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First Order Probability

I’m a strong believer in motivation when it comes to instruction. I do not mean here the importance of motivation as a general desire to learn; that is indeed critically important, and the techniques for instilling a desire to learn are discussed endlessly in education. Here I mean something much smaller: the importance of explaining why a particular technique is useful before drilling a student in the technique.

Too much math instruction begins with considering an Abelian group A and its many properties, or the volume swept out by a cone rotating about its vertex, or some equally arbitrary exercise. A student keen on math might perform the exercise for the fun of it, but most will do so simply for fear of a bad grade, thinking all the while, “So what?” Keep that up long enough, and kids become convinced that math isn’t worth the effort. They may recognize that some egghead somewhere uses math to make better mousetraps, but will come to conclude that that kind of math must lie immeasurably far beyond what they’re asked to do, that useful math is unattainable to all but a select few. That’s a toxic attitude. Besides, homework is a lot less enjoyable when it feels like mere busywork.

Factorial math is important to an understanding of probability. For a positive integer N, the factorial of N (written as “N!”) equals the product of N times N-1 times N-2 times…all the way down to 1. So 6! equals 6 x 5 x 4 x 3 x 2 x 1, which equals 720. Factorials are important to probability because they often count the number of possible cases of a given event. Find the total number of cases, and find the number of cases satisfying a given condition—also often a factorial—and you can calculate the odds of the condition being true.

Traditionally, the first lessons in probability are preceded by a discussion of factorials. Also traditionally, but very foolishly, lessons and exercises in factorials are rarely offered with the motivation for learning them at all: factorials allow you to count certain kinds of things easily, and that lets you find probabilities. Anyone from high-powered stockbroker to street punk shooting craps in a back alley can appreciate the value of understanding probability. Only math geeks canappreciate the value of understanding factorials as an exercise in itself. It makes factorials seem pointless. Ultimately, of course, the week passes and the class reaches probability, and the students begin using factorial math for concrete purposes. But by then, the damage is done.

For this reason, I’m breaking with tradition. I’m opening my lectures on factorials with the first, basic problems in probability. The kids won’t be able to solve those problems right away, but they will be able to see somewhere on the conceptual horizon that we’re engaged in this bizarre form of counting for a reason. We’ll end up skipping back and forth between concepts in probability and concepts in arithmetic, but I think they can handle it, and their homework will make a lot more sense.

Wish me luck. Sometimes traditions exist for good reasons, and I may be about to find out why this one exists.

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