Being able to use math terminology freely and appropriately is a big goal in elementary math education. In my opinion, there's no way you can really use math vocabulary intelligently unless you've first had a chance to experience the concepts in a concrete manner. This is a huge part of what happens during a Math in Your Feet residency -- I am never surprised when an astonished teacher tells me: "I can't believe how easily they're talking about math!" It would not be an exaggeration to say that most of us need to do math with our bodies in some way before we can talk about it.
Much of the math my daughter and I do together is hands-on and verbal; as a learner, she is best served by conversation. We'll be in the car on the way to somewhere and she'll spurt out some new computation she's been working on in her head and with her fingers. She may only be working with numbers one through twenty but, based on the frequency and variety of these kinds interchanges, she seems to be going pretty deep into her questioning.
The other day I wrote Maria Drujkova from Natural Math with a (very) basic math question:
"My daughter is telling me all the different ways she can 'make ten'," I wrote, "what's that called in math language? You know, 2+8, 6+4, 9+1...?"
Maria, being the awesome person she is, got right back to me:
"These are called 'number friends' by elementary teachers. 10 is important as the base of our number system. She is playing with a newly-discovered idea of an 'equivalence class.' You can try it with other things. For example, numbers whose difference is her favorite number (10, 2, whatever) - like 15-5 or 12-2 or 99-89. It's a road to ratio and proportion, too - which come up a lot in music signatures and dance!"
Thanks to Maria, it appears my daughter and I will have a lot more to talk about, well in to the future! There are other ways we are talking math these days, as well. You can find out more in Conversational Math: Part Two.