Addition

How do I reconcile these two things?

Sam was looking at a picture of six birds on a box, which were in two groups of three, and within the groups, there were two small birds and one large bird. She counted, pointing:

One
One, two
One
One, two
One, two, three, four, five, six
So one plus two plus one plus two equals six. And one plus one plus two plus two equals six.

Wow! But the other day, we had this conversation:

Me: Sam, what is four plus zero.
Sam: Ummmm, five.
Me: No. Remember, zero means nothing.
Sam: Ummm, six.
Me: No. Look, I'm holding up four fingers, and I'm going to add nothing to them. Zero. So how many do I have?
Sam: Four!
Me: Yes! Now, what is seven plus zero.
Sam: Eight.
[repeat same explanation]
Me: Now, what is one plus zero?
Sam: Two.
[and so on]

It's really hard to get an idea of what she understands and what she doesn't. I'm thinking this means that she still has to have physical objects to count. The zero conversation happened in the car and the only time she could get it was when I held up fingers.

Anyway, it's all fun stuff! I love to try to figure out what's going on in that budding little brain of hers.