Kitchen > PrivateWebHome > SubjectArea > LongDivision

select another subject area

# Entries from LongDivision

StrugglesWithLongDivision 07 Jul 2005 - 20:37 CarolynJohnston

I remember very clearly the problems I had with certain topics in mathematics. I remember getting confused on the day that my fourth grade teacher taught us how to multiply two-digit numbers by two-digit numbers (I had spaced off during the critical fifteen minutes when she explained the moves to us -- I was permanently spaced out as a kid, actually). That confusion was with me for a long time. So I thought I had a particular rapport with any kid who was struggling to learn math, having once been a kid who couldn't do math to save her life. My then going on to be a math Ph.D., and a math professor and researcher, made me what I thought was a pretty decent role model for struggling kids.

I was pretty good at teaching any topic, in fact, as long as Ben could learn it easily. We hit our first big bottleneck at long division. Multidigit multiplication was actually pretty easy for him; particularly since, in Everyday Math, Ben had learned this slick trick for multiplying multidigit numbers called lattice multiplication and was going to town with it. But long division was a different story. Ben had trouble lining up the columns, remembering to pull down the next digit after every step, and knowing where to finish his calculation and what to do with the remainder. Long after he had demonstrated that he knew what to do at every stage, he still couldn't reliably get the right answer.

I couldn't see that anything would help him master long division but long practice. He had learned all the steps and could apply them, but being methodical about it wasn't part of his nature. So, every night for a couple of months, I would give him several long division problems to do; it would always require several revisions before he would be done for the night. I could be what I needed to be -- a brick wall demanding that he apply care to his computations before he could consider himself done. What was doing me no good at all, just then, was my appreciation of the beauty of higher math.

The long division algorithm we all learned is actually just a repeated application of the Division Algorithm, which in its naked form, once understood, sounds obvious to the point of stupidity. The repeated application of the simple division algorithm with divisors that are decreasing powers of ten is just a thing of beauty, though, something written in The Celestial Great Book of Math. A lot of good it did us, though, in helping Ben to learn to apply long division. It took him a long time to learn to do that reliably, but we stuck with it until he got it.

There is the question of whether we even need to do this -- to torment students by making them practice the tedious long division algorithm -- especially now that computers and calculators are everywhere. It's claimed that such drilling kills the joy of math, and that we can teach children to love math better if we don't force them to do computations. I'm claiming (but not yet from any position of certain knowledge) that we do need to teach computation. I'm going by the fact that, in my association with mathematicians and physicists and engineers and computer scientists and finance people in my schooling and various jobs, I've known many people who could apply the long division algorithm, and some few who could appreciate its beauty; but I've never known a single soul who could appreciate its beauty without being able to apply it.

MathInTheBlood
ForgivingDivision
ForgivingDivisionPart2
TryThisWithForgivingDivision
TeacherGuideEverydayMath
EverydayMathEpilogue
ThirteenQuartersInTerc
HowNotToTeachMath
WhoSaysLongDivisionIsHard

SwoopAndSwoopPart3 23 Jun 2006 - 13:25 CatherineJohnson

As a child, I was never taught the reason why the cross-multiplying ‘trick’ worked when you're comparing fractions.

So when I read Carolyn's explanation (SwoopAndSwoop), I didn’t understand what she was talking about until I wrote out her fractions myself, and put in the missing steps.

TeacherGuideEverydayMath 07 Oct 2006 - 13:19 CatherineJohnson

Wow.

Speaking of sneaking a peak at the teacher's guide, it just so happens that I have open, on my desktop, a bunch of pdf files from the Everyday Mathematics Teacher's Reference Manual, Grades 4-6, The University of Chicago School Mathematics Project, Everyday Learning Corporation, Chicago, IL, 1999, ISBN 1-57039-515-2, pages 127-139, courtesy of one Tsewei Wang, Ph.D., Associate Professor, Department of Chemical Engineering, University of Tennessee and Concerned Parent.

Have I mentioned how much I love the internet?

Interesting to see that Everyday Math teaches the same Guess-and-Check algorithm for long division that's in Trailblazers.

Only, Trailblazers calls it 'Forgiving Division' (pdf file; search for 'forgiving division'):

Forgiving Division Method
(URG Unit 4 pp. 5, 6, 53; SG p. 113)

A paper-and-pencil method for division in which successive partial quotients are chosen and subtracted from the dividend, until the remainder is less than the divisor. The sum of the partial quotients is the quotient.

+ + +

So say you're dividing 239 by 3.

Instead of using math facts to know that 3 goes into 23 seven times, you start by guessing how many times 3 goes into 239.

+ + +

OK, let's divide 239 by 3 using forgiving division!

I'm going to start by guessing the number . . . 7!

I guess 7!

3 x 7 is . . . 21!

I write down 21 underneath 239, then I subtract, and I get . . . 218.

Whoa.

That's a lot.

OK, I'm going to use a strategy.

I'm going to guess . . . 10, because 10 is a friendly number.

10 x 3 is . . . 30!

I write 30 underneath 218, then I subtract----188.

Wow.

188 is big.

OK. 188. I'm down to 188.

. . . I'm going to try 10 again.

10 x 3 is 30, subtract 30 from 188, get . . . 158.

158?

Wait.

Wait.

I'm lost.

What number am I down to?

Oh. 158. I'm at 158.

OK, I'm going to try 20.

20 x 3 is 60, subtract from 158, get . . . 98.

Oh good! 98! That's really good! 98 is below 100!

Maybe I could try 30 this time.

30 x 3 is 90, subtract from 98, get 8!

Fantastic!

8!

8 is a really friendly number!

Now I can use my math facts and find that 8 divided by 3 is 2.

2 x 3 is 6, subtract from 8, get 2; 2 is less than 3, I'm done!

Yay!

Finally!

Now I add up all my partial quotients and the answer is------

7 + 10 + 10 + 20 + 30 + 2 = 79 remainder 2.

79 remainder 2!

That's it!

All done!

Bye Bye!

The end!

Forgiving Division

see:
The Many Faces of the Bitter Single Guy

and:

BlameTheTeacher
ProfoundUnderstandingFundamentalMathematics
ForgivingDivision
ForgivingDivisionPart2
TryThisWithForgivingDivision
ILoveTheWorldWideWeb
TeacherGuideEverydayMath
EverydayMathEpilogue
ThirteenQuartersInTerc
HowNotToTeachMath
StrugglesWithLongDivision
MathInTheBlood
WhoSaysLongDivisionIsHard
Everyday Math alternate division algorithm

keywords: Sponge Bob Bitter Single Guy

AboutLongDivision 13 Sep 2005 - 01:35 CarolynJohnston

(I actually wrote this post a couple of days ago, when my internet connection was down!).

Ben's half-brother is visiting for Memorial Day Weekend. It's always wonderful when Colin comes; in spite of their size difference (Colin, who is 16 and about 6'2", is more than a foot taller than Ben) there is a lot that they can do together; watch movies, play Nintendo, play basketball.

But, of course, learning still has to go on, and last night I insisted that Ben had to get some long division practice in. He knows the long division algorithm, and a few months ago I taught him how to divide by decimals. So now I am trying to get Ben to overlearn decimal long division, and the best way to do that is to get him to practice it.

So I handed him a sheet of paper with some long division problems on it and asked him to do them. He did them too fast -- too eager to get back to Colin and the Nintendo game -- and got most of them wrong. Not surprising, perhaps, but I'm looking for his long division skills to be so automatic that he can do them when most of his conscious attention is elsewhere.

I want long division to be a no-brainer for him, literally. It should be in his fingers.

He did the problems over again this morning; I stood looking over his shoulder to try to figure out what had gone wrong the night before. I was surprised at how good he actually is at the long division algorithm. He is, in fact, working out the few bugs left before he achieves mastery, and the distraction of Colin's presence had driven them out into the light.

If your kid is at or near the mastery point in long division, here are a few problems to look out for, and some sample problems that might help diagnose them.

• Uncertainty about what to do if the divisor does not divide the current number, after you bring the next digit down. For example, this occurs in the second step of dividing 92.0 by 9. The answer to this problem is 10.2222... a child who does not have this down cold will typically get 12.222222 for an answer, skipping over that lone zero.

• Uncertainty about what to do with a problem where the dividend has fewer decimal places than the divisor. One example of this is the problem 34/.21. In setting up this long division algorithm, the divisor and dividend should both be multiplied by 100: i.e., the decimal should move to the right by two places for both values, and the division problem should become 3400/21. A kid who does not completely have this nailed may get confused about what to do with the 34.

• Uncertainty about where to stop the long division process. Division problems that do not terminate should read, in general, something like "find the value of 213/14 to the nearest tenth (or hundredth, or whole number) ". A kid needs to be taught explicitly how to handle answering these questions. For example, suppose a problem reads: find 92.17 divided by 13 to the nearest tenth. Then the child should actually calculate the quotient out to the hundredth place, and round the answer to the nearest tenth. In the case of this problem, the child will get 7.09 as an answer through long division, and should round this answer to 7.1.

I would strongly advise against doing what I did last night -- that is, handing him ten juicy long division problems to do in a chunk. When faced with a lot of problems like that, my kid tends to lose hope of ever finishing, and despair makes him careless. Better to give him only three or four at a time, which I plan to do from now until he has long division down cold.

StrugglesWithLongDivision
MathInTheBlood
ForgivingDivision
ForgivingDivisionPart2
TryThisWithForgivingDivision
TeacherGuideEverydayMath
EverydayMathEpilogue
ThirteenQuartersInTerc
HowNotToTeachMath
WhoSaysLongDivisionIsHard

ForgivingDivision 10 Oct 2006 - 01:55 CatherineJohnson

It's official.

TRAILBLAZERS does not teach the standard algorithm for long division at all:

``` The paper-and-pencil method that Math TrailblazersTM uses to do long division is somewhat different from the way long division is traditionally taught in the United States. This method, called the forgiving division method, is often easier for students to learn. They do not have to erase as much, and they learn more about division and estimation. ```

from:
Letter Home (pdf file)
page 6
Division and Data

+ + +

If you were wanting to see what forgiving division looks like, page six shows a forgiving division of 644 by 7.

I'm surprised they actually tell parents this is what they're doing.

Of course, by the time you get the Division and Data letter you've been receiving TRAILBLAZERS PARENT LETTERS for years and you're still in the school. They probably figure they've worn you down.

StrugglesWithLongDivision
MathInTheBlood
ForgivingDivisionPart2
TryThisWithForgivingDivision
TeacherGuideEverydayMath
ILoveTheWorldWideWeb
EverydayMathEpilogue
ThirteenQuartersInTerc
HowNotToTeachMath
WhoSaysLongDivisionIsHard

ForgivingDivisionPart2 10 Oct 2006 - 02:30 CatherineJohnson

This is pretty droll.

Here's a parent asking Math Forum for help on his son's forgiving division homework:

``` From: Dan Bruce Subject: Solving division problems using the "forgiving" method My son has been asked to solve his division problems using the forgiving method, but he doesn't recall what this process is, and judging by the answers he's arriving at, he's way off base. Have you ever heard of this method and could you demonstrate it using the example 100/6? Thanks. ```

``` Date: 05/15/2002 at 09:49:17 From: Doctor Mitteldorf Subject: Re: Solving division problems using the "forgiving" method I'd never heard of the forgiving method, and couldn't find references to it in our archives. From a reference that I found in a discussion group on the net, I gather that it's about piecing together whatever multiplication facts you are comfortable with to solve the problem at hand. Suppose you want to know how many 6's there are in 100. You can remember that 7*6=42, so you write down the 7 as part of your answer, then take the 42 away from 100 and have 58 left. Next step: you might say the same thing. There's another 42 in there, so there's another 7 sixes. Write down another 7 under the first one, and subtract 42 from 58. Now you've got 16 left, and you know you can squeeze 2 sixes out of 16, but not 3. So you write down the 2 under your 7's and add them up: 7+7+2=16. You've pulled 16 sixes out of 100 (with 4 left over that wasn't enough to make another 6). You did it in groups of 7, 7 and 2, but someone else might have done 5 and 5 and 5 and 1, and the "standard" method would have been to do 10 + 6. The method is forgiving in the sense that your partial guesses don't have to be anything in particular, as long as you don't overshoot. - Doctor Mitteldorf, The Math Forum `````` ```

+ + +

Yup.

I can just see all the extra learning about division and estimation that's going on here.

TryThisWithForgivingDivision 10 Oct 2006 - 02:30 CatherineJohnson

Try it.

WhoSaysLongDivisionIsHard 13 Sep 2005 - 01:35 CatherineJohnson

EverydayMathLongDivision 13 Sep 2005 - 15:06 CatherineJohnson

Thanks to NYC HOLD I have a graphic of Everyday Math's substitute division algorithm. TRAILBLAZERS teaches the same approach, which it calls 'forgiving division.'

...instead of teaching long division, students are taught to divide numbers using the partial products method, a technique where children guess how many times a number goes into another and keep subtracting the guesses until they come up with the answer (see box). This method works, but it takes more time and doesn't allow the student to divide past the decimal point.

[snip]

Isaacs and others defend the alternative algorithms by explaining that they teach students how math works. The partial product method of division, for example, is a lot more transparent to students than the long division method.

I'm sure he's wrong about this. I found partial product division quite confusing myself when I used it.

otoh, I think partial product division might work as a teaching tool when used on simple demonstration problems. (I tried it on a complicated division problem and got completely lost mid-stream.) I might use a problem like 16 divided by 2 to show that division is repeated subtraction, analogous to multiplication being repeated addition.

I haven't tried it with any children just learning long division, but if I ever get a chance to, I'll take notes.

### the honeymoon

Some parents like the program as well. "It's sort of incredible," said Susan Pottinger, whose son Theo attends kindergarten at P.S. 261 in the Cobble Hill section of Brooklyn. "For him it's great fun. He's fascinated by numbers. He sees patterns everywhere," she said. "He'll put shoes away and alternate shoes with sneakers and say, 'See I'm making a pattern with my shoes.' "

We parents (well, some of us) spend those early elementary school years in a wonderland. Then the you-know-what hits the fan in 5th grade.

source:
Weighing the Factors Does the City's Standardized Math Curriculum Measure Up? By Amy Sara Clark

### update

Lone Ranger supplies this link to lattice multiplication, the method Everyday Math teaches children when they cover multiplication. Carolyn points out that lattice multiplication is distinctly opaque; it obscures rather than reveals the fact that multiplication depends on the distributive property.

Here's another link to lattice multiplication at Math Forum Carolyn posted awhile back.

why long division?

Everyday Math's alternative division algorithm
forgiving division
forgiving division, part 2
try this with forgiving division
who says long division is hard?
Everyday Math division algorithm fighting innumeracy at CO
conceptual understanding vs numbers

keywords: Columbiajournalismstudent EverdayMatharticle

ForgivingDivisionIsEasier 10 Oct 2006 - 02:33 CatherineJohnson

TRAILBLAZERS' rationale for replacing the long division algorithm with forgiving division:

Given the vast amount of time and the frustration involved in learning the long division algorithm traditionally taught in the United States, we instead use what we call the “forgiving method.” Sometimes it is referred to as the “subtraction method.” While this method may seem new, written record of it appears in a book published in 1729 while the first record of the traditional method appears in a publication dating from 1491 (Hazekamp, 1978). As with the traditional method, the forgiving method requires students to estimate quotients. The forgiving method is different in two ways. First, the student starts by estimating the entire quotient instead of the first digit. Secondly, if the estimate is too small, the student can continue with the procedure. This greatly alleviates the frustration of having to erase, and to some extent, allows one to get around a forgotten multiplication fact. (page 145, grade 4)

[snip]

Research has shown that low-ability students show better retention and understanding when taught division with this method and become better estimators of quotients. Students who were taught the forgiving method were better at solving unfamiliar problems and were better able to explain the meaning of the steps (van Engen and Gibb, 1956). Another study found that students who were taught both the forgiving and traditional methods did not confuse the methods and that the total amount of time needed to learn both was the same as the amount of time needed to learn one of the methods (Scott, 1963). Understanding rote procedures enables students to perform mathematical tasks with confidence and meaning. When children understand the mathematics they do, they come to believe that mathematics makes sense, and they are better able to think and reason flexibly. (page 146, grade 5)

[snip]

In this unit, an alternative division method is presented, rather than the one traditionally used in the United States. This method, which we call the forgiving division method, does not require that the greatest quotient be found at each step, eliminating the frequent erasing encountered with the standard algorithm. Research shows that students who are taught the forgiving division method are better at solving unfamiliar problems and are better able to explain the meaning of the steps in the method than those taught the traditional method (van Engen and Gibb, 1956). The forgiving division method also gives students the opportunity to practice mental math. (page 166, grade 5)

source:
TRAILBLAZERS background, grades K - 5 (pdf file)
page 167

We have quite a lot going on here.

First of all, we have an explicit statement that TRAILBLAZERS content is geared toward low-ability students. Not high-ability, not average-ability. Low-ability.

Do parents know this?

Second, we have an explicit statement that the authors of TRAILBLAZERS have opted to replace the standard algorithm with the forgiving version because the standard algorithm takes too long too teach ("a vast amount of time") and is too hard ("the frustration involved").

These observations strike me as correct. From what I gather, it does take quite a lot of time & frustration to teach the standard algorithm, although I question how much frustration would be involved using Singapore Math, Saxon Math, or Direct Instruction.

The problem with this line of reasoning is that the standard of diminishing returns has not been applied to activities like Antopolis.

Thirdly, and mystifyingly, we have the inevitable Research Shows passage in which we are assured that in fact it takes no more time to teach forgiving division and long division than to teach either one on its own. That strikes me as unlikely, regardless of what 'research' does or does not show. Under normal circumstances, learning two things takes more time than learning just one. But, supposing the research is right, the obvious question is: Then why aren't you doing it? If it takes no more time to learn both algorithms, and if it's a good idea to learn both algorithms, then—hey! Teach both algorithms!

(For me it almost certainly would have been helpful to have studied both algorithms, though it would not have been helpful to practice both to mastery.)

question

I could probably think my way through this one, but in the interests of efficiency I'll ask you.

Can you do decimal division using forgiving division?

I'm not instantly seeing how that would work....

update

Which means you can't use forgiving division to convert a fraction to a decimal. The Trailblazers grade 5 Student Guide tells children to use their calculators to accomplish this task.

wit and wisdom

This is funny.

TRAILBLAZERS grade 4 has a lesson called, "Oh, No! My Calculator is Broken."

This is Lesson 3 in Unit 7, Patterns in Multiplication.

The Key Content in "Oh, No! My Calculator is Broken" is:

• Recognizing that there are many strategies for doing simple multiplication problems
• Using efficient strategies to do multiplication problems involving the last six facts
• Using the calculator efficiently in problem solving
• Communication problem-solving strategies

I'm wondering how you use a broken calculator efficiently in problem solving.

why long division?

JDOnLongDivision 29 Nov 2005 - 16:43 CatherineJohnson

Another incredible post from J.D. at Math and Text—this time on long division. (Speaking of synchronicity.)

Here's the final image:

SubtractionDivision 29 Nov 2005 - 16:03 CatherineJohnson

24
÷2
12

My kids are doing division this way, and one of them asked how to divide 92 by 2. He thought you couldn't do it, because 2 doesn't go evenly into 9.

update

The TRAILBLAZERS scope and sequence chart shows that TRAILBLAZERS does not teach an algorithm for long division until the end of 4th grade.

NyuMathMajor 03 Oct 2006 - 01:13 CatherineJohnson

Ed talked to an undergraduate majoring in math today.

I guess the kid spontaneously told Ed that, "Calculators are the worst thing that ever happened to math students."

Ed said he almost burst out laughing, because next this student went on to say that nobody who used calculators as a kid can do fractions, and if you can't do fractions you can't do calculus.

Ed said this guy could have been channelling me.

The student also said that, in high school, his calculus teacher had told the students who were having trouble, "You're having trouble because you used calculators in grade school and you never learned to do fractions." It was obvious to her. He spent quite a lot of time describing automaticity to Ed, and how important it is.

Ed asked why he hadn't used calculators as a child, when everyone else was, and the answer was chilling: he hadn't used calculators because he 'was into' math, he liked it, and he wanted to do the calculations by hand.

What that tells me is that only the natural born Math Brains are going to make it through these days — natural born Math Brains who know they're natural born Math Brains.

Your basic kid is going to use the calculator if the teacher hands it to him.

Then he's going to regret it later on.

That's what happened to the other kids in his high school calculus class.

Ed asked him whether the kids who'd used calculators could catch up.

The kid didn't think so. At least, he hadn't seen it happen.

Math is hard, he said. It's hard, it takes a long time to learn, and he didn't think a high school student who'd lost that much time could make it up.

That's what James Milgram said, too.

no calculators in Irvington

I don't think any of the grade school kids here are using calculators.

One of main criterion for choosing a new math curriculum was (paraphrasing) 'constructivist approach.'

One of the other main criterion was emphasis on math facts & computation.

TRAILBLAZERS was the only constructivist curriculum they considered that stressed fluency in math facts.

(I assume they're teaching the traditional long division algorithm in spite of the fact that TRAILBLAZERS teaches 'forgiving division,' but I don't know. Nevertheless, nobody's passing out baskets of calculators.)

Good for them.

which reminds me

I had to buy Christopher an expensive graphing calculator (or something) last fall, for Middle School.

He never used it once, and then finally lost the thing.

Good riddance!

His teacher is letting them use calculators for the first time this year, to calculate circumference & area of circles. I'm not even sure that's such a good idea.

Since he's doing KUMON, though, I figure it's OK. He's incredibly fast & accurate on the KUMON sheets.

Of course, the two "Fraction Levels" - E & F - are killers.

-- CatherineJohnson - 14 Feb 2006

KippGoesToKindergarten 04 Oct 2006 - 16:11 CatherineJohnson

Trying to track down a Jay Matthews column on St. Anne's school in Brooklyn, I came across this column saying KIPP has started an elementary school in Houston.

That's good news.

And check this out.

They're combining Saxon Math with Everyday Math:

At SHINE, Brenner says, he is blending the more modern Everyday Math with the more traditional Saxon Math for first-graders. The proponents of those two teaching programs have been at war for 20 years; can combining them really work? I'd predict that joining such radically different elements would cause an explosion, like when I used to toss manganese shavings into the surf to illuminate beach parties.

Brenner seemed unfazed by my doubts. "Our kids are off the charts in math," he says. I haven't surrendered my skepticism, but I will visit his school, and then watch what happens when Laura Bowen brings all this here, where Washington can get a really good look at it.

I'm not surprised.

My friend with the kids in the fantastic private school told me her school combines Everyday Math with traditional math. They seem to do nothing but EM for the first couple of years; then they shift.

I was shocked when she told me this, and assumed that her kids were getting shortchanged.

Then she faxed me her son's math homework.

WAY past anything kids are doing in public schools. This boy was doing long division with a gazillion digits; no forgiving division anywhere in sight. The word problems were serious and challenging - challenging at his level. My friend was shocked that we have to reteach math at night. She and her husband never reteach any subjects at all. The kids in her school are way up at the top of U.S. kids, and they're learning everything they know at school.

Barry has mentioned before that James Milgrim thinks Everyday Math would be a good supplemental program when used with a traditional math curriculum.

Looks like he's right.

-- CatherineJohnson - 12 Apr 2006

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson

I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006