Sunday, October 21, 2012

Right Answer?

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.


  1. To effectively teach this, you have to start with the idea of bases using pictures or bubbles or some other visual way for the kids to realize that a certain quantity of something can be represented by something else.

  2. Hmm. I solved this problem by immediately noticing that 42 is 5 more than 37. 500 - 100 = 400, so you take 5 from 400 and end up with 395.

    But it's still good to teach the basic algorithm before getting into shortcuts like this (although the book Math Magic by Scott Flansburg applies "tricks" like this to math education to show how complicated calculations can be done mentally in a short time).

  3. Benjamin: I'm not sure where your point is directed. Do you mean before they can understand numbers, or do you mean before they can understand place-value? If you mean before they understand numbers, well, yeah. If you mean before they can understand place value, yes. Definitely. Blocks and manipulatives are very important.

    Jeff: I'm not sure that's true. (Indeed, I could argue that it isn't.) Do you have arguments in support of that? My question would be: Why is that method a short-cut? What are we cutting short?

    The metaphor is of a path that the students are supposed to be on, that they get off of because they've found a shorter way. But I'm not convinced there is a "the path". Why not conceive of the math problem as a field we're trying to cross, with many possible routes across it. The algorithm gives one way of always getting across, that if you follow to the letter, you'll always get to the other side (though you may have no clue whatsoever what you did). But why is that a preferred way across, so that finding another way is a short-cut?

    And, if someone explored a field (or a town) by always sticking to the book, he'd never really get a feel for the town. Wouldn't that be true of numbers too? If someone always "explores" by following the prescribed rout, he won't really get to know numbers at all. All he'll learn is to follow the prescribed path.

  4. That said, Jeff, you came up with a particular way to do this particular problem. It may work in lots of situations, though I'm not sure if there's a particular method to what you did. But can you come up with a *method* that works, always, and show it did? (I realize this is a very difficult problem since we already have one method, and it's hard to think of something really different from it. I know of several algorithms for multiplication. Can you come up with one, and can you show it works?)

    I'll put up another post on this.

  5. I don't like commenting three times in a row, but I realized what you were saying Benjamin. I'm not exactly arguing for using different bases. That can just be confusing. I'm arguing for teaching students to understand our number system, and more importantly, teaching students to do the sort of thing mathematicians do, and not something nearly irrelevant to what mathematicians actually do. A part of that may be teaching them to use different bases--we could talk about whether it is--but it isn't the core of what I'm advocating.