The writing of mathematics is very efficient. Perhaps the teaching of mathematics can afford a little more redundancy. Here is a key example:
Mathematical induction-the champion of efficiency.
The beautiful idea:
Induction is often used (and usually taught) to prove formulas that hold for all non-negative integers. For example, the sum of the first n non-negative powers of 2, is equal to one less than the next power of 2. If n=4 it looks like this:
Our next instructions are to add 16 to both sides:
and we see that the pattern continues!
Let’s do it one more time. Start with our newly discovered equation:
add 32 to both sides, and voila:
It’s pretty neat, and also really clear from this kind of discussion…unless of course…
Unless of course, our discussion only uses one equation, which corresponds to n=1, the so-called base-case, and you write:
The base case is just too small to see what’s going on or to notice any patterns. Your students may even feel stupid writing this down, as if it’s a secret handshake and they really don’t know why it’s necessary.
You guessed it; this is the way it’s almost always taught. But wait!–it gets worse; the instructions for seeing how the “pattern” continues are too cryptic for many beginners to follow. They are hidden in the formal details of the so-called inductive step. So it’s a pedagogical lose-lose.
It’s done this way because the base case is the smallest example that will make the proof work both completely and rigorously. After all, a full proof would have to include the case where n=1, and a formal proof works just fine with that as our main example. So let’s be efficient and … actually let’s not.
Focus on the beautiful idea
The beautiful idea behind the principle of mathematical induction should be given priority. That’s the first part of this post, and the first thing to be communicated regarding mathematical induction. Formal proof-writing etiquette can come a little later at the university level, or not at all in elementary school.
(Yes—I’m suggesting that elementary teachers should introduce math induction!)
So let’s use n=4 and n=5 for our examples, then look at the inductive step, keeping in mind how they relate to these examples, and then rewrite the proof the next day.
In other words, let’s give up some of the efficiency requirement.