J Prasad Senesi jsenesi@uottawa.ca

Curriculum Vitae Academics Teaching Personal Home


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.  


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 ends.  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.


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...).     
ach 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?)


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.


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.