Bullet's right... 8.5 doesn't have smooth mesh preview I'm pretty sure.
Create a nurbs sphere, and hit 1,2,3, and 1 again. That's what smooth mesh preview does for polygons in newer versions of maya. I think it's probably not a good idea to have 3 be the shortcut for mesh > smooth, since this is an actual operation on the mesh, instead of just a preview that can easily be reverted.

Not sure how to replicate what the tutorial is doing, you could just hit undo after every smooth, but be careful not to forget and also delete the history. You can change the number of divisions the smooth does, which might get you closer to what you're seeing in the tut. Hope that helps.