Is it possible to build a paper airplane that will enter a fully developed spin?
Besides the location of the center of gravity, two things determine how an aircraft spins, and both are not well shaped for spinning in paper airplanes:


The wings (obviously): They should ideally be straight with an aspect ratio between 5 and 10. Swept wings will also spin, but straight wings make spinning and recovery easier.
The forward fuselage. Here a rounded shape which is higher than wide is ideal and will promote flat spins. The two-dimensional shape of paper airplanes will not produce the vortices which drive the yaw rate in spins.


Also, for spinning it needs a stalled wing, some asymmetry and some time until the spin fully develops. To create a strong pitch-up moment, masses at the tip and the end of the fuselage are also needed. A pure paper airplane will have too much yaw damping and too little inertial pitch up to spin.

Size should be less of an issue because separated flow changes less with Reynolds Number, and spin tunnels regularly use scales of 1:20 or 1:30 with good results. But I wonder whether the low wing loading of a paper airplane will prevent spins - the models used in spin tunnels are carefully sized not only for geometric agreement, but also for their masses and mass distribution.

Maybe it will be better to show your students videos from spin tunnel tests.


Spin tunnel footage of fighter aircraft
Spin tunnel footage of GA aircraft
Description of a spin tunnel