This debate surfaces from time to time, and often the debatiness of it stems from particular interpretations of what "art" and "science" are. Or rather, what they are not, as people often bring these terms up in order to dismiss them.
"Science" is subject to a great deal of misinterpretation, with folks tending toward the notion that science is a matter of cold, hard settled facts. Science, this theory goes, tells us exactly how things always work. We see this conception of science every time someone talks about the science of reading as if SOR tells us that if we do exactly X with every student, every student will learn to read, every time.
The appeal of this view is understandable. It's human to desperately want a set of instructions that tells us exactly how the world works, that shows us that if we put in X, we always get Y. Some people turn to religion for this kind of certain set of rules, and some people turn to science.
But this mechanistic view of the universe was pushed out a hundred years ago, replaced by chaos theories and quantum physics and relativity. And science itself is better understood as a way to look for answers, to test and revise and work slowly toward the Truth without necessarily the expectation that you're ever going to get there. The universe is shifting and moving and fuzzy.
And so is a classroom. You can't put in X and always get Y, because the ground shifts every day. Not just different students in the room, but students who change from day to day, as does the teacher. If teaching can be reduced to an equation, it's an equation in which most of the variables change every single day.
Yet teachers do work scientifically. Every lesson involves hypothesizing (this should help the concept make sense), testing (let's see how they react), and revising (bloody hell--I need to take those sentences off the worksheet). Teaching is methodical, and teachers strive to reduce the number of variables in play (I'm going to teach this lesson as if my girlfriend didn't dump me last night, as if that kid in the third row doesn't annoy me, as if last period wasn't a disaster).
So if by "science" you mean that teaching is a set of settled, lab-proven inputs that always get you the desired outputs (so simple that it can be reduced to, say, a bunch of computer algorithms), the no, teaching is not a science. But if you mean that teaching is a process that involves an unbiased development, testing, and revision of various techniques and tactics to achieve a desired measurable outcome, then sure, teaching is a science (with a huge caveat around that "measurable outcome part), albeit a science occurring within a chaotic system that is so complex that currently only a human brain can navigate it.
When people bring up the idea of teaching as art, they may be signaling the idea that teaching is just about getting all touchy feely and following your muse, that a great teacher must follow their gut.
But the thing about art is that there is no art without skill. You can't paint a great painting if you don't know how to manipulate paint. You can't play a jazz solo if you don't know your way around your instrument. Skills do not guarantee great achievement--you can be flawless and dull. But you can't make art without some sort of skill set. In fact, the kind of art that you are able to envision is usually limited by your actual skill set.
Good teachers do not wander into their classroom every day and just wing it. They have to know the material. They have to know the skills of presenting the material, of gauging reaction, of spotting results. At the same time, they need to be flexible, adaptable, expressive, human. For me, teaching was always very much like performance--engaging, working with, responding to an audience. But even in the world of jazz, you don't just grab an instrument and do whatever. You have to be able to make notes, and while there is room to be creative and artful, there are also structures to follow. Excuse the language, but I always think of this story (relayed by Winton Marsalis, among others)- excuse the language:
Saxophonist Frank Foster called for a blues in B-flat during a street concert with other musicians when a young tenor player began to play "sounds that had no relationship to the harmonic progression or rhythmic setting," causing Foster to stop him:"What are you doing?"
"Just playing what I feel."
"Well, feel something in B-flat, motherfucker."