The more I work with math and my children, the more I "get" why manipulatives work as well as they do. G just added 4332 and 2464 without even having the slightest idea when she sat down that she could do that. She sort of resisted at first, because although I presented the place value stampers as a game, she saw bigantous numbers and knew I was up to mischief. I am sure me telling her she was going to tell me the sum of those numbers didn't help, but I was going for the shock value of her success. Well, mischief managed. She swiftly took over the stamping and told me the sum. Then she smiled.
I've read about children in Montessori environments doing advanced calculations swiftly and accurately, but I never really understood how that works-- they're just beads-- until I began to show my own kids how it works. It's a little unnerving to go into the unknown, but so satisfying when it works. And oh, how it works! I've not wanted to make / purchase the beads because I have felt that storing them in a home situation would be too unwieldy. Today, working with stamps of a similar design, I changed my mind. This was fabulous, and fast.
I didn't spend a lot of time with manipulatives when I was a child, and math became a source of extreme difficulty for me. If you miss one small kernel of understanding, you lose the option of understanding most of what comes after. It makes me happy to see my kids not only just soak it up, but hold "math" in a sort of cavalier regard.
Edited to add links for BBB:
Base Mathematic Lesson Plans (this is not what I used for this post, we were working with a Montessori three-part lesson)
You can purchase stamps here
but one of my favorite math / science equipment purveyors is here.
I did a quick google search and found this for you, to better explain how we began what we were doing. We have the actual bead materials shown in her initial photograph, but instead of the beads described for the lesson, we were using the stampers. The current lesson, which is related today, is based on this presentation.