I move across the board, from one side of the room to the other, toward students and around them, in an effort to maintain the momentum of the conversation and, if necessary, to prevent the students from daydreaming away (it's harder for them to ignore a moving object). I speak with volume and clarity, and I always try to remain mindful of language (see below).

Each lecture should have an arc - a beginning, middle, and end - yet no lecture should stand isolated. In the beginning I remind the students how we arrived here (Last time we learned how to classify all finite objects X, and now we'll use this to understand...), and in the end I will foreshadow the future (But what about those X that are infinite?)