It wasn't until some time later that the reason it didn't sit well with me was because it was fallacious reasoning, based on bad exegesis. Paul is not saying that disagreement is divisiveness. If someone comes up with a 'new' way of doing worship music (for example), and I don't find the new way to be scriptural, then I speak up and give a reason why it isn't scriptural, I'm not being divisive. The one holding to orthodox teaching is never divisive in the defense of orthodoxy. He's saying (look closely) that the one who teaches "contrary to the doctrine that you have been taught" is the divisive one.
In my current reading, one of the books I'm partway through is The Creedal Imperative by Carl Trueman. Trueman gives a good example of this on pages 67-68. In giving us a propositional rendering of belief in Romans 10:9-10, Paul states we should confess with our mouths that Jesus is Lord and believe in our hearts that God raised him from the dead. Trueman writes this:
Words and content are thus significant. What Paul does not say is: if you have a warm, incommunicable feeling in your heart and express this by incoherent sounds from your mouth, you will be saved. No. There is propositional content here-- publicly expressed in a manner comprehensible to others.
I've heard just this bit of bad reasoning used to accuse John MacArthur of divisiveness for his Strange Fire conference recently. But Dr. MacArthur is the one defending orthodoxy; the nuevo-spritism is the divisive party.
I'd like to think this idea transfers over to the secular realm. If you hold to a well-proven idea, and someone comes along and challenges it without any grounds other than, "I said so", you are not being divisive when you argue in favor of the established idea you held. But in today's culture, you'll be accused of all sorts of things for defending orthodoxy, whether religious or secular.
Funny how those seem to go together in a postmodern mindset.