Thoughts on carrying and teaching

Today I helped a number of students out with percent problems in study hall. After helping one student set up a problem correctly, I proceeded to watch him multiply "25 x 5" using the traditional pen-and-paper carrying algorithm and come up with 110. At first I tried to help him figure out where he'd gone in the algorithm, but it wasn't immediately clear to me, and it finally occurred to me why this particular error was so frustrating. "If you have 5 quarters," I asked him, "how much money do you have?" Not surprisingly, he knew the answer at once and was a bit surprised at himself for not seeing it. He then spent a puzzled minute trying to figure out where he'd gone wrong carrying.

The pen-and-paper algorithms for doing addition, subtraction, multiplication and long division are incredibly frustrating for the untidy. When I was in school, I remember turning lined paper on its side to help me keep my columns straight (that was the same year I had to make cut-outs out of note cards to help make sure I read one problem at a time). Yet somehow a great number of untidy students remain slavishly attached to these algorithms, using them as their only method for multiplying, even with very easy numbers.

It occurs to me that the basic algorithms we all learned for simple math are a form of early, somewhat clumsy technology: we use pencil, paper, and columns to help work around our memory limitations. Obviously, we have better technology now: the calculator built into every computer, cell phone and so on. Having students rely on one form of technology (pen and paper) is not fundamentally better than having them rely on another (calculator).

There is of course a very real drawback to both of these methods: you need to have the technology in hand to do the math. Stand a student in a department store evaluating 35% off of $40 and neither technology is very convenient. I would hope, though, that within a second or so even a student daunted by the 2-digit multiplication would know that the discount is between $12 (3*4) and $16 (4*4). It wouldn't take much to then say it must be half way between ($14).

For teachers, though, the pen-and-paper algorithms have a key advantage: they make all problems look the same and all students' work look the same. Errors in computation are relatively easy to spot and correct with these methods. They also are likely to point out (rightly) that mental math is prone to errors and impossible to revise. All of this makes algorithms better to teach.

That said, I wonder how much is sacrificed in terms of flexible thinking, real understanding of what's going on when you carry, etc., by teaching simple rote methods. Might it not be better if most teaching focused on either flexible mental math models for small numbers and estimates for all numbers (that took into account human memory limitations) or calculator-based math for precise numbers (that eliminated problems of human sloppiness). In such a world, students might learn the pen-and-paper algorithms in much the same way they learn to compute square roots using tables -- as an artifact from pre-calculator times. Most of their effort, though, would go into quickly evaluating problems like "30 * 49" (30 less than 1500, thus 1470), 491*35 (slightly less than halfway between 15k and 20k, thus a bit less than 17,500), and 48 * 48 (between 1600 and 2500, closer to 2500). Students should be able to do at least these quick estimates quickly, I'd think, as a means of having general number sense when reading an article or out and about, and as a means of checking for typos or column-os when calculating them precisely by hand or by calculator. But, I doubt that they can and I doubt that they've been asked to with anything like the frequency with which they've been asked to calculate sums on paper.

It strikes me that throughout education, decisions may be driven by these same factors, favoring work that is reproduceable with little variation across students, easily represented on paper, and possible to complete by following simple steps against work that tends to support multiple solutions and happens mostly in your head. I can think here of other "tidy" processes often enforced en masse, such as notecard systems for research, 5 paragraph structures for essays, and so on. It may be that such systems produce the most reliable results as far as work from students, but isn't our goal for results in their minds. And, if that is where our goals lie, how can we possibly measure success there?