Training my middle school singers to listen for and work to produce overtones as they sing changed the tone quality my singers produced in magnificent ways, but it also had an awesome and unexpected result.
Students who suffer from ADHD and other attention deficit issues suddenly pop to attention when they hear the overtone. Once they've achieved it for the first time, all I have to do is to remind them to listen for them and work for them in future lessons. It's truly amazing to watch. Choral singing improves, but even better, behavior improves. ...And we all know that, in the choral music classroom, we will take all of that sort of result that we can get!
The overtones are difficult to accomplish with inexperienced singers, but here are three of the ways I help my singers achieve them:
a) I use solfege for all sight singing. The vowel work that you achieve with solfege is awesome. You can correct so many things through it.
b) I talk constantly about using the "tall mouth". I say things like "North and south with your mouth. East and West is not the best."
c) I use the vowel "oo" when I am trying to achieve overtones for the first time. Once they know a particular melody really well, I ask them to sing the melody on "oo". Make sure they are completely confident with the rhythms and the pitches before you try this with young singers.