Arithmetic or mathematics?

I stopped over to Amish Ervin's two nights ago and one of the girls was working on homework. She had 30 problems of division and had asked her mother, Rachel, for help. Rachel looked at me and asked if I was good with numbers. I looked at the problem: 239.68290000 divided by 999. Now it has been a looooooong time since I worked a problem like that. Do they even do this kind of long division in school now? I said something to Ida (the scholar) about math and she looked at me funny and that is when I realized she'd never heard of mathematics, just the word arithmetic. That got me to thinking, what is the difference? I have been out of school long enough that I'd forgotten. Had to look that one up! Basically arithmetic is all about numbers and mathematics is all about theory. I gave her alittle help and she was workin away. Saw her yesterday and she came up to tell me she had gotten all 30 right! She learned arithmetic and I learned something also!