At the start of the Math for Elementary Teachers class I'm mentoring in, we asked the students how many of them thought they knew enough math to teach elementary mathematics. Almost all of them said they did. Everyone knows subtraction, right?
Actually, not really. Everyone can do the following problem, but it's a really difficult algorithm to understand (E. Thanheiser "Preservice Elementary Teachers' Conceptions of Multidigit Whole Numbers, Journal for Research in Mathematics Education 40, 251-281.):
In order to follow best, you should probably get out a piece of paper and work the problem. (Comment below!) Or, at least, figure out how you wold work the problem.
When you did it, you probably did something like the following: 7 minus 2 is 5. You can't take 4 from 3, so you have to borrow from the 5. So you cross out the 5, write a 4, and a little one next to the 3. 13 minus 4 is nine. Four minus 1 is 3. So the answer is 395.
Ok, now the hard part--the essay question part: What does that little 1 you wrote next to the 3 mean? Why can you cross out the 5 and write a 4, and why can you treat the second column as 13-4? How does that make sense? And do the 3 the 9 and the 5 say "Let our powers combine" and summon three hundred ninety-five? Could you explain it to a first grader? (Again, feel free to leave a comment below explaining your answers.)
Surprisingly, these latter questions are more closely related to the actual practice of being a mathematician, than the rote application of the algorithm is. Or perhaps even more closely related: Come up with a way of doing the subtraction problem, and show it will always give you the right answer. (Presumably, asked of someone who doesn't know the standard algorithm.) Again, if you'd like to try to come up with an algorithm, and to explain why it will always work, feel free to leave a comment. (Though, if it's an easy question for you, don't spoil it for everyone.) Oh, and there's definitely more than one right answer to this last question.