Cirrus SR20 Flight Instructor and student reported near miss with another aircraft while on approach at a non-towered airport. Flight Instructor took evasive action to avoid a collision.

Narrative

I was the CFI flying with a student pilot practicing traffic pattern work on Runway XX at ZZZ in Aircraft X. In a base to final turn while my student was flying and making a radio call; I noticed Aircraft Y (who had made a regular radio call 2 minutes earlier reporting that they joined the downwind behind us) was now at the same altitude as us while they were in the downwind. Observing that this was unusual because we were only 500 feet AGL I began visually looking for them now that we were on about a 1 mile final descending with wings level. We made a radio call for 'short final Runway XX' into ZZZ and I heard a followed radio call from Aircraft Y stating 'uhh how short on final are you?' No other calls were made by Aircraft Y that I can recall. As the pilot made that call he abruptly cut in front of us without seeing us and continued his approach very low to the ground with a very steep bank and no regard for traffic spacing with us. I; as the CFI; immediately took controls and performed a go around to avoid a collision. I then came in to land after he cleared the runway. No accidents or injuries. But a very close call with a pilot that was not paying attention and flew very recklessly without a regard for other planes in the air. I have flight radar recordings I'd be happy to share if necessary.