In my experience, most people who go through math classes at a University, not to mention classes in High School, are mostly "blindly" following a template of how to do induction. They don't really understand the principle behind it, why it's a valid proof technique, etc. Put another way, if I used induction incorrectly, I could probably still trick them into thinking it's a valid proof because it looks like induction.

The same goes for a lot of things in Mathematics. The same also goes for recursion - most people don't really understand recursion (unless it's used at a "simple" level, like in Tree Recursion).