How DO you multiply?

I really enjoy tutoring students at LearnToBe.org, although I always get a little nervous as I begin a new session that I'm not going to know the answer, or that I'll know the answer but not be able to explain it in a way that's meaningful to a ____-grader. Today a third-grader asked me the very open-ended and deep question: How do I multiply?

First I tried to figure out what she was really asking. She's home-schooled and said she'd soon have a test where she would need to multiply numbers from 1 to 100 together, but she didn't seem to have even the basic 1-to-12 times tables down. If I remember correctly, when I was in 3rd grade we always had a timed multiplication test when we came in from recess, and I think the numbers didn't get any bigger than 12. Anyway, I wasn't really sure where to begin, so I started with a general explanation of what multiplication is, then showed two ways I could think of to multiply if you had a lot of time on your hands -- write out the numbers and add them, or draw a picture with dots. But I said she really needed to ask her home-schooler for some flashcards, because only after she gets comfortable with 1-through-9 will she be able to do tricky two-digit numbers like 48 x 73.

But, really, I don't remember how I learned multiplication. So many building blocks ...