Today’s post is a little bit off the main topic of this series of posts… but I wanted to give some pedagogical thoughts on yesterday’s post concerning the following proof by induction.

Theorem: If is a positive integer, then is a multiple of 4.

Proof. By induction on .

: , which is clearly a multiple of 4.

: Assume that is a multiple of 4, so that , where is an integer. We can also write this as .

. We wish to show that is equal to for some (different) integer . To do this, notice that

by the induction hypothesis

.

So if we let , then , where is an integer because is also an integer.

My primary observation is that even very strong math students tend to have a weak spot when it comes to simplifying exponential expressions (as opposed to polynomial expressions). For example, I find that even very good math students can struggle through the logic of this sequence of equalities:

.

The first step is using the main stumbling block. Students who are completely comfortable with simplifying as can be perplexed by simplifying as . I attribute this to lack of practice with this kind of simplification in lower grade levels.

Here’s another algoebraic stumbling block that I’ve often seen: at the beginning of the case, some students will make the following mistake:

.

Because these students end with a multiple of 4, they fail to notice that the second equality is incorrect since

.

Again, I attribute this to lack of practice with simplifying exponential expressions in lower grade levels… as well as being a little bit over-excited upon seeing and wishing to use the induction hypothesis as soon as possible.