Division, Not Long Division

Division is the most powerful arithmetic operation. It makes comparisons. When the numerator and denominator have the same units, the comparison makes a dimensionless number, the only kind that the universe cares about. Long division, however, is something else entirely. In my post “Dump algebra,” many commentators objected to my loathing of long division. But long division is not division! Long division is just one way to do the computation, and is far from the most useful way.

I’ll illustrate with an actual example of division. For my environmental-protection lawsuit, now in the Massachusetts Supreme Court, I needed to divide 142,500 by 4655. Here is the long-division calculation, my first use of the method in 30 years:

The calculation took me a few minutes with paper and pencil, some of the time to reconstruct the algorithm details and to get the bookkeeping straight — even though I already knew the answer quite accurately.

I knew the answer because I had already applied a more enjoyable method: skillful lying. I turned the numerator 142,500 into the nearby and convenient 150,000; and turned the denominator 4655 into the nearby and convenient 5000. Dividing 150,000 by 5000 gives 30. It’s likely to be an accurate estimate, because the two errors (increasing the numerator and increasing the denominator) partly compensate.

The next correction is not too hard, and comes from estimating the two errors. The actual numerator of 142,500 is 7500 less than 150,000, a decrease of 5 percent. To fix this error, decrease the estimate by 5 percent. The actual denominator of 4655 is 345 less than 5000, a decrease of, let’s say, 7 percent (7 percent of 5000 is 350). To fix this error, increase the estimate by approximately 7 percent. The two fixes together require increasing the estimate by 2 percent (7 percent minus 5 percent). So, 30 becomes 30.60—which is very close to the actual quotient of 30.6122…

This whole calculation took about 10 seconds in my head. There’s no need for long division, and I hope that I live another 30 years without using it again.

For students, learning long division mostly means learning like a parrot. A classic example is from the National Assessment of Educational Progress (NAEP) results reported in 1983 (Carpenter, T., et. al., “Results of the Third NAEP Mathematics Assessment: Secondary School,” The Mathematics Teacher, 76:652-659). Thirteen-year-olds across the country were asked:

An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed?

70 percent of the students did the long division correctly (the result of 1128/36 is 31 and 1/3). From doing the division correctly, the most popular answer, chosen by 29 percent, was the meaningless 31 R 12 (31 with a remainder of 12) buses. Another 23 percent chose 31 buses, leaving 12 soldiers stranded. Only 18 percent chose the correct answer of 32 buses. Even then it’s not clear how many of the 18 percent were sure of their answer or were just guessing between 31 and 32.

Here is a flow diagram illustrating the answer distribution:

It’s easy to learn long division yet understand little.

 

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  1. Bart says:

    I agree and disagree. I think that we have to teach long division to get them to understand the precision – and this is where the stories come in – we use the stories to clarify what is most appropriate and correct for the situation. Once they understand the process then we can help them understand how precise they need to be – or not. Depends whether you are moving troops or working with quantum particles. It also depends where you work. I do a lot of field work. I do have a calculator. I have had that calculator fail on me at the most inopportune moments. So, to pen and paper I went. For some stuff. For the precision.

    For other stuff, like in demolition, figuring out how far to cut around the base of a column to make it fall over, we just used Pi = 3 plus a little. But if were weren’t going to let this tower fall, rather lift it with a crane, then an approximation tells you if you are in the ballpark of your cranes limits, and you may need to get to the precision to make sure.

    Critical thinking stems from the process.

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  2. Corwin Delight says:

    Until last year in calculus, I hadn’t used long division in about 10 years either. But my Calc teacher showed us a trick to using long division to simplify complex polynomials:

    3X^3- 2X^2+15
    ——————–
    X^2+5X

    set it up like long division,

    _____________
    X^2+5X |3X^3-2X^2+15

    then solve like long division. You’re left with a regular non-fraction polynomial, and a smaller, more simplified fraction (the remainder from long division) for the answer.

    3X +17
    ____________
    X^2+5X |3X^3-2X^2+15
    -(3X^3+15X^2)
    _____________
    0X^3-17X^2 +15
    -(17X^2+85X)
    ______________
    0X^2 -85X +15

    Answer:
    3X^3- 2X^2+15 85X +15
    ——————– = 3X+17 – ———–
    X^2+5X X^2+5X

    Note: This may or may not have actually made things easier, as I made these numbers up on the spot, but the important thing here, is that this works not just for cubics, but for any polynomial of any degree.

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    • Corwin Delight says:

      Bah, the white space that I entered to make the formatting look like long division symbols didn’t really work. Hopefully the whole thing is still understandable.

      the final answer btw, is 3X+17 – ((85X+15)/(X^2+5X))

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  3. Alan T. says:

    Back when I was in eighth grade, calculators were rare and clunky. Of course we all knew how to do long division. My superb eighth grade math teacher, Mr. A. Suber, to whom I am forever grateful, taught us how to extract square and cube roots by hand, using a process similar to long division. He must have done something right; one of the students in the class eventually went to Harvard, and two went to Stanford.

    Over the years, I forgot the details of how to extract square roots, but I can reconstruct the algorithm if I want to because I understand the underlying principles. This morning I calculated the square root of 7 to four decimal places.

    I value understanding the principles used to extract square roots even though I never need to do so. In the same way, I think it is important to know how to perform long division, even though it can be done using a calculator as a mental prosthesis. Understanding concrete algorithms such as long division and extraction of roots reinforces and may even be necessary for the understanding of general principles.

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  4. James says:

    “I knew the answer because I had already applied a more enjoyable method: skillful lying.”

    Humm… What are the odds that you’ll encounter a judge or opposing counsel who actually knows long division? You lose your case, and are lucky if your “skillful lying” isn’t counted as perjury :-)

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  5. James says:

    “I knew the answer because I had already applied a more enjoyable method: skillful lying.”

    Humm… What are the odds that you’ll encounter a judge or opposing counsel who actually knows long division? You lose your case, and are lucky if your “skillful lying” isn’t counted as perjury :-)

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  6. JPB says:

    The saddest thing about American education is that typically if a student does division in this manner showing your work as teachers will ask you to do, you will be marked wrong on the test, even if you get the right answer.

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    • Seminymous Coward says:

      Barring a class on estimation where that would likely be acceptable work, the reason it would be marked down is that it’s a test on dividing numbers, not roughly dividing them.

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  7. truzo says:

    Sanjoy, you are missing the point of Long Division, which is to systematically figure out the proportionality of various numbers by isolating the relevant terms through a process. All you are doing with the “142,500 is 7500 less than 150,000, a decrease of 5 percent” is isolating calculations in your head to a base 100 scale (per cent) in an abstract fashion by introducing “long” addition and subtraction, percentages, and a bunch of ambiguous rounding. Frankly, and as someone who uses math every day, I would rather follow the concrete, paper process than remember the rules and placeholder percentages used to increase/decrease multiple values. In fact, i first skipped over that convoluted paragraph after the third sentence just to get to your point–which seems to be that your preferred process took you 10 second, and that a process you don’t use took you longer. Guesstimating is something you adapt at a later time to improve speed and efficiency, not as a basis for understanding.

    Furthermore, the army bus example is a word-problem issue. It has little to do with the fact that long division spits out an answer with a remainder, and more to do with a lack of learning “context”. This type of issue could be applied to any situation where a process (computer) spits out an answer and the user (human) does not apply the knowledge. Yes, it has to do with fractions and wholes, but cannot be blamed on long division.

    You seem to be engaging in calculated omission, distraction and divergent tactics. The overwhelming response to your “Dump ALGEBRA” article was that Algebra is a crucial learning tool that teaches skills that can be summed up, to be succinct, in one word: variables. Why you chose to focus a follow-up post on long division, which takes a whole 2 weeks to learn in third grade (i’m sorry, my teacher dwelled on it for 2 weeks and 2 days, so 3 weeks to learn) is beyond me, and seems like filler. If Freakonomics needs more content I know some people who could provide recommendations.

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  8. Brian says:

    The problem is that the shortcuts you use to make calculations in your head are harder to learn and different situations call for different approaches. It is a skill that needs to be developed over time to get the experience working with numbers in different ways. Even then, you aren’t always going to be very precise in a quick amount of time.

    Long division is a pain, but you use the same formula on every division problem and it gives a precise result. It is easy to teach and easy to test for.

    In the real world, you would use a calculator but that would be even more parrot learning for students starting out in math. At least doing the steps in long division will give you experience that will help you mentally calculate things later on.

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