One of my proudest moments as a teacher occurred in the fall of 2002. I was teaching “Geometry for Educators” with only two students enrolled in the course. On this particular night, we were reviewing for an exam and so I posed a problem, wrote it on the board and asked the students how we might go about proving it. I was astounded. Not only did each of the students come up with a correct proof, their proofs were fundamentally different from each other’s as well as the proof that I had in mind. At that moment I felt like I had really accomplished something; the students weren’t just parroting back information that I’d given to them, they were using the information and thinking for themselves.
The most important thing to convey, the one thing that educators must convey is understanding. This is particularly true of teaching mathematics. Poorly understood facts and formulas are fleeting and will be forgotten as soon as they are not used on a regular basis. Things that are understood are permanent. A formula or process that is understood can be rediscovered or reinvented even if it is forgotten.
Because of this, the most important question for us to answer is "why?” Specifically; why are things done the way that they are done and why do certain things happen the way that they happen? We must answer this question even though it may not be asked.
If we answer the question "Why?" effectively enough, the answer to the question "How?" should be self-evident by the time we present it.
Unfortunately, by the time many students reach college they have eschewed real understanding for lists of steps and memorized facts. "Just give me the formula I need to get the answer," they assert. This clearly helped them earn reasonably good grades in high school, but it is insufficient in a reasonably rigorous college course. Thus, the challenge becomes not only to help the students to understand, but to convince them that understanding is the superior way of doing things. If they understand why something happens they can always recreate it or figure it out if necessary. To put it very simply, if you understand what multiplication means you can always add seven to itself six times even if you've forgotten that 6 x 7 = 42.
Two essential ingredients for conveying this understanding are to present things in a way that the students can grasp and in a way that they will remember.
One of the events that led me to this conclusion was the realization of just how differently I think and reason when compared to the typical student in one of my classes. For example, suppose you flip a coin ten times in succession, and you want to find out how many of the possible outcomes will contain exactly three heads (H). My reasoning has always been as follows. We have a set of ten blanks that we can consider being numbered 1 to 10. If we choose a subset of this set with three elements in it and place an H in each of those spaces, then when we fill the remaining blanks with T's, what we have is precisely the sort of outcome that we are trying to count. Thus there must be C(10,3) = 10!/3!7! = 120 such outcomes. We don’t need to concern ourselves with the order in which we chose the spaces because the H's are indistinguishable.
Although this has always seemed intuitively obvious to me, it can be confusing to students. Before this point they are taught that C(n,k) is used to count things in which "order does not matter" but we care about the order of the H's and T's. The difference between the importance of the order of the H's and T's and the order of the spaces that we put the H's and T's into is subtle enough that students do not grasp it easily. After 12 years of thinking about this problem in this way, it occurred to me to try to explain it in the following manner. If we flip a coin ten times in succession, the number of possible outcomes that contain exactly three heads is the same as the number of ways we can rearrange the word “HHHTTTTTTT”. It seemed as though most of the students understood this explanation right away. This and other instances like it have convinced me of the importance of understanding how students reason so that things can be discussed in these terms.
The other challenge is to present things in a way that the students will remember.
There are several broad principles that are important here. Intuitive ideas are more powerful than rote processes. English sentences are easier to remember than formulas or equations. Plain speaking clarifies where fancy terminology obfuscates. Doing is as important as listening.
Students are most likely to remember things when they are actively engaged in learning. In a classroom setting, students should be putting concepts to work immediately. Because of this, most of my “lectures” are run like workshops. Short introductions of broad principles are followed by discussions where the class as a whole thinks through an example or two followed by the students having an opportunity to try things out for themselves. This kind of approach can be most effective in a technology rich environment, where computer work can be interspersed with other types of activities or where students spend a good percentage of their time in a computer lab type of environment. The use of technology in teaching mathematics must be approached with caution however. It is important that such activities are carefully thought out and developmental in nature, leading students to think more deeply about certain topics or to conjecture and discover new ideas. Poorly constructed activities can undermine true understanding and cause the tool to supplant understanding rather than supporting it.
Thus, I would say that the crux of effective teaching of mathematics lies in presenting material in a way that students will understand and remember. Students who understand the concepts that lie beneath the techniques and algorithms will not only be able to use the techniques, they will be able to refine them and extrapolate new techniques from the old ones. Teachers who require only memorization and regurgitation do everyone a disservice.