Saturday, February 21, 2009

Why teach long division: part I

Three weeks ago, I asked Why teach long division? So far, I've received 8 thought-provoking replies.

My thoughts are provoked into thinking that we should teach long division.

But first, there are really two questions here, and one has an easier answer than the other.

The one with the easier answer considers any general algorithm for multi-digit division to be an instance of "long division." Besides the long division algorithm that most of us learned as students, we now have the partial quotients method, popular, for example, with Everyday Math.

So the first question is, should we renounce (or minimize) general multidigit division algorithms and rely on a combination of calculators and estimation skills?

Here, the answer is a resounding "no."

Reason #1 pertains to the general problem with doing math primarily via calculators and estimation skills. This is the problem of "poverty of feedback." Briefly put, if your estimation skills tell you that the result on your calculator screen is wrong, there's no feedback about what you did wrong. You don't know if you accidentally pushed the wrong key, or whether you're accurately typing in the wrong sequence of keys, or whether you made a mistake in setting up the problem in the first place (e.g., mistranslated a word problem into the wrong equation). There's simply no paper trail to go back over.

If, on the other hand, you work out the problem by pen and paper, you can review your calculations and see if you made an arithmetic mistake or an algorithm error; or if, instead, you'd better go back further and see if you set up the problem wrong.

I'm a linguist by training, and I see this "poverty of feedback" problem, as well, in programs that purport to teach grammar, like Rosetta Stone and Laureate Learning. Here, as with calculators, the learner simply clicks (on the picture that goes with a sentence). Because the software code can't tell from a simple click what the basis for the person's mistake was (e.g., whether it stemmed from deficiencies in vocabulary, or deficiencies in grammatical parsing), all the program can do in response is to indicate that an answer is "right" or "wrong". As an alternative, I've developed a program called GrammarTrainer. Here, you construct a sentence rather than clicking on a picture, which enables the software code to analyze your answer and give you detailed linguistic feedback about what exactly you're doing wrong. Preliminary empirical results suggest that this feedback is crucial.

As with language, so, too, with math: simple clicks make meaningful feedback difficult; meaningful feedback is key.

3 comments:

Lsquared said...

I don't remember if I was one of your responders to the last one, or if I thought everyone else had already said what I would say, but I would have been in the pro-long-division camp. It's a) useful for understanding division and b) gets you ready for polynomial division. So, I'm going to say something I expect will be surprising.

I think the partial quotients method is fine. It's not that new--it's been part of the teaching-long-division bag of tricks for decades, and it does exactly the same stuff that long division does, only a little less efficiently. In particular, if you understand it, learn it, practice it, you will understand division better than before you learned it, and you will be prepared to learn how to divide polynomials (and, for that matter, Abstract Algebra).

A fascinating thing I picked up from a Math for Elementary teachers book is that the long division algorithm you and I know takes up a lot more space and paper (is less efficient) than the one our grandparents probably learned (something I've seen called short division). You need a certain amount of efficiency in your computations, but because computations build on each other, it's more important to be efficient with the more basic computations (addition) than the later ones (division). So, if you have a slow algorithm for adding multi-digit numbers, that's bad, because it's going to slow your multiplication down even more, but if you have a slow division algorithm, there's not too much further that goes, because you don't get repeated long divisions as steps in later algorithms--most of the time they are the later algorithms.

Now this doesn't mean I'm against teaching and learning the standard algorithm--it's awfully convenient when all of my college students learned how to do things the same way--but I'd much rather have them understand partial quotients division than not know how to long divide at all (which happens, and is most inconvenient).

bky said...

I did not know what partial quotients was until I followed the link, but I recognize that as what I call doing it by hand. If you want to convince a skeptic that a certain quotient is what it is, you would do partial quotients (or multiply to check). It has a common sense appeal. Also, when it comes to dividing polynomials, people usually do something like that rather long division, or synthetic division as I believe it is sometimes called. At any rate, partial quotients computes the division with remainder and people using it will develop the operational number sense that reliance on calculators short-circuits.

However, partial quotients is a method but not really an algorithm in the sense that two people doing it on the same problem might have different sets of intermediate calculations and a different number of steps while still arriving at the correct result. This is only relevant to the advantage of the standard algorithm that it is an opportunity to teach algorithmic thinking. I like that way that it exactly calculates the number digit-by-digit in a deterministic route.

Another advantage of the standard algorithm is that it easily lends itself to operating on dividends that have a decimal part (e.g. 23.706 divided by 5) and thus is useful for finding decimal expansions of fractions, digit-by-digit. Getting at the result digit-by-digit is very useful when you want to calculate a certain number of decimal places. I haven't thought it through, but it would seem that partial quotients falters when you go past that last integer remainder. Or not?

Lsquared said...

it would seem that partial quotients falters when you go past that last integer remainder. Or not?

No, it works just as well as it does with integers. Not maximally efficient, but it does always work, in the same way, and with a finite number of steps.