How NOT to graph a shifted function; or, The nature of mathematical research

It happens to us all at some point: we're confronted with a problem to which we simply don't know the answer. It's a problem never posed to us before, one we've never seen...it may not even look much like anything in our prior experience.

It's happened to me. Many times.

For example, for the past year or so I (and, off and on, several undergraduate researchers and a couple of colleagues) have been struggling to find answer to a seemingly simple question: where on Earth does the mode of the independence polynomial of an arbitrary 2-regular caterpillar lie?

Okay, so maybe it's not that simple of a question...but it's one that's resisted analysis of every kind we've attempted for well over a year.

However, undaunted, we have tried several different means of cracking this muthah. We've tried geometric methods, combinatorial methods, algebraic methods...even analytical methods. We've tried it all, to no ava...well, to some avail: we've learned a lot about the structure of the objects we're studying, and though we don't yet know what method will work to solve the problem, we can tell you several methods that won't work. Hey, we've tried.

And that's what matters: the fact that we've tried. In the end, it's okay to not know what the answer to a particular question is. After all, none of us are born with inherent knowledge of algebra and calculus and combinatorics: we're going to be asked questions the answers to which we simply don't know.

Put another way, ignorance is inevitable; what matters most is how we confront that ignorance. Inaction gets us nowhere. Action of any organized kind is preferable, and more preferable still is action of a sort our experience suggests will give us a means of responding to the problem we're posed. This kind of confrontation with ignorance is called learning...or even research.

Yes, research: it doesn't cheapen that lofty term at all to use it to refer to the simple actions we undertake when we, for example, try to graph a simple function we're unfamiliar with.

Allow me to demonstrate.

When confronted with a truly unfamiliar function, here's what not to do:


You can't be expected to be familiar with every function ever invented...there are too many of them! But don't just sit there! Don't let ignorance get you down! If you're not sure of what to do, maybe try something that's worked in the past, like...


Ah...now we've got some traction...a few more values...


...and we're starting to see results...now let's plot some points...


Huzzah!


Ignorance dispelled! Or at least held at bay for a bit. Congratulations: you've now learned something new. Put another way, you've completed a miniature research project. Seriously, you've just done research, applying known methods to approach an unknown problem. That's how you do it.

At the risk of sounding repetitive, let me exhort you once more: please, don't just sit there. It won't simply "come to you" if you're not doing anything at all, but it might if you try something out.

P.S.: photo credits go to my former student, and awesome stats major, Karl. Thanks, Karl!