Saturday, February 21, 2009

The seduction of induction

Here's a question I hope some of my 280 folks will be willing to answer in the comments section (other onlookers are encouraged to reply as well): what's so hard about inductive proofs?

I've noticed that every time I've taught a proofs class, three times here now, and once at the University of Illinois, the homework set the students found most challenging was the one dealing with induction. Fellow teachers, have you seen this phenomenon at work as well?

Students simply seem to find induction tricky, intellectually irksome.

Here's one reason this proof method might prove (no pun intended...okay, kinda intended) more difficult for most than others: the method relies on the proof of a conditional and not an absolute statement, and aside from the base case, therefore, one is never proving a statement directly. The very reason the proof method succeeds is the "if it's true for n then it's true for n+1" clause. Understanding the sufficiency of this clause (together with the base case) is itself a major leap, and therefore most beginning provers are uncomfortable with the very validity of an inductive proof, and therefore are less motivated to complete all of its components.

Maybe that's it.

Ideas, anyone? Throw me a frickin' bone here.

As it is, though, this semester's crop did pretty solidly on their induction homework. Of the twenty-five or so homework sets I graded over five hours this morning, although there were only two As, everyone who managed to turn in a completed homework set did halfway decently, and the class average couldn't have been much below 80%. I'm pleased. Though this term's students are a bit shyer in class than their Fall 2007 counterparts, their dogged pursuit of homework perfection is easily comparable to that strong semester's work ethic. Kudos!

Moreover, their LaTeX is coming along beautifully. A handful have really gotten the hang of it (Trixie, Tish, and Siegfried, y'all rock!) already and are executing nearly flawless TeX masterpieces, and several others (Omar's comin' atcha!) are using it regularly and their proficiency isn't far behind. More importantly, three or four people have already commented on how having to type the math up makes them think much more carefully about what they're writing, which is ultimately the primary goal of the class.

Keep it up!

1 comment:

prototrix7 said...

I think two things make induction hard. The first being the Inductive Hypothesis. Its so simple that sometimes we over-complicate it, which messes up the entire proof. I didn't realize until just today that it isn't a direct proof, but more of a transition if you will. Its hard to explain, but I think you are right, because the inductive step isn't really a direct proof, it messes people up.