Astudent and a physics teacher hold opposite
ends of a horizontal spring stretched from west
to east along a tabletop. identify the directions
in which the student should vibrate the end of
the spring to produce transverse periodic waves.

A transverse wave is a wave in which the vibrations of the wave are perpendicular to the direction of energy transfer. The energy transfer is from one end to the other, so to ellicit this sort of wave on a tabletop they will need to vibrate the end of the spring left and right.

To produce transverse waves, the student should wiggle his end
of the spring either left-right, or up-down, or upper-left/lower right,
or lower-left/upper right, etc. ... back and forth along any line that's
perpendicular to the spring itself. 

I think he can create a transverse wave and a longitudinal wave
both at the same time.  f his wiggles have any component at all
that's parallel to the spring, then he'll put a longitudinal wave into
the spring.  If he only does push-pull-push-pull, then the wave
will be purely longitudinal, with no transverse wave at all.