By correct I mean from a biomechanical perspective to minimize the risk of injury. I have seen some advice saying that strumming should be controlled by the big muscles such as the shoulder. My teacher says it is simply a rotation of the wrist i.e. A small muscle movement. What's the safest technique for avoiding RSI, carpal tunnel and similar?
Actually both parts are correct - you have different kinds of strum, and different sizes of movement.
Often, on an un-mic'ed acoustic playing to a large room, you would use a large movement from the elbow, with a little shoulder movement added in, to ensure you get significant power to the strings - translating to volume.
A strum with partial string coverage (think a basic D or C shape where you are only playing 4 or 5 strings) it can give you much more control to keep the arm relatively still and rotate at the wrist.
Safest is probably to make sure you vary things, sometimes use the wrist, sometimes the elbow, sometimes a bit of the shoulder - this helps spread the load on your joints, muscles etc.
Of course the extreme version is the windmill (but be aware it can hurt a lot if you get it wrong :-)
while playing accompaniment for traditional Irish music i found that strumming with the elbow can cause a lot of problems, me and a few local fellow guitar players have had issues with "tennis elbow" like injuries causing the muscles in my upper forearm to become knotted and extremely painful, one friend even developed tendinitis and had to stop playing until he built his muscles up again.
this can be avoided by performing the correct stretches before picking up a guitar. also with that style of music the guitar turns into quite a percussive instrument with similarities to the bodhrán (an Irish percussion instrument). these similarities mean that players often play a lot harder than is needed. great influences in the world of Irish music include Steve Cooney (an Australian man), another great musician who really plays hard is Donal Lunny (bouzouki player) and a great overall player is a guy called Arty McGlynn.
i was always taught to use my wrist and forearm instead of my elbow to strum the guitar and was always told to be careful not to play too hard while at the same time, not being afraid to "hit" the thing to get a bit of sound out of it.
I use a combination of a rotation of the forearm and movement of the forearm (up and down) with just a little of the upper arm. You probably should avoid bending the wrist too much. Overall you want to be relaxed, loose and natural feeling - this last part can be misleading because whatever you get in the habit of doing will start to feel natural to you.
Hold your arm up, bend your wrist forward maybe 20 degrees, certainly not as much as 45. This is about the amount of play you need to do the work mainly with the rotation of the forearm and a little up and down.