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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COMMENTS: 52


  1. Eric M. Jones. says:

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    • Dave says:

      Incidentally, a zillion years ago people understood logarithms. I used to tutor a lot of kids in algebra 1, 2, and calculus, and I haven’t ever bumped into a kid with a solid understanding of logs.

      Long division in itself isn’t a difficult skill, or particularly useful, but long division is a great indicator of whether someone has their basic division abilities down pat. The 142,500 / 4655 is definitely nothing that most students will ever get easily, but there is no problem making sure kids can do 1027/3.

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      • Eric M. Jones. says:

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  2. YX says:

    That seems to suggest kids need to learn MORE math, not less… that’s pathetic.

    Long division is second grade math in Russia, China and India.

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    • Matthew says:

      In Soviet Russia, math divides YOU. (Sorry, I couldn’t resist.)

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    • Enter your name... says:

      I believe that it is commonly introduced at the end of second grade here in the US these days, too. So?

      Doing something earlier doesn’t make the students learn it any better. In fact, Finland doesn’t even start school until age 7—when American kids are in second grade—and they perform better on nearly all measures than Russia, China, India, and the US.

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    • Cindy says:

      What
      Isn’t long division that thing you do with polynomials? They learn that in second grade?
      They learnt that far into algebra in second grade?
      When did they learn all the basic math stuff?
      Are those people like born with basic math skills or something?

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  3. BL1Y says:

    “This whole calculation took about 10 seconds in my head.”

    It took him about 10 seconds to divide 150,000 by 5,000, and then to figure that 7,500 is about 5% of 150,000 and that 345 is about 7% of 5,000, and then figure what 2% of 30 was?

    I don’t buy it.

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    • Ryan says:

      And what if you need more than 2 digits? Do you just keep bootstrapping your way up (in your head mind you)? At some point you start writing these steps down, you create a nice bookkeeping method and you have … long division.

      I will say that too much emphasis is put on the mechanics of long division and not enough on understanding why it works in the first place. The latter is an important, fundamental property of numbers and division in general, the former is just a conveniently implemented algorithm. Reproducing dozens of pages of division exercises is a good way to make some young kid hate what the teacher calls ‘math’.

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      • Eric H says:

        “And what if you need more than 2 digits?”

        Use a calculator!!!

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    • Modi says:

      1% of 150,000 is 1500, 1% of 5000 is 50, 1% of 30 is .3. Is it still hard to calculate out those percentages in less than 10 seconds now?

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    • Charles L. says:

      Yeah. You’d think an economist would be able to do that in 5 at most.

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  4. Clancy says:

    I think you might get more traction with this one than dumping algebra. Not only is long division tedious and no one ever uses it, but it doesn’t even teach you anything new about math. Do I have a deeper understanding of the process of division because I know long division?
    No.
    I have to divide large and sometimes odd numbers all the time. I try to have a calculator with me as much as possible, but if I don’t then I have to work it out. My reflex is to do the long division, but I have to stop myself and say “That’ll take an hour and you’ll probably screw it up anyway. Just estimate it.”

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    • Seminymous Coward says:

      It was supposed to teach you that subtracting multiples of the divisor from the dividend is an effective way to break down the problem of dividing things. Furthermore, it was supposed to teach you that you can select those multiples perfectly in a single pass.

      It also provides one of the only clear algorithmic solutions many people ever see. That should be illustrating to you the value of a defined, methodical process for gradually attacking seemingly intractable tasks. It is also setting up a common reference point for explaining what an algorithm is to you later.

      It’s unfortunate that in your case it seems to have damaged your self-esteem to the point of giving up on elementary arithmetical calculations. That’s a pity when it could have been an opportunity for your teacher to let you see yourself overcoming what was previously an insurmountable large challenge via learning and a bit of persistence.

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      • Clancy says:

        Well, I’m glad you learned long division as part of your Introduction to Recursive Algorithms class. But I learned it in third grade as “Memorize these steps: Divide, Multiply, Subtract, Bring Down, Repeat.” As I imagine most kids did.
        I don’t know what makes you think I’d given up on arithmetic (or that the power of positive thinking could bring it back). I mentioned I do arithmetic on a daily basis, often without a calculator available. You may have confused the word “estimate” with the word “guess.” My method of estimating is similar to Mr. Mahajan’s. It involves rounding the numbers and thinking of them as a fraction, then reducing the fraction, then estimating the error coming from the rounding. It’s much faster and less prone to error than long division. Since I’m not a Rain Man-esque savant, I occasionally make arithmetic errors. Since there are many more steps in long division, there’s a greater chance of a mistake.

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      • Seminymous Coward says:

        Not every lesson has to be explicit. Though I’d say that actually discussing long division in the broader context of systematic problem solving would be superior to your teacher’s apparently narrow approach. The problem with that approach is in the way long division is taught not in its being taught at all.

        Smaller Issues:
        Long division can be interpreted recursively or iteratively equally well.
        You are obviously supposed to make easy reductions before beginning long division. Learning when to apply a technique is part of learning a technique, so I’d count it as a failure if your teacher didn’t teach you that e.g. long dividing an even number into another is silly.
        Giving up on something is, in fact, one of the few problems easily resolved by positive thinking.
        If you need to get into estimating the error in your estimation, you shouldn’t be doing an estimate. For a substantial fraction of mental computations all you want is a bound on the answer, anyway.

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  5. Joe says:

    Hence, the problem with asking people who don’t use math if their kids need to learn math (ie. should your kids learn algebra).

    Math education in the US is a royal mess. Tinkering around the edges, adding or dropping topics, isn’t going to fix things.

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  6. Iljitsch van Beijnum says:

    But it’s much harder to not learn long division and yet understand much.

    Long division is the only way to do division accurately with pen and paper. Using tricks to simplify the calculation enough to perform them in your head is nice, but it’s very error prone and it’s not a universal solution.

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    • Clancy says:

      Long division is also error prone. There are dozens of calculation steps where you could make mistakes. If you really need to be precise to the sixth decimal place, (e.g. calculating the landing speed of a Mars Probe) then you should use a calculator or a computer.

      Thinking about reducing fractions provides much more insight into the process of division and is an important step towards learning algebra.

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    • D.Z says:

      “Long division is the only way to do division accurately with pen and paper. ”

      That will sound nitpicky but I remember learning something called Fourier Division a long time ago, that was also accurate and doable with pen and paper. It wasn’t any less painful though.

      I’d argue in favor of long division because it could be used as an introduction to algorithmic thinking but for it to have any interest, kids would have to understand the underlying principle of the long division to begin with. So scratch that.

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  7. Mike says:

    What I’m struggling with here is that you have effectively done in your second approach is essentially the long division in your head.

    Unless I’m totally missing your point (or I have a modified strategy for long division), the first step of long division is inevitably the “skillful lying” step / estimation. You don’t get to put 30 above the line until you have figured out the number of times 4655 can go into 142500 completely. A long divider is going to try something like one of the following possibilities to get the whole number: 4000 into 140000 or 5000 into 145000. That would get you your starting place and add or subtract a 1 until you get to the right “whole” number.

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    • zepplin says:

      Exactly. If they let you use negative numbers, it’ll be even more synonymous.
      e.g. 72/15 = 5.(-2) = 4.8

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  8. David O. Cushman says:

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    • Seminymous Coward says:

      If you want to pick the problem apart, it’d be a lot more entertaining to just say “You only need 1 bus …for 32 trips.” That at least presents a real-world objection to the simplistic logic of this word problem for children. Furthermore, it could even lead to a discussion of the optimal number of buses for the army training camp to have around.

      I will indulge you for a moment, though. The externally imposed 1128 soldier condition tells you the optimal number of soldiers to transport. There’s no way having a bunch of untrained soldiers sitting at the bus stop for their full enlistments makes any sense.

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    • Enter your name... says:

      No, the answer follows from an assumption that when the problem statement said that “1128 soldiers are being bused”, it actually meant that “1128 soldiers are being bused”, not “some, most, or all of the 1128 soldiers are being bused, depending on all sorts of unstated factors”.

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    • Mike says:

      Agreed. It’s an unwritten assumption that in these types of problems, the “correct” answer for the test is the next highest whole number. However, you wouldn’t know that unless you’ve been taught the test. The math isn’t faulty, the way of testing is.

      If we’re assuming buses can be partially empty, the actual answer is undetermined. A much better question would be to ask the *minimum* number of buses necessary.

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      • James says:

        No, you don’t need to have been “taught the test” to answer the question correctly, you just have to think about it.

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      • Ryan says:

        I agree with James. If you don’t want to think at all, and you only mechanically reproduce methodologies, yes, you have to be taught to the test. But if you think about it even a little, you’ll come to the correct answer.

        This is a classic problem with people’s understanding of mathematics. As soon as they start thinking about ‘numbers’, it gets all scary to them and they completely disregard the fact that the numbers actually mean something.

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      • SunnyvaleCA says:

        The minimum number of busses necessary is obviously just one. Busses are not single-use transportation. A single bus can take all the people by making several trips.

        Maybe a better question, which still requires the use of division, would be: how many trips would a single bus be required to take.

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  9. 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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  10. 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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  11. 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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  12. 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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  13. 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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  14. 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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  15. 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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  16. 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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  17. Bob says:

    They really need to improve how they teach estimates to our children in schools.

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  18. Philip Crooks says:

    You rounded and estimated to get a good approximation to the answer. Do you realize that what you did requires sound number skills if you did not know how to do long division you may not have been able to estimate so weell.
    By the way there is research that seems to show a link between being able to multiply and divide and understanding algebra.

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  19. ajb8 says:

    If you really want kids to learn you can’t get bogged down with cheap tricks, you have to teach the true foundations of mathematics. Replacing long division with this “division by estimation” is an insult to pure math.

    “We have to reinvent the wheel every once in a while, not because we need a lot of wheels; but because we need a lot of inventors.”

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  20. jagiela says:

    Years ago there was this nifty invention called a calculator that would do all these computations for you. Then years later, they came up with web access for your cell phone and now your phone can do all this for you. You don’t even need to guess anymore.

    There is just so much worthless education out there: algebra, multiplication tables, cursive writing. Its all so silly but it keeps teachers employed

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  21. Drloko says:

    So you discovered estimation? Great. Next, let’s just set pi to 3 and the square root of two to 1.5. The point of long division is to provide a method to get the exact answer. Because to some people it’s important to have the correct answer, even if that means keeping track of all if those tedious decimal places.

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  22. Not A Math Nerd says:

    Wasn’t that same “rough guess” stuff used on one of the landers send to Mars a few years ago – you know, one of the ones that went SPLAT. Well, I guess not – that was due to unclear communications and inaccuracies in units. However, it is that type of rounding/estimations that causes errors that can create damaging situations.

    I can appreciate your use of rounding and estimations, and how they can be more accurate than leaving 12 soldiers with an extra 20 mile hike to reach their training site. But the students do need to learn long division – along with logic to use that long division wisely.

    Please do not bemoan the use of a tool, however tedious (and long division can be tedious) because folks haven’t been taught the logic of HOW to use it properly.

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    • Sanjoy Mahajan says:

      The Mars Climate Orbiter crashed because the engineers spoke in different languages (units): “The peer review preliminary findings indicate that one team used English units (e.g., inches, feet and pounds) while the other used metric units for a key spacecraft operation. This information was critical to the maneuvers required to place the spacecraft in the proper Mars orbit. ” See http://mars.jpl.nasa.gov/msp98/news/mco990930.html

      Extra accuracy would’t have fixed this problem. But if the engineers had been really lucky, a really inaccurate calculation could have canceled out the error from ignoring the differing units!

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  23. Gwen Nicodemus says:

    I love seeing articles that explain different ways to do things. My son is dyslexic. He can’t work out a 2-digit multiplication or division problem on paper, but if I ask him verbally, he generally gets the answer. Or really close. He has his own “cheats” like this, but he can’t quite articulate them to me.

    As as far as the 10-second thing, I can believe that. I’ve watched my kid do it. I’ve watched my husband do it. Actually, I can do it too.

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  24. Holli says:

    Long division becomes useful later when learning how to divide polynomials, especially ones that have remainders. You have to learn the process in the simpler realm of numbers first. Besides that…how many of these kids can do percents, much less in their heads? Or know how to compensate for errors? Kids are more concrete thinkers, not abstract. They need to do it on paper, not in their heads! And the example you give about the buses is not an error in long division, but applying it logically. It’s more understanding the concept, than the actual computation. Even using your method, they might not have gotten it right…

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  25. Charles says:

    I taught math in junior and senior high for 32 years, retiring in 1997. I taught basic math up to and including Calculus in that time.

    It’s my opinion that the number one thing which has destroyed mathematics education is the calculator. Theoretically, it can be useful, but it’s not used the right way… all it’s used for is to do simple arithmetic that should be learned in elementary school. Too many kids, even in calculus, can’t do simple arithmetic, but rely on the almight crutch – the calculator.

    I taught my calculus students without a calculator; the other teachers taught the calculator. EVERY year, on the departmental final exam, my students averaged 10 points higher than the other classes. The only difference was the absence of calculators. The other teachers were equally capable with the material, if not more capable.

    Take the calculator out of school…. learn the tables at an early age… learn the algorithms.. KNOW HOW to get the answer by yourself…

    Give a class a basic multiplication of two 5-digit whole numbers using the four-function calculator and most will get it wrong… virtually all of them. If they learned the multiplication table and multiplication algorithm, they’d do much better.

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    • Sanjoy Mahajan says:

      I agree about calculators. I never used one in school myself. My favorite solution these days is the QAMA calculator (see my Freakonomics post about it), which forces the user to estimate first before it will give an answer!

      -Sanjoy

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  26. Phillip Bryan says:

    Look up the topic of ‘Euclidean Domain’ (an object of study in advanced algebra.) You will learn that the ‘loathed’ long division algorithm is a form of the ‘Euclidean Algorithm’ the existence of which gives the name to the object of study, and results in a rich body of interesting and useful results, both in Mathematics, itself, and in applications.

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  27. Phillip Bryan says:

    Have you tried to program your method? Long division is easy to program.

    After one example of what happens when one neglects a remainder, the students will know to include them for the next problems. This error also provides good motivation for introducing the greatest integer function, so we can make a ‘silk purse’ out of this ‘cow’s ear.’

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