The other morning, right after I woke my daughter for school, she announced, "Mommy, I know that six divided by three is two. AND I know that six divided by two is three. I just figured it out last night, and Daddy said it was right." I agreed that this was indeed correct. She claimed that this was the only division that she knew, and I proved her wrong there by asking her what ten divided by two would be. She counted by twos on her fingers and came up with five. We also discussed the fact that since anything divided by one is itself, she actually knows things like 1,900,017 / 1.
Asked after this what she wanted for breakfast she said "More math!" So, I set out twelve pieces of cereal next to her plate and I shot division and multiplication problems at her while she ate her bagel. She understands intuitively that if twelve divided by four is three, then twelve divided by three must be four. She's starting to understand that in that case, four times three must equal twelve. We went a bit beyond the 12 (twenty-four divided by six, etc.), but it started to be less fun, and we eventually moved on to reading picture books.
What I'm finding interesting, something that I hadn't expected, is seeing HOW she solves problems. She doesn't always get to the answer the way that I would, or even in a way that makes immediate sense to me. But she gets there. For example, she knew that 12 times two was 24 because she had memorized that 12 + 13 = 25, and calculated it would be one less than that. Whatever works, I say.
With Common Core at school, there is considerable emphasis on showing how you come up with a solution, so I'm encouraging her to share her thinking with me. She's still working in school on addition, with numbers that add up to 10 or less. This is actually ok, because she's learning to memorize those sums, instead of having to calculate them each time. In the meantime, I figure it can't hurt for her to practice multiplying and dividing at home, if that's something she finds interesting.