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.
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.
My first job as a math teacher was as an undergraduate
mathematics and science tutor for the athletics department of Missouri
State University. I was later hired for a similar position
in the MSU mathematics department study lab, and subsequently I taught
three semesters of college algebra as a primary instructor there.
While attending UC Riverside as a graduate student, I worked as a
teaching assistant for a wide variety of courses, ranging from
entry-level college algebra to upper level undergraduate abstract
algebra and number theory. As a primary instructor at UC Riverside and the University of Ottawa, I have taught
college algebra, differential and integral calculus, and linear
algebra, as well as a graduate-level abstract algebra seminar.
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.
Working as a tutor and teaching to just one or a
few students affords us the ability to listen carefully to everthing
the students have to say. Students usually feel more uninhibited
to voice their suspicions and guesses with a smaller number of
people. At this level it is important to create an environment in
which the student feels free to make guesses and mistakes and to see
the consequences of those mistakes. To foster this environment it
is helpful to sometimes describe those mistakes which we may
feel tempted to make and to allow the student to pursue their
It is also helpful to approach the presentation and eventual solution
of a problem as a non-linear story. Too often the students will
open a textbook and find the solution to a problem provided in one fell
swoop; or even worse, find a solution with no attached problem, or a
theorem with no motivation. So we can begin by describing a
problem which is easy to understand, and by phrasing it as a question
that seems natural to
ask: where are all of the solutions? How many of them are
there? Are objects X and Y
secretly the same thing? From
here we can describe some ideas that don't work and describe the
subsequent journey - pitfalls and dead ends sometimes included - that
leads to a solution.
With a larger audience and a course which begins
and ends in a single term, this approach must be compromised.
There is never enough time to explore every blind alley and curious
guess that occurs to us. It is still the responsibility of the
instructor to ask questions and to tell an engaging story, but with the
constraints of time they must choose with care the details to be
omitted or simplified. Careful preparation and organization are
crucial at this level. Questions from the students should be
encouraged and entertained, but restricted. This is the tension
between the slow and patient processing of an idea and the directed
pursuit of the final goal. Too much time spent pondering detailed
questions and we run the risk of failing to complete the arc of an
idea. This has the dangerous consequence of misleading students
into thinking that mathematics is about continually sinking into a
quagmire of disconnected, difficult computations (see
the three-point lecture below). I have also learned,
after teaching several large courses of over 160 students in
California and in Ottawa, a very practical lesson. My ability to
inspire students by providing a bird's eye point of view of the
mathematics is only as good my game on the ground, i.e., my
effectiveness in efficiently managing the course. This includes
all of the non-mathematical stuff like effectively handling a large and
noisy class, writing clearly on the board, responding promptly and
helpfully to large numbers of student emails, managing a presence in my
office to help students, grading material with constructive criticism,
and maintaining a frequently updated website for the course.
My responsibilities as a teaching assistant were
found between these two extremes. The nature of the job varied
depending on the primary instructor, but usually involved
leading recitation sessions in which (ideally) the students came to
class with questions about homework or material recently covered by
their primary instructor. I then spent some time elaborating
on the finer points of the lecture by, for example, providing examples
or counter-examples to a given theorem or closely examining why certain
hypotheses are necessary. It wass important for me
to be able to gauge the progress of the students, and I often did this
by asking a series of questions, beginning with the easier general
themes (who can tell me what an integral means algebraically and geometrically?) and moving on to more difficult ones (when can two distinct functions have the same integral, and what does this mean?).
For some classes I gave 5-minute quizzes at the end of every discussion
session for a low point value. These frequent quizzes took little
time to grade but revealed much about the students' comprehension.
METHOD AND STRATEGY
ESTABLISHING PRESENCE IN THE CLASSROOM
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).
Perhaps the simplest structure to employ in a
lecture is given by the standard three-point lecture: tell them what
you're going to tell them, tell them, and tell them what you told
them. It is obvious and effective, but too often neglected.
I begin the hour by sketching the big picture with a few words (Today
we will try to find all of the roots of an arbitrary quadratic
function, and see some examples for which those roots aren't even real
numbers...). I continue with the lecture and conclude by reminding the students what we've done (So today we saw that all of the roots of a quadratic function are of the form...).
Another effective broad format is one mentioned above: mathematics as a
story with an initial conflict and a final resolution (Conflict: some stubborn quadratic expressions resist factorization! Resolution: so we can complete the square...).
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?)
MINDFULNESS OF LANGUAGE
At all levels of instruction I try always to be
conscious of and consistent with the language I use, and to impress
this importance upon my students. I want to make sure that college algebra students know the difference between a term and a factor,
and that they correctly use these words. When teaching
undergraduate abstract algebra I encourage the students to carefully
state their questions out loud - even to themselves, and to obey the
rules of grammar when they do so. Sentences must be
complete. Students must strive to make these question-sentences grammatically
correct. If the students claim to understand a proof, I ask them
to speak the proof to me. This fine-tuning of language must be
emphasized, though, only after a more general motivational discussion
and after students are given the opportunity to speak spontaneously and
freely about a new idea.
TEACHING AND LEARNING
I believe neither teaching nor learning can exist
without the other. I encourage students to help other students,
and this forces them to structure a small lecture in their minds.
The proper organization of ideas required of and resulting from this
process is a large part of understanding mathematics.
Mathematics is about questions (and about
answering those questions!), and students should be reminded of
this. Comprehension evolves as follows: first nothing makes
sense, and we don't even know why
we are confused. Then our understanding develops and deepens side
by side with our ability to ask meaningful questions. Often the
answer to a question is not so far away if we can just figure out how
to ask the question.