Fifty years ago, I took lessons at an uncontrolled airport and this happened to me, fortunately my instructor took over and as cool as a cucumber corrected the situation. His detailing of what he was doing fell on deaf ears.
This is called a "tail slide" or "whip stall". It is an aerobatic maneuver when performed on purpose, but can be dangerous if done by accident because it causes damaging forces to control surfaces. Some aircraft if the CG is in the wrong location, will tend to tail slide before they fully stall (my C-177 is prone to this). – Ron Beyer
Aerobatically, to enter a tail slide you would enter from level flight at full speed (maneuvering speed) and pull straight up (1/4 loop) and wait for your airspeed to bleed off to zero. The aircraft will then slide backwards and eventually the nose will drop. You stay in the dive until you have enough airspeed to recover.
Doing this on accident is a little more complicated, and is surprising that the J-3 could even do it. What happens is that you pull up while adding power just as you would be practicing a power-on stall, only the tail stalls before the wings do. The tail drops and the aircraft slides backwards. This is especially dangerous at low altitude because the only way to recover from a full tail-slide is to allow the nose to drop and gain airspeed in the dive.
The recovery action for an incipient tail-slide is similar to recovering from a stall. Nose down to the horizon and full power. Newer training requirements don't let the student go into a full stall, it teaches to recover the aircraft as soon as the aircraft begins to buffet, or at stall speed + 10kts. I'm not sure I agree with the new requirements but that is a topic for another question.
Tail Slides are aerobatic maneuvers with two recovery techniques; stick forward, and stick back. After the aircraft has been flown vertically until airspeed reaches zero and power is reduced, the aircraft slides backwards until the elevator ( now acting as a canard) pitches the aircraft quickly, either backward ( forward stick), or forward (back stick). By holding this elevator input, the pilot causes the aircraft to "whip" into the down line, setting up for another maneuver, or simply recovering to level flight. Only aircraft certificated for aerobatic flight should be used to demonstrate the Tail Slide or "Whip Stall", as the loads imposed on the elevator and tail assembly of standard category aircraft could exceed design limitations, resulting in partial or catastrophic structural failure.