Off the top of my head:
-teachers shooting students will be a problem
-who is paying for these guns and training? Teachers make practically nothing and most school districts are already cutting basic stuff to get by
-a teacher's responsibility is to protect their students. That means either locking their classroom down or getting their students evacuated. You can't do either of these things running down the hallway trying be Mr cowboy and trying to shoot a shooter.
-A lot of schools already have armed security. Clearly the threat of other armed people do not stop school shootings.
-Along with this, afaik, most school shooters go into the shooting prepared to either die or kill themselves. So again, having more armed people does nothing to deter these people.
-School shootings tend to last a matter of minutes. 17 people were killed in 6 minutes in Florida. At best having armed teachers would have cut down the shooting time and saved a few more lives. It wouldn't have stopped the actual attack.
-teachers are now forced to be primary targets for shooters. They can't risk having a teacher shoot them and as a bonus, if they kill a teacher that had a gun, they get another gun to shoot even more people with
-A lot of people say that "gun free zones" are easy targets, but I'll remind people that places like capital hill and the white house are gun free zones and I can't remember any instance of innocent people getting gunned down there. It's amazing what happens when you actually have security.
-In order for the teacher's gun to be of any use at all in a school shooting, it will need to be easily accessible, which means it is easily accessible to students as well, which is potentially creating even more school shootings as the access to firearms now becomes a given.
-You're teaching kids that guns=power and authority, and this is supposed to help curb school shootings how exactly?
-There are plenty of examples of unarmed people stopping armed threats.
I can start actually looking up more/better arguments if this list isn't long enough, but my most important point is that arming teachers isn't an answer to the question "How do we stop school shootings?" It is an answer to the question "How do we kill a school shooters faster?"