[seanflyon]

Perhaps the counting up method makes it easier for most kids to grasp the underlying concepts. The borrowing method can also teach kids the underlying concepts and follows nicely from addition as it is the inverse of "carrying the 1".

> You just follow the rules with no real reason.

This is the real problem here. We should not teach our kids to memorize a set of rules but to understand the concepts. When I was in school I had a rule that I would not memorize rules that I did not understand or formulas that I could not derive. I would probably be faster at arithmetic if I had memorized my multiplication table like I was supposed to, but I think that rule served me well.

[adamtj]

When you're in 2nd or 3rd grade, it's more important to learn the mechanics of subtraction than to understand why it works. Kids at that age need to fill their brains with facts so that they have the raw materials for developing understanding when the are more mature and more able to understand.

In this case, borrowing is more compact and efficient. That makes it faster and easier, both on paper and in your head. If you write out the borrowing method as verbosely as this counting-up method, it's almost as easy to understand. However, that's not important in elementary school and shouldn't be done. The counting-up method has the disadvantage that you can't write it more compactly.

Students who learn the counting-up method will be hobbled in algebra: they'll be wasting their limited brainpower on the mechanics of subtracting the hard way when they could be using an easier method and devoting more brainpower to learning the concepts of algebra.

Of course it's important to understand how subtraction works, but by the time you get to algebra, that should be easy. Anybody who is uncomfortable in their ignorance can either just think about it, or ask a teacher. It's not hard. Just note that 325 = 300 + 20 + 5. Then line everything up and go. That understanding isn't worth a lifetime of pain.

[seanflyon]

I think that understanding basic concepts is foundational. It not only gives you an understanding you can build off of, but also teaches you how to think, how to approach a new problem. You say that kids need to fill their heads with facts, but I think they should fill their heads with concepts that they actually understand. Simple concepts that they understand, not facts, are the raw material for developing more complex understanding when they are more mature.

I realize that I cannot prove what I just said, so perhaps I stated it too strongly. It's possible even, that what I claim is true for some children and what you claim is true for others. I know that an emphasis on understanding has served me well, and that an emphasis on rote memorization has worked poorly for several people I know.