1 + 0 = 0

How dumb can one math teacher be? Leave it to me to assign two different enormous take-home tests and a variety of extra homework problems that were do in the final week before a two-week Christmas break of 107 students, guaranteeing my unavailability to my family or any other activity, because of the perpetual movement of my red didactical wand that is my grading pen over the holiday.

But why did I assign this to begin with? Was it because the students needed the practice? Did I want them to feel an academic pinch (something many of them are not used to). Yes, but primarily, it was because they had angered me by their continual reluctance and to study and recalcitrance to change their poor, lazy habits which were obviously not getting the results I demanded.

So this ultimate “self-flagellation and alienation from loved ones” was because students at the PreAP and AP level are not taking care of business. After grading the final batch of quizzes from the previous week with pathetic scores, when it became obvious to me that my single effort in assessing the students was greater than their collective efforts in producing them, I really began to examine the sum of my frustrations teaching this level of math for the past eight years. “Why is it so hard? Why do so many not get it? My level of explanation has been rewarded and applauded in my short career by students, parents, teachers, administrators, and university professors alike. It had to have gone deeper than just my inability to “teach” the concepts. But what? Here’s what I have finally discovered, although the reality of it is disheartening.

I teach math. I teach math skills. I teach theorems. But, I also teach math humanistically. The implications of math history and philosophy are part of my regular curriculum. I frequently discuss the “discovery” of math, and often structure my questions so that students have the opportunity to make the same “discovery” on their own. Some do. I balance my instruction between pure and applied mathematics: pure being a game involving the manipulation of symbols according to the rules of math, and applied being the study of the beautiful laws of nature. Both of these possess beauty and power. But seldomly in high school does anyone ask, “WHY?” It’s just there: another class they have to take. Some get to appreciate me, without truly appreciating the math at the level I am hoping for.

So why are some more successful than others? Although there is scientific evidence that there is no math “gene,” there is mathematical intuition. You can see it in the students. Some have more, some have less (irrespective of their work ethic.) It can be developed, but it takes toil and sweat and determination. Math is patterns. If a person has no insight into the visual, numerical, and symbolic sense of math, they are handicapped. Math is replete with arcane symbols. “Real” mathematicians are regarded as genius. People respect them for their insight (if not for their abilities to be fodder for an Academy Award winning script) and are cowed at their knowledge and ability to “pry” open the secrets of the natural world.

But, being a mathematician AND a math teacher, I am continually confronted with sequential problems that have nothing to do with math at all. All I can best hope for is to “sell” math to the students on the thin basis that it is “fun.” It is only in this manner that I get through the week.

After discussing my frustrations with colleagues from across the nation (via the AP Listserv), by problems are representative of a classic classroom crisis of understanding versus pedagogy.

The grim esotericism, in which even the best of us sometimes fall, the preponderance, in our current writing, of those dreary textbooks which bad teaching concepts have put in place of true synthesis, the curious modest, which, as soon as we are outside the study, seems to forbid us to expose the honest groping of our methods before a profane public . . .”—Marc Bloch, The Historian’s Craft

Invariably, a certain fraction of my instruction is devoted to proving theorems. This fraction is certain to be greater and deeper than that of a basic algebra class, and increases with the level of mathematics. One of the true objectives of teaching/showing proofs is to hope to convince students by unimpassioned reason, by psychology, by intuition, of the truth of a statement. Frequently in more elementary-level classes (those below Precal)—and it is experienced by all educators—that some confused student interrupts the proof in earnest with the cry (should they be so bold and eloquent): “I don’t see why you did what you did. I don’t even understand why what you say is so, is so. Additionally, I don’t comprehend how you came to do what you did!”

This is where we teachers of higher-level mathematics are confronted with a crisis of understanding. How should we/do we deal with it? Unfortunately, not very well (those of us with patience, that is.) Maybe we go over the sore spot again in a less technical, slightly different way, using terms and analogies, OR perhaps, anxious out of a mandated requirement to cover a certain amount of material, we overtly brush the student aside with a cursory explanation stated with the additional advice that understanding will surely come if only the student will go over the material later on his own, including reading, (that’s right, reading!!!) the math textbook.

OR (as dedicated teachers, such as I and my many colleagues throughout the world), we alter the course of the lecture/lesson completely, going for depth and the expense of breadth, and hoping for no fire drill and other frequent modifications to the schedule that severely limit our already sacred 3 to 4 hours with the students each week) until we have explored the mathematical difficulty until the students understand.

My first reaction to a crisis in understanding is to think: why are they confused right here at this spot? This is only a definition! This isn’t even the “hard” part, yet! This is the straw that breaks the camel’s back. We have been piling up sophistications upon theorems upon corollaries upon lemmas upon conjectures. Are they confused with the notation? The hypothesis? The conclusion? Do they know the difference between necessary and sufficient? At best, I have moved on, bludgeoning my class into, at least, accepting the proof without understanding it. But is has become clear that, having created an initial barrier; the theorem/concept/idea would not be cleared up by simply restating it in excruciatingly fine detail. Another approach was required.

It is a general consensus in the math educational community that the road to understanding is the way of the initial discovery: if one knows and sees the way some idea or concept was originally though out, then this is also a good way for presenting it in the classroom, but this is not entirely true. The initial discovery may have been obscure, buried within a specific context, or made unnecessarily difficult. Many of the presentations in a textbook are glossy, up-to-date versions that can, themselves, be obscured in their conciseness and generality. In such a case, a presentation that retraces the original problem and the historical, humanistic framework can offer more insight.

Ultimately, a proper education is the collaboration of efforts from teacher and students working, thinking, exploring together. Both must be willing to admit faults, make errors, and exchange information and ideas in a safe, non-judgmental environment. The true beauty of learning is the process itself.