There's a difference between having a police officer, having an armed guard, and having armed staff. Having cops stationed at schools is not a bad idea, assuming it can be afforded. As far as I know, there aren't many districts in this country that can take a cop or two off the streets every day for every school there. However, lots of places use off-duty or retired police, with which I have no problem. Almost every time we hear of an armed shooter being stopped, it's by a responding police officer, so sure.
I'm not overjoyed about the idea of armed guards - that is, non-cops - because, well, they're not cops. Security job positions are typically very low-paid jobs with very little training and - even considering that the guard would have classes in how to use a firearm safely - I would not want to entrust the next generation in the hands of people with no experience in crisis management. I recognize that many other disagree with me on this point, which is fine.
Arming teachers is the worst idea. Do we really think it's a good idea to introduce thousands more guns into the schoolyards, in the hands of teachers, a job which is already overworked, underpaid, and highly stressful? How long would it be before aggressive students wrest them from the hands of those teachers? Before a few teachers themselves lose their own shit and open fire? Or before students develop a full-on bunker mentality after having to report to an armed camp every day to learn their times tables?
Adam Lanza, like most spree shooters, was not a criminal - he was a college student who went crazy. The best way to have prevented his shooting would have been if his dumbshit mother wouldn't have kept a roomful of guns in a house where a mental patient lived; same goes for Kip Kinkel in Oregon in 1998 and Jeffrey Weise on the Red Lake Indian Reservation. The Virginia Tech shooting would have been best prevented if the mental patient hadn't been allowed to buy all of those guns; similarly for Jared Laughner and James Holmes. The Columbine shooting wouldn't have happened if their straw purchasers hadn't been able to get all of their guns at the gun show and that incredibly seedy gun merchant (also, if their parents had been awake). The Fort Hood shooter bought thousand-dollar handguns with 30-round clips with no red flags raised.
These all could have been prevented, but in a lot of different ways. Arming teachers isn't one of them.