In a few weeks I will complete my forty-first year teaching high school mathematics. Why have I kept at it for all these years? It certainly isn’t for the money or the glory. I continue to teach for several important reasons: the students remain engaging, my colleagues are interesting, and the material is exciting and challenging, even after four decades. Mathematics, not everyone’s cup of tea, is especially appealing to me because there is still so much to learn. Nearly every day brings a surprise; a little piece of information about algebra or trig or the life of a mathematician that I didn’t know, or a student’s unexpected method of solving a problem or “off-the-wall” question. These surprises are enormously uplifting; they energize me. The expectation of these “wow” moments sustains me through the predictable “been there, done that” parts of teaching; this is why I keep coming back to the classroom.
Just yesterday, I came across a fact that nearly knocked me off my chair. I was writing a test that included problems on the Binomial Theorem. This theorem, as I’m sure you know, provides a formula for expanding expressions such as (3x + 2y)5. It turns out — and I am amazed that I didn’t already know this — that the sum of all the coefficients in the expansion is 55, 5 being the sum of the coefficients of x and y, 3 and 2. This is a simple, yet profound relationship. I was so astonished that I was moved to try to prove this relationship (successfully) and to go around my office showing this result to my colleagues, none of whom had seen it before. I am also planning to base a cooperative learning lesson for some of my classes on this result. Another fact that continues to astound me, and my students, every time I mention it (and I do talk about it in all my classes every year) is that .99999… = .9 = 1, exactly. The discussions we have about why .9 = 1 is true tend toward the rollicking. These conversations also give my students important insight into the nature of the set of real numbers.
It is intriguing to me that mathematics, a subject thought by so many to be formulaic and dry, has the capacity to amaze me and my students on a regular basis. The challenge for me is to keep from becoming so focused on the usual that I overlook the gaps between the theorems and definitions, those areas that contain relationships one might never imagine. I must always allow students the space to think in creative and alternative ways. Mathematical wonderments may not occur every day but they do happen often enough to ensure that teaching is anything but humdrum. Unexpected conclusions attract the attention of many students, even those for whom math is not their favorite subject. It helps, too, that many of these surprises come from the students themselves, thus establishing a closer connection between them and the subject. The basic curriculum may remain mostly the same year to year, but the subject topics are merely the basic ingredients that students and teachers work with every day as they generate the real work of education. Teaching mathematics continues to be exciting and fun for me as long as I remember to encourage as much student input in our daily work as possible. How can teaching be anything but exhilarating when groups of intelligent individuals get together each day to discuss complicated, interesting, and challenging material?
This issue of Independent Teacher is loaded with interesting and useful articles. The topics range from Michael Thompson’s insights into the psychology of teenage boys, to Trey Colvin’s use of “wikis” in his English class, to Kevin Ruth’s discussion of how and why his school’s language department revised its curriculum, to Tracy Cole’s thoughts on how classroom desk arrangement affects her teaching, plus much more. We hope that the ideas included in these articles will make a real difference in your teaching. Please write to us at [email protected] with your comments on any article or suggestions.
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