In heavy, deep slush, just about every student I've had tells me something like, "My skis get stuck in this stuff and I just can't turn them". As the previous posters have said, this is because the only turns most people showing up at ski school for lessons know how to make are skidded turns on a packed, smooth surface. So, the most important task is to teach the person how to make carved turns.
Unfortunately, this isn't as easy to do in slush as it is on a packed surface because of the re-emergence of fear. If a carving exercise doesn't work in a lesson on a packed surface, the student can always momentarily revert back to skidding to get themselves under control. Unfortunately, in heavy, deep slush, this familiar fall-back option (ie, skidding) simply doesn't exist, so even on very gradual pitches, the student feels like a beginner again, and fears of flying down the mountain like a run-away garbage truck re-surface in spite of all the speed-controlling friction from the slush.
In order to successfully teach any new skill, you first have to give the student something which makes them feel safe and in control. While it may sound counterintuitive, IMHO, one of the best ways to do this is by taking the student back to the good ol' PSIA-approved, narrow angle gliding wedge on green or easy blue terrain in these slushy conditions.
Just a few degrees of wedge in heavy wet snow will slow one waaaay down. The student thinks to himself, "Hell, this is easy – it’s just a beginner move. If I get going too fast, I'll just open up the wedge even more and come to a complete stop". Thus, the fear of lack of control and/or speed is instantly eliminated. If you sense that there still is some residual fear, have the student do some wedge change-ups straight down the fall line to convince them that this beginner-like technique really will keep them safe.
Now, you are at a psychlogical point where you can actually teach them how to do a proper, narrow-angle gliding wedge turn through the slush. I usually demonstrate that with the proper technique (eg, both knees move in the same direction), I can make a 180 degree turn in about a ski length or two, even at sub-walking speeds in slush.
This usually *really* gets their attention because this is exactly what they couldn't do a few minutes ago when they felt that their skis were "stuck". Now the stage is set for them to learn - they feel safe AND they see something very desirable, very "do-able" and non-tiring that they want to learn.
The rest of the lesson is a piece of cake. When they can all do very short radius, extremely slow gliding wedge turns (in the slush), I'll ramp up the speed a bit (cuz now they have the confidence to do so), let the skis spontaneously match, and away they go.
Tom / PM
PS - Note that just as in modern carving technique on packed snow, the importance of the big unweighting moves mentioned in previous posts is vastly decreased. They might be needed as the student progresses to really funky snow and really steep slopes, but for the usual spring slush on anything up to a black groomer, one can usually ski it extremely smoothly and non-athletically.
PPS - While I'm a big fan of fats for slush, the progression I described above works very well with the usual, normal width rental equipment. OTOH, as the new slush skier advances and wants to go faster, especially in cut-up conditions, fat skis definitely makes things a lot easier for them.