Two tricks for introducing pedal (overlapping pedal):
1) Teach a two-octave crossed-hands arpeggio in C Major. LH C E G RH C E G LH C
(No pedal yet). That was ascending. Then teach the same notes descending. Then put them both together, going up and down. Then have them play the whole thing with the pedal down.
It forces them to get the right notes, because with the pedal down, any wrong note in an arpeggio is going to sound really bad, and this will get them to start listening.
2) Teach a five-chord progression using common tone in C Major: (RH) C chord in root position; F chord in second inversion; C chord in root position; G7 chord in first inversion; C chord in root position. Then have the student play the first chord and hold it down with the fingers, and while holding the chord down, have them depress the damper pedal. Now, without lifting the pedal, have the student play the second chord and hold those fingers down. Both chords are now mixed together, which sounds "muddy". Now have the student, with the second chord's fingers still down, lift and depress the damper pedal, thus "clearing" the pedal of the first chord's tones. I have my students do this with each chord of the five-chord progression above. (I call that progression the "Grand Progression", which I got from Piano Adventures. It makes it easy for them to remember.)
Both of these tricks get the student to listen to what the pedal can do between notes or chords, rather than what most people do with the pedal when they are just messing around on the piano, which is: they think something like "Oooh! The pedal makes the tones last a long time and sound really cool!" but then they don't take it to the next step and really address all the cool things that the pedal can be used for.
I have gotten into a space where I only (almost) teach overlapping pedal. If they can get overlapping pedal, which is coordinating chord changes or note changes with the pedal, then they can do the easier kind of pedalling in which you are holding a chord down and you depress the damper pedal just for a moment, to help the sound out. That kind is much easier to do, because it doesn't have to have precise split-second timing.