This is a fantastic question and I apologize for taking so long to get around to answering it. The big thing, the major important thing I am trying to teach Morgan that does not come naturally to me, is emotional awareness and regulation. I knew that little kids have tantrums, and that bigger kids eventually grow out of that sort of thing, but it turns out that this is not entirely a matter of time. Emotional regulation is a learned process (with some developmental inputs) and there are lots of things I can do to help Morgan get through a tantrum, help reduce the frequency and intensity of tantrums, and teach her other ways to deal with the feelings that lead to tantrums. And every single one of them is bizarrely non-intuitive.
The primary thing I'm doing now is giving her vocabulary words about feelings. We've gotten books specifically about feelings from the library, with photos and illustrations of faces expressing different emotions, or illustrated situations in which she's supposed to guess what the character is feeling (I think my favorite is How Does Baby Feel? by Karen Katz -- it has several different positive emotions, which is a little unusual in this type of book). I try to point out bits in other books where characters are illustrated with clear emotions. And when she's experiencing a strong emotion, I try to label it for her, or to provide several possible emotional explanations if I'm not sure what it is she's actually feeling. This is, sometimes, magic:
T: No, you have three stickers already. You may not have any more stickers until after dinner.
T: Are you sad because you can't have any more stickers? You can say "I'm sad."
M: I'M SAAAAAAAD. ::abruptly stops crying, as though a switch has been thrown::
We also talk a bit about the emotions other people are feeling, like if a kid on the playground starts crying we will talk a bit about what he might be feeling and why, or we'll talk about how the dog is feeling when we're walking him.
The other backwards-seeming tantrum stopper is to agree with her about how cool it would be if she could do or have whatever it is that she is upset that she can't do or have. She's not developmentally to the point where this is as magic as I've read it can be, but she's verbal enough that it does work now, slowly. To some extent this reduces tantrums for the same reason that learning a bit of baby sign reduces crying (ATTN ALL NEW PARENTS: LEARN SOME BABY SIGN) -- much of what causes the upset is the feeling that she hasn't communicated her desire clearly, that I don't understand what it is she wants. So if I clearly indicate that I do understand the desire, and that I don't think wanting the thing is a problem, then she feels better about the situation even if she still can't have the thing. (Also I tend to talk about when she can have the thing -- after dinner, maybe next week, when you're a grown-up, whatever the appropriate time frame is.)
There is a fine balance between, one the one hand, ignoring Morgan's emotions, and on the other hand making them too big a deal. Neither of those is great. The ideal is kind of what we aim for when she falls down: notice, give her a moment to have her own reaction, then make a neutral informational sort of comment ("You fell down." "You look upset.") and stand ready to provide help or comfort if she needs it. Morgan specifically does not want as much physical comfort through emotional upsets as a lot of kids seem to, which is a little weird for me, so I am trying to practice being more verbally supportive rather than scooping her up for a hug, because if she's actually tantruming hugs really do not help.
Actually, speaking of informational comments, that's another cool new non-intuitive kid-herding technique I've been trying out lately. But that may be another post -- it is time to get ready for swim class!