I remember very clearly the problems I had with certain topics in mathematics. I remember getting confused on the day that my fourth grade teacher taught us how to multiply two-digit numbers by two-digit numbers (I had spaced off during the critical fifteen minutes when she explained the moves to us -- I was permanently spaced out as a kid, actually). That confusion was with me for a long time. So I thought I had a particular rapport with any kid who was struggling to learn math, having once been a kid who couldn't do math to save her life. My then going on to be a math Ph.D., and a math professor and researcher, made me what I thought was a pretty decent role model for struggling kids.
I was pretty good at teaching any topic, in fact, as long as Ben could learn it easily. We hit our first big bottleneck at long division. Multidigit multiplication was actually pretty easy for him; particularly since, in Everyday Math, Ben had learned this slick trick for multiplying multidigit numbers called lattice multiplication and was going to town with it. But long division was a different story. Ben had trouble lining up the columns, remembering to pull down the next digit after every step, and knowing where to finish his calculation and what to do with the remainder. Long after he had demonstrated that he knew what to do at every stage, he still couldn't reliably get the right answer.
I couldn't see that anything would help him master long division but long practice. He had learned all the steps and could apply them, but being methodical about it wasn't part of his nature. So, every night for a couple of months, I would give him several long division problems to do; it would always require several revisions before he would be done for the night. I could be what I needed to be -- a brick wall demanding that he apply care to his computations before he could consider himself done. What was doing me no good at all, just then, was my appreciation of the beauty of higher math.
The long division algorithm we all learned is actually just a repeated application of the Division Algorithm, which in its naked form, once understood, sounds obvious to the point of stupidity. The repeated application of the simple division algorithm with divisors that are decreasing powers of ten is just a thing of beauty, though, something written in The Celestial Great Book of Math. A lot of good it did us, though, in helping Ben to learn to apply long division. It took him a long time to learn to do that reliably, but we stuck with it until he got it.
There is the question of whether we even need to do this -- to torment students by making them practice the tedious long division algorithm -- especially now that computers and calculators are everywhere. It's claimed that such drilling kills the joy of math, and that we can teach children to love math better if we don't force them to do computations. I'm claiming (but not yet from any position of certain knowledge) that we do need to teach computation. I'm going by the fact that, in my association with mathematicians and physicists and engineers and computer scientists and finance people in my schooling and various jobs, I've known many people who could apply the long division algorithm, and some few who could appreciate its beauty; but I've never known a single soul who could appreciate its beauty without being able to apply it.