Addendum to 'Knowing Numbers'

After posting Knowing Numbers & UNO Update yesterday there has been a development around subtraction, the addition of division (word play intended), an observation about the thinking-backward mind, and the arrival of the perfect book at the perfect time.

Today, the kid asked for another math test on the drive home from the Columbus Children's Museum.  I gave her equations: 18+2, 16+4, 13+7, 10+10. 

"Hey, those are a lot of different ways to make twenty!" she exclaimed.

"How many more ways can you think to make twenty?" I asked, thinking only of additional sets of addition.  Here's what she came up with: 22-2.

We stopped to get gas.  We went into the service station with me quizzing her: "Twenty take away one is...  twenty take away two is..."  and came back to the car having easily figured out the pattern all the way to zero. 

I think that all of this has been made possible because her brain spontaneously clicked into counting backward a couple weeks ago.  I am triply convinced that this is a brain development phenomena because earlier this week she wrote five sentences 100% in reverse (right to left sentences and words plus all letters written backward).  I had never seen backward writing like this from her of this magnitude.  And, just tonight, she was working on singing the alphabet song backward.  This is not a coincidence.  A supportive learning environment is key, but sometimes you just have to be patient for the brain to be ready.

Back in the van she asked, "Do you think I'm ready for the next level?"

"Sure," I said, with mind racing to figure out what the next level was, "What's 33 take away four?"

Done.  "Do you want a puzzle to solve?" I asked, thinking of something Raising a Happy Child had written in comments to yesterday's post.

"Yes!"

We'd been doing multiplication using visual groupings, why not invert the process?  Voila!  The kid's first introduction to division.  "You and three friends have 20 candies.  If you share them equally, how many candies will each of them get?  You can use pictures like we do when we play Star Count, you know, make circles and put in the dots."

She started easily, but it took a little work.  No matter, we've got mental and visual math firmly in place now, and she is so proud of herself.  I think she likes the feeling of a good challenge -- solving a puzzle that's hard enough, but not overwhelming.

And, to close, here's the book that came through the library system to my door tonight:

That's right, Young Children Continue to Reinvent Arithmetic, Implications of Piaget's Theory by Constance Kamii. 

I am on some strange yet wonderful journey of discovery.  What's next?!