## Saturday, January 31, 2009

### Why teach long division?

Seriously, why teach it?

This is a genuine question, not a rhetorical one, and one that I've been wondering about lately.

Mrs. H said...

I teach algebra II and must teach long division of polynomials. I always begin the lesson with a review of long division of a 4 digit number divided by a 2 digit number. If a student never learned basic long division, they're going to have a very difficult time learning to divide polynomials.

bky said...

There are two good reasons for teaching the standard algorithm for dividing integers, if that's what you mean by long division:

(1) so kids can calculate the quotient of two numbers(and have an exact form for the remainder if that is wanted, rather than a decimal expansion), and

(2) it is an introduction to algorithms. It is odd that many people who deride the teaching of the traditional algorithms cite the availability of calculators as a reason not to learn the algorithm. I find it useful to regularly (every 6 months or so) have my kids (homeschooled) go over the operations of it, with the idea of helping them not only have confidence that what they are doing makes sense but also preparing them for them to understand the concept of algorithm by familiarity with a few specific examples (also on the list: standard stacking algorithms for multiplying, adding, subtracting).

A good idea is to practice occasionally with something like money: show 734 as 7 dollars, 3 dimes, 4 pennies. If you divide by, say 3, you start with 3 piles, each with 2 dollars; the left-over dollar is exchanged for dimes; etc. It is also useful to do the same problem on paper based on writing out the expansion 734 = 700 + 30 + 4 and then successively dividing each place value with remainder, and throwing the remainder downhill:

734 = 7x100 + 3x10 + 4
= 3x(2x100) + 1x100 + 30 + 4
= 3x(2x100) + 13x10 + 4
= 3x(2x100) + 3x(4x10) + 14
= 3x(2x100) + 3x(4x10) + 3x4 + 2
= 3x(2x100 + 4x10 + 4) + 2
= 3x244 + 2

This is "doing it by hand". The algorithm is a formalization. The algorithm is based on repeated division-with-remainder; you never really need to know what place value you are working with, or which side of the decimal point -- just divide and throw the remainder on the next lower place value. If kids understand this for integers, then dividing in the presence of a decimal point is just as easy. Note also how distribution is vital to long division, and since distribution is difficult for grade school kids this also gives practice recognizing and using that vital aspect of arithmetic.

Some critics say that the LD algorithm doesn't teach place value. Of course, it's not suppose to: it's supposed to divide numbers. But looking at it as an algorithm does indeed reinforce the concept of place value. Long Division is a keeper.

Niels Henrik Abel said...

Obi-Wandreas, The Funky Viking said...

There are two main problems in my line of work in this regard.

The first is a practical matter. The first book of the New York State Exams for 3-8 grade forbids the use of calculators. Granted, this book is also multiple choice, but that doesn't help if you don't have a clue what the answer is.

The more important matter, however, is that the overuse of calculators (along with math programs that bear only a passing relationship to math and administrators too afraid of poor matriculation numbers to hold back students who don't learn) has contributed to the demolishing of my students' number sense.

If you don't know how to figure out the answer yourself, then you have no idea where it came from. You simply trust a magic box to put out the right answer. If you input the numbers in the wrong order, you have no idea that your answer doesn't make sense.

Case in point: an assignment yesterday required students to fill out a table of data which was growing exponentially. Several students had the following numbers: 2500, 2650, 2809, 2977.54, 315.61924(misplaced decimal point), 3345.563944, etc. They stared at me blankly when I pointed out that having \$315 in the middle of those numbers made absolutely no sense. Not to mention the number of students I had to question individually for at least 45-60 seconds straight before I could get them to realize that those numbers referred to dollars and that they therefore should not have more than two decimal places.

They're too used to using the calculator as a crutch to actually know what the numbers mean.

It's amazing that in the midst of "teaching the program with fidelity" that I actually find time to occasionally teach some math.

lgm said...

If you teach long division, your student may eventually understand what the calculator is doing. In fourth and fifth grade, it's more likely that the student will come to understand place value, especially if he is allowed to progress to figuring out what the relationship between the remainder and the fractional or decimal form of the quotient is.

lefty said...

Thanks for all the great comments, and to Niels Henrik Abel for the great article by Klein & Milgram, which a friend of mine with whom I shared it may introduce into her teacher training course!

Niels Henrik Abel said...