I'm strange about my treats (usually apples or carrots, although sometimes I'll put a bit of molasses and sugar on a slice of apple for her). I like to use them as a reward for Indie if she's behaving.. I know people say that treats cause a pushy horse, but in the past few weeks, Indie has become much more respectful of me. Not because of the treats, but because I continue to correct her if she gets pushy. Treats are just there for when she shows improvement and I usually give her a bit of an apple for after I do her feet since I'm still treating them and it's nice that she stands still for me.
As for cutting them up, I usually cut them up into 1/8s so that they last longer. Lately, I've just been leaving 1/4 slice for after a ride and then I save the 1/8s for when she shows improvements in things on the ground. She doesn't seem to mind that it's only a little slice either.