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.

 


Eric M. Jones.

A zillion years ago (estimate) logarithms were no longer calculated by hand, they were looked up in a table. Same with trig functions...remember the Taylor and Maclaurin series?

Extracting roots, long division and multiplication, percentages and even addition and subtraction (above a couple digits), should be relegated to the dustbin of history too. Just grab a calculator. If you really need to, look up the manual method. This is useful if you need to write a software program where numbers are manipulated in some way.

Same goes for those whacky "fractions". Take it from your Uncle Eric kids...THERE ARE NO FRACTIONS, just one number divided by another. Manipulating them is a good intro into similar manipulations for pre-Algebra, but that's all.

Not something that many people need to know. There are far better things to learn. and BTW I am thoroughly FOR Algebra and Trig, and if you're going into a technical field, Calculus.

Read more...

Dave

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.

Eric M. Jones.

(partial repost) Here's a way for your students to think of Logarithms–

Let’s say we have a string of digits like 1234567890. We can specify where we want to put a decimal place to get many different values e.g. 1234567890 decimal-place 3 would be 123.4 etc. The decimal place callout also can be zero or even negative: 1234567890 decimal-place -1 would be 0.1234....

But we can represent ANY value if we don’t restrict the decimal place to integral values….e.g. 1234567890 decimal-place 3.678=(whatever). The decimal point can float. It doesn't just stick into an integral position.

It should be clear to the student that we can use ANY number for the base (e.g. 1234567890) and any number as the exponent e.g. 3.678. Certain bases like 2, 10, or “e” confer solving ease in particular types of problems.

Simple!

YX

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.

Matthew

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

BL1Y

"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.

Ryan

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'.

Eric H

"And what if you need more than 2 digits?"

Use a calculator!!!

Clancy

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.”

Seminymous Coward

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.

Read more...

Clancy

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.

Read more...

Joe

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.

Iljitsch van Beijnum

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.

Clancy

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.

Mike

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.

zepplin

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

David O. Cushman

The answer of "32" follows from an assumption that is not explicitly stated, that the cost of stranding 12 exceeds the cost of using the 32nd bus. If not, "31" or "31 R 12" is a better answer. (I do doubt that many of the students giving those answers thought of it this way.) A more fundamental question is what is the optimal number of soldiers to transport. To me, when the question moved beyond just being arithmetic, the question's wording became inadequate. It also shows that just learning to do arithmetic is inadequate.

Seminymous Coward

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.

Bart

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.

Read more...

Corwin Delight

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.

Read more...

Corwin Delight

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))

Alan T.

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.

Read more...

James

"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 :-)

James

"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 :-)

JPB

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.

Seminymous Coward

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.

truzo

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.

Read more...

Brian

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.

Bob

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

Philip Crooks

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.

ajb8

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."

jagiela

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