New Method for Teaching Division

[Unregistered]

Double Division - a method for doing division
http://www.DoubleDivision.org

I found a new method for doing manual division that ?may? be better than long division in a number of ways. It might be useful to teach this method BEFORE teaching long division, or maybe INSTEAD of long division.

I think it's simpler and more intuitive - but I may be wrong. Please check this out and give me feedback. Feel free to forward this to anyone who may be interested - or suggest other places that I might post.

Thanks,
Jeff

[Lisa's Hotscakes]

So, if students are taught this method instead, how well will they do when they get to division by polynomials in algebra?

I would caution anyone from teaching methods that are off the beaten path. Many of the techniques we use in the higher grade levels depend on these subskills.

BTW, I found the method long and convoluted. Besides, what is wrong with guessing and trial & error? You can develop a good feel for numbers this way.

[Unregistered]

http://www.doubledivision.org

You raise some good points.

Double division is a new method, but that doesn't mean it is not better. We have to be open to new ideas.

I would agree that the method is longer, but definitely not convoluted. I think it is much more obvious exactly what is happening.

Long division teaches how to follow a long procedure and it gives some practice multiplying and subtracting. I'm not sure how many people understand what is happening when you bring down the next digit, or understand why you have to add a zero to the answer when the "number after subtracting" is less than the divisor.

About polynomial division, on the one hand long division is more similar to polynomial division because in both cases you are choosing a multiple of the divisor from out of the blue. But on the other hand, double division does a better job of reinforcing the idea that division is really subtracting off multiples of the divisor. This may prepare students for polynomial division better.

I would sum it up like this:

Reasons To Teach Division:
- to teach mathematics
- a method to actually use in rare instances

Double Division Advantages (compared to long division):
- simpler procedure
- teaches how division works
- no trial and error
- gives practice doubling numbers

Double Division Disadvantage (compared to long division):
- more subtraction
- more steps (see note below)
- requires more space on the paper
- may not lead as directly to polynomial division - arguable

Your thoughts?


IS DOUBLE DIVISION LONGER?

If we assume there is an equal chance of all ten digits being in the answer then on average there will be 1.5X as many "multiple and subtract" steps. For example a "7" in the answer requires 3 steps, and a zero in the answer requires no steps.

Also remember that the multiply part of the "multiply and subtract" steps is already done for you. So this part will be faster. Of coarse you have to pre-multiple the divisor three times in the beginning.

In the end I think it is longer, but not as much as you might think initially.

[Lisa's Hotscakes]

You raise some good points.

You don't even realize how many enemies you made with this statement. :D

About polynomial division, on the one hand long division is more similar to polynomial division because in both cases you are choosing a multiple of the divisor from out of the blue. But on the other hand, double division does a better job of reinforcing the idea that division is really subtracting off multiples of the divisor. This may prepare students for polynomial division better.

Well, I disagree. I did some examples using double division and found it a little confusing. I don't see how it helps understand polynomial long division, as I am not sure how the concept of repeated subtraction is so important in polynomial long division. Sure, you can repeatedly subtract polynomials from other polynomials until you have a remainder, but I am not sure how important that concept is to an algebra student. This would be a good issue to research.

On the other hand, the idea of the remainder I think is clearer with double division, which would help with polynomial long division. Students need to understand why the remainder has the same polynomial denominator as the original rational polynomial -- a crucial point in complex algebra).

However, we certainly don't want algebra teachers throwing their hands up in disgust when they try to teach polynomial long division to students who are not familiar with traditional long division, so it may not be worth it no matter how many advantages it provides.

[Unregistered]

Hi Lisa,

Well if you've done a few examples, they you have done more double division than anyone in the world. I've probably done it four times - so you might be second.

How did I make enemies by saying you raise some good points? I don't mean to make enemies.

To me the process of guessing and trying different multiples is uncomfortable. How many times does 372 go into 2711 - I don't know. I also never know what where to start writing the answer.

It honestly seems easier but a little longer to me. You just double, double, double, then pick the best ones to subtract, then add up your answer.

The notes in the long division example below seem pretty confusing, "714 is really 714,000 because we really multiplied back by 2000..." "How many 300's are in 1942? Try 5 or 6..."

long division:
http://members.aol.com/loydlin3/Ldivide.jpg

double division:
http://doubledivision.org/Ddivide.gif

To me there is very little understanding of how long division works, whereas double division is very obvious - but I've already said that.

I am not a teacher. I thought of this after a friend of mine was asked in a job interview to write a program to do division using only addition, subtraction, and multiplication. When I did the same (for fun) I had to think about how division works, and then I tried to create a simpler method for kids. My only goal is for enough people to consider it so that if it's helpful people can use it.

If nothing else this method teaches that division is subtracting off multiples of the divisor. That's what your doing in long division also, it's just harder to tell.

Do you think this could be useful to teach BEFORE long division?

Jeff

[Unregistered]

I don't see long division and double division being that different.

1) In long division you guess which multiple to subtract where as in double division you pick from four options.

2) In double division you write out the zeros so it's clear how big the numbers are.

3) In double division you write your answer on the side where you have room to accumulate many parts of the answer. In long division you have to get each digit of the answer exactly right. Where as when you write parts of the answer on the side then you have the option of choosing exact correct digits of the answer, or choosing smaller digits and adding them up.

It seems to me that you could do polynomial division and write the answer on the side if you wanted. Long and Double division are just not that different. Either one can be extended to polynomial division.

Another thing to consider is many students will never get to polynomial division. If it's 5% more difficult for them, but many more students get a better understanding of division and the distributive property - then the trade of may be worth while.

Jeff

[Lisa's Hotscakes]

I don't mean to make enemies.

Then repeat after me...

Lisa, you ignorant slut! IT IS JUST LIKE YOU TO CRITICIZE!!! You're typical of the corporate whore-monger capitalists that want to round teachers up and shoot them in the back of the head!

I am not a teacher. I thought of this after a friend of mine was asked in a job interview to write a program to do division using only addition, subtraction, and multiplication. When I did the same (for fun) I had to think about how division works, and then I tried to create a simpler method for kids. My only goal is for enough people to consider it so that if it's helpful people can use it.

Well, I commend you for conjuring a clever approach. I think it has its merits, but in a different way than you. To me, the real value in double division is it reinforces the concept of the remainder, which often perplexes students. But with double division it seems to me that the remainder is simply what you are left with after the repeated subtraction.

My concern is that algebra teachers will object since they currently rely on long division skills to teach polynomial long division.

If you want teachers using it, just tell them that I hated it. The other posters in this forum will be lining up to learn it. :D

Do you think this could be useful to teach BEFORE long division?

Yes, if teachers can stay on their pacing guide.

[Unregistered]

When I saw your double division method, I was very hestitant. I've dreaded teaching my 6th graders long division the past five years because it is usually a painful process. They learn bits and pieces in 4th and 5th but there seemed to be a huge jump that I feel that most of my class really and truly gets. After two review lessons on it, I taught my class the double division method and I found myself checking work where there were more right answers than wrong! Even the kids who are still struggling with double-digit division were really excited about this method and tended to go through this problem with fewer errors (although their parents were bewildered by the method) although three of my students have chosen not to use this method because it was "more work". I think this method will make into my teaching practice regularly even though I must teach the traditional algorithm.

[Unregistered]

http://www.doubledivision.org


That's cool.

Please email me and tell me how you taught it.

My email address is:

wilson [at] silcom [dot] com

Thanks,
Jeff

[Unregistered]

My niece was taught a similar method of doing division. When she got into sixth grade she had a terrible time with division because her teacher would not allow them to use the new method. After learning the new method, she had trouble learning long division. By the way, she has been the same school system since she stared school. She is also a very bright girl and gets very good grades (liong division being the exception). I think she finally got the hang of it but there was a lot of frustration. Also, in the school system she is in most students take Algebra, many in the seventh or eighth grade ( I should know I graduated from there and have three nieces in the school system. My nices are in the 7th, 8th and 10th grades).

Brandi

P.S. One of my own daughters learned long division last year when she was in third grade and had no problems.

[Unregistered]

It sounds like the math curriculum at this school was not planned out as well as it should have been, and your niece suffered for it - that's unfortunate. I'm glad she got the hang of it in the end.

The fact is that long division is built into most math curriculums and that fact is not going to change any time soon. The question is whether double division (or "1-2-4-8 Division") can help in teaching long division by reinforcing the principles of division and giving students success with a less frustrating alternative.

My only goal is to make it available for people to consider, and so far the feedback has been mostly positive - but not all positive.

Jeff
http://doubledivision.org

[Unregistered]

Depends on what you mean by "positive." Does positive mean, "They agree that this is the way to go?" or does it mean, "It was constructive"?

My opinion is quite simple: It is a clever approach that could be used to supplement long division, but it is no substitute for long division. Is that positive?

[Unregistered]

Yes, by "positive" I mean constructive. No one has thought that this method should replace long division.

Given that long division is built so strongly into our system, the question is whether this method has a place at all? I don't really know.

I've seen that people teach this method without doing the doubling at the beginning - the goal being to show that division is subtracting off multiples of the divisor.

Doubling the divisor in the beginning makes it into a more practical method - that is also instructive.

If it's done right, I could image kids learning double division first and then learning long division as a faster way - and in the end understanding long division better.

Jeff
double division - long division teaching aid (http://doubledivision.org)

[Unregistered]

I think we can agree.

[slowwalker]

I have been teaching a similar method for 6 years now and it really is quick to understand and learn. I will try this double, double "Tim Horton's Coffee" method tomorrow! (Ah Canada)

Division is, really in our lives, the act of taking groups out of a pile and spreading it around. As in.....
I have 5 678 candies to share with 7 people.(lucky them!).

I want to deal with friendly numbers that end in zeros because they give me 'zero' problems to subtract.

If I know my multiplication facts, I know that 7X8= 56, so I could take out 5600 candies and give each person 800. 7X800=5600. I have taught the students to multiply 7X800 to be "56 and 2 zeros". Who ever says the full mane when calculating anyway! If I do not know my facts then I could take out any lesser multiple that I do know! Teaching multiplication before division MUST cover the Multiples of 10, 100, 1000 first!

Then 5600 is a snap to subtract FROM 5678 and find there are only 78 remaining candies to divide up!

Well 7X10 is 70,,, so take those out and give each person 10.

This leaves only 8 candies so I give each person 1 more. The remaining one candy is a problem for fairness, so I eat it!!!

How many candies did each person get? Well, I gave each person 800 and then 10 and then 1. Must be 811 each!

So simple you can do it in your head.

And of course double digit division is the same. Just nibble away at the pile and subtract 'zero' numbers. (multiples)

The student does not need to know his/her facts as well as in the long division method. But they will see that if they learn their basics, they can do the questions faster!

The long division method seems to teach rules but this other way seems to teach understanding!

Just my thoughts!

[Jeff Wilson]

(I could go for a Tim Horton double double right now.)

Here is a paper on teaching long division that you may find interesting.

http://www.math.technion.ac.il/~ra/englongdivision.doc

Let me know how it goes doubling the divisor. What age children are you teaching?

Jeff
(I finally registered)
double division (http://doubledivision.org)

[Unregistered]

Our school just adopted the EVERYDAY MATH series. It is published through the University of Chicago.

The series teaches this method as an alternative way to divide (called partial quotient division). They also teach partial sum addition, partial difference subtraction and partial product multiplication.

Basically, it gives children the option to use this method or the traditional method.....I get mixed reviews from my students, but I stress that either way is acceptable.

[Unregistered]

But again, teachers in the future may require that students know long division.

We need to keep this in mind.

[Unregistered]

From what I've seen of the UofChicago partial-quotient division method--it serves to help understanding but sadly it is not helpful in a practical sense (higher levels of mathematics). Some students who are taught this method and prefer it over standard division are totally incapable of doing "regular" division and continue to revert back to partial-quotient, simply because they have not been taught to estimate with large numbers. Similar to the lattice method in multiplication, this method serves to delay learning of fundamental operations.

[Unregistered]

I teach numeracy and literacy in an Internet exchange centre, one of the stipulations is that we pass the very test that we are teaching. Your practical solution to long division is fantastic. It makes so much sense. A number of my learners have complained that there must be a logical technique to long division that does not involve the random multiplication guesswork.
This technique is a God send

Double Division - a method for doing division
http://www.DoubleDivision.org

I found a new method for doing manual division that ?may? be better than long division in a number of ways. It might be useful to teach this method BEFORE teaching long division, or maybe INSTEAD of long division.

I think it's simpler and more intuitive - but I may be wrong. Please check this out and give me feedback. Feel free to forward this to anyone who may be interested - or suggest other places that I might post.

Thanks,
Jeff

[Unregistered]

A number of my learners have complained that there must be a logical technique to long division that does not involve the random multiplication guesswork.

I think students learn more about numbers when they continually guess the products. Long division is a better teaching tool, in my opinion, and more versatile.

[owen]

You said:
"I found a new method for doing manual division that ?may? be better than long division in a number of ways. It might be useful to teach this method BEFORE teaching long division, or maybe INSTEAD of long division. "


I looked at your idea but I did not think it was any better. In fact I found it very hard to follow. If I have trouble with it I think children will have a problem with it as well. It just has too many more steps.

[Unregistered]

Hello Owen,

I said "may be" better because it is new and I don't know if it's better or not.

I am actually teaching it for the first time to my daughter now, and I'm learning a few things in the process. (We've only spend a half hour on it so far.)

Observations:

1. She has to know regular long division because that what they are teaching her in school.

2. She does not really "understand" either method now. Now I am just showing her the steps; double three times, subtract, add up the answer. I am thinking of ways to help her understand division.

3. The method itself (without considering understanding) is no more or less complicated than long division. It's just different. She is picking up this method fine. I don't know how long it has taken her in school to learn the long division method.

4. HERE IS THE BIG DIFFERENCE. I'm seeing the new method is longer BUT does NOT require knowing your basic division facts - which she doesn't know very well. It's a longer but easier procedure. It also give practice doubling numbers - which is a very timely thing for her to be practicing. (Of coarse she does need to know her division facts.) It will be interesting to see what she thinks of each method in the end.

So I don't know if it's better or not. I do think that knowing many ways to solve a problem is good. I think (but haven't experienced yet) that this will help her learn more about how numbers work. I also think the advantage of not having to do the individual divisions will get bigger and bigger as the problems get bigger and bigger.

jeff

[Unregistered]

Just a few thoughts on this matter.

Why do we want students to be able to do "long division" or "double division" or any number of other possible methods for doing division? Is it to be able to divide 2 numbers? If that is the case then we should strive to teach the method that is "easiest". But I don't think the point of teaching students to divide is so that they can compute an answer, especially in "non-easy" cases. When presented with a problem like 176352476/9967387 are you more likely to get out a piece of paper and compute by hand or just to grab a calculator. I think most people would just grab a caclulator, that then tells me that the purpose of teaching division is not to teach students to divide but to perhapes teach them how to think... how to problem solve... not how to compute. Our classical "long division" algorithem has some very nice advantages for future math topics that have little to nothing to do with division. The fact that you have to "guess" the "correct" number to divide nicely is nice practice for estimation techniques, for finding factors of higher order polynomials, for picking a theorem or lemma to enable one to do a geometry proof, for decideing which integration tequniqe to use, to find a sufficent delta given an epsilon to show continuity, to pick a relavent test to show that a series converges, the list goes on. We have so many methods in math that don't have a set method to do, instead there is a list of suggestions of things to try at best and just blind guessing by the students at worst. Long divison is in my opinion a lovely first expossure to a part of math where the procedure to follow is not so clear that there is no guess work.

On the argument for "long division" because of later links to polynomial division I don't think there is a need to have had the one before the other. Although the sturcture of the divsions is the same the method is actually somewhat different. When doing "long divison" we "guess" the number that "best" divides. With polynomial division there is no guessing, if we want 3x into 6x^3-4x our first step is to ask what do we multiply 3x by to get exactally 6x^3. We ignore the rest of the polynomial. We could very well do the same thing when dividing numbers by modifying our standard "long divison" tecnique by instead or adding the factors ....a*1,000+b*100+c*10+d*1+e*.1+... to get our answer alowing subtration as well. Of course there is a reason that we don't teach this method it gets rid of the guess work but forces us to at times use fractional multiples of powers of ten as our answer. So to get a "nice" number as our answer we would at times have to add 3/7*10,000 and 3/9*100 and other such terms.

Anyway that's just a few more thoughts on the subject.

[Unregistered]

When doing "long divison" we "guess" the number that "best" divides. With polynomial division there is no guessing...

True, but other than the fact that the leading coefficients of the quotient are fairly easy to discern, the two methods are very much the same.

I do agree with your assessment of the utility of long division.

The fact that you have to "guess" the "correct" number to divide nicely is nice practice for estimation techniques, for finding factors of higher order polynomials, for picking a theorem or lemma to enable one to do a geometry proof, for decideing which integration tequniqe to use, to find a sufficent delta given an epsilon to show continuity, to pick a relavent test to show that a series converges, the list goes on.

Agreed, 100%. A large part of the field of numerical analysis is based on a "feel" for numbers, and methods that automate the process circumvent this skill.

[Unregistered]

You both raise some good thoughts, thank you for your input. I would agree with almost everything that you said, except I would add a couple points.

First, I do think there is some value in teaching a method that people can actually use to get an answer - because there are times when a calculator is not available. My guess is that people would have an easier time doing double division 15 years out of school than long division.

Even so I agree that the main purpose of learning division is to teach about numbers and how to solve problems. And I agree that long division helps teach future math concepts like how to estimate and the others you mentioned - most of which I've barely heard of even being engineer.

The point I would like to add is that double division may teach and reinforce the more basic concepts of what division is and of the distributive property of multiplication. These are the concepts that elementary school students need to grasp first, and these are the concepts that regular people need in regular life.

No doubt my daughter needs to be able to do long division, and long division will help prepare her for higher math. But double division will help her with lower math. At least that's my prediction and I'll know better in a year or so.

Jeff
http://www.doubledivision.org

[Princess]

is dis a chat rrom

[Unregistered]

I gave up after a few minutes of trying to figure out how to "do" the double division! I am truly interested in it but I don't have the time to spend!
This just goes to show how many students feel about mathematics in general. If it is doesn't look easy, then don't bother.
I will look at it again. I am glad some others have posted that it is not easy to do, it makes me feel better.
A comment about the "need for long division": I agree with the posts in general. Long division takes up too much time in elementary curriculum but it is a good algorithm to know for future mathematics. However, if someone is having problems with learning long division, then they will 'probably' be having problems with future mathematics so they will not be going on into advanced mathemathics anyway... Why bother?
I have argued this on other websites and get the response "we can't predict if someone will become interested and good at maths so we need to teach it to them". I agree, education is "equal opportunity".

[Unregistered]

I gave up after a few minutes of trying to figure out how to "do" the double division!

I made Double Divison Calulator to make it easier for people to see how the procedure works. Just click "Start" and then "Next Step", "Next Step", "Next Step" ...

See:
Double Divison Calulator (http://www.doubledivision.org/)

Jeff

[Unregistered]

Thank you for posting the calculator.
It makes sense now. Since I don't teach
elementary, I won't be using the method
much but it will be good to have secondary
students look at it to see what they think
about it...

[Unregistered]

Thanks Jeff, I'm going to try this. It is logical and makes sense for me to (re?)introduce it to year (Grade) 7s, remediate some Year/Grade 8s and some Year 9s who have a mental block about long division. I think it should work. Don't take too seriously what the 'negative' people are saying - perhaps they are afraid of something new or something they don't understand. Me, I'd try standing on my head if that may work!
Cheers,
Laura :)

[Unregistered]

Don't take too seriously what the 'negative' people are saying - perhaps they are afraid of something new or something they don't understand.

It is all about what is best for the students. Our own egos are irrelevant. I don't knock the new method of division because I don't like it; rather, I knock it because I don't see the value in it for my students.

[Unregistered]

It will be interesting to see if this method catches on at all.

Over the last month about 50 people per day have found the site in the search engines - 2/3 from Google. Another 20 find the site by following links from message boards like this. I can see that some are following links from within email messages.

Most people from search engines are searching for "long division," "teaching long division," etc. - so they do not stay on the site too long - but most (1/2 to 2/3) people will go thru one example on the calculator.

I can see what buttons people press and how long they stay. It's interesting that a few people per day spend a half hour or more doing various problems on the calculator. - So people are learning how to do it.

I imagine more and more teachers will actually try teaching it to their students and we will eventually learn whether it's valuable or not. (So far one teacher has taught it to a sixth grade glass and thought it was valuable.)

Jeff
http://www.doubledivision.org

[Unregistered]

I learned a way of division that is similiar to double division the only difference is that you use multiples of 10 and subtract and keep guessing until you have 0 or begin repeat. Then you add up the numbers on the right and that is your answer. Alot easier than doubling the divisor.

[Unregistered]

Using just multiples of ten is an option. People teach this method to help demonstrate how division works and to reinforce the distributive property of multiplication.

I don't think this method is taught as a practical method because there are so many steps. For example dividing 85434 by 37 to get 2309 would take 14 steps, as opposed to 5 for double division, and 3 for long division.

The idea of double division is that it can teach how division works like the "multiples of 10 only" method, but is also a practical way do to division - that doesn't require trial and error guessing.

In general double division will require 1-1/2 times as many subtractions as long division, where as "multiples of 10 only" will require 5 times as many subtractions.

The calculator at http://www.doubledivison.org shows how to do it step-by-step.

Jeff

[Unregistered]

Why do they make kids do this, and where did you learn it? My 4th graders are learning partial products, a differnt way to divide.

[Unregistered]

I was taught a method of division so very similar to this 20 years ago it's not funny, by my father who learned it from a text book called The Trachtenberg Speed System of Basic Mathematics because I was unable to master the normal division method. I went on to master algebra, trig, and calculus without any permanent damange from learning an alternitive method, and have taught my son alternitive methods that cause his teachers to call me and ask me what he is doing on the page because they are clueless.

Math is the manipulation of numbers... there is no one path, only one right answer.

Lugh

[Unregistered]

It's wonderful to hear that your son's teachers call to try and figure out why he is doing what he is doing. There have been many instances of teachers not being aware of "alternative" (but still correct!) way's to do problems and just marking students wrong and assuming that they arn't learning how to do math. Where what really happened is the student went home and got help from mom/dad/grandpa/whoever else and was taught some other method. This is especially a "danger" in area's with large populations of people origonating from non-american cultures. Many people are supprised to learn that the standard algrithums that we (US) use for addition, subtraction, multiplication and division are not always the same as the standard algorythms that other cultures teach.

One feels sorry for the kid's that came over from another country and are struglling to fit in already, they don't understand the math in class and go home and spend time trying to learn it from a parent who then teaches them in the manner that they learned. The kid thinks all is well and his teacher will be so happy that they learned and instead the teacher marks all of their problems wrong. How devestating for this child!

The people reading this topic are likelly not the type of teacher to not be aware of alternative algorythms (after all that is what this topic is all about) but a great number of US elementry teachers struggle with math and have little confidence in their own ability to deal with math. (The idea of adding fractions with dissimilar basses has struck fear into the hearts of many young teachers when faced with the fact that now they must teach this idea, when they don't even know why it works). This lack of confidence leads to the overuse of teaching manuals and answerbooks and inevitablly marking answers wrong without any thought other than it not being exactally as in the answer key.

[Mister D]

I teach 5th grade, and am willing to use whatever method students are able to grasp. Many have a hard time with the traditional algorithm.

I'm finding the partial quotient method very helpful. I found a short demo in wmv format here. http://mb.msdpt.k12.in.us/Math/PartialQuotients.wmv, and will be trying out the doubling method too.

I see the key in this is building the students' number sense, then pushing them on toward more efficient strategies. Eventually, I want my students to be able to do the "regular" division algorithm.

[Unregistered]

Thats really great! I forun this @ http://www.forumvadisi.com. Thanks...

[Unregistered]

Wow! What a great way to teach division to the kid who is not "getting it" the "traditional" way!
I disagree with those who are not able to see how this can help! It's a shame that some people and teachers are not open to other alternative methods to teach our children concepts. People in general, just get stuck in there way of thinking, and feel that there is no other method, or concept that can work. I applaud you for undertaking this method and creating a web page that we can easily understand. To those who think it's difficult to understand at first, take a few minutes of your time to learn this method- it took me under 5 minutes. (I am someone who has difficulty with math concepts, myself.)
You are truly innovative! I for one, thank you for the creative web page for teaching this concept as a different approach for a difficult concept (long division) for most upper elementary school children.
Thank you from my home school son, has been struggling with long division for the last 3 years! Your method has helped him, enormously! He is a bright child and grasp's concepts rather quickly (with the exception of long division). Long division has been an area that he has not been able to easily able to grasp since 3rd grade (at public school). We have struggled with it in 4th and 5th grade (home school). All of the traditional methods have just been more confusing to him Imagine the difference for him if his 3rd grade math teacher had taken the step to teach him this alternative method. I wish I had found your page sooner! In the past 2 years I have been all over his supervising teacher (at with his home school program), the Internet, and have spent over $100 on manipulative's and teaching division books, to no avail. In a afternoon, with your method he mastered his dread of long division!
This upcoming year he starts a private school, and this is the one area that he has been having difficulty with, and embarrassed about. Not being to do long division confidently (he is a perfectionist). Now, with your method he feels able to tactical a difficult issue (Long Division now converted to Double Division) with confidence.
For all the naysayers out there, this might be the one method that can help a child having a difficult time grasping the traditional method of long division. Imagine what a wonderful feeling it is to see the children understand something that has been a thorn in their side, and come out the other end with a better understanding of math concepts all together! It may not be for every child, but neither are traditional methods, in teaching all children, all the time. Students are individuals and some grasp different concepts in different ways. As teachers we should be open to every possible idea, and method. Sometimes one size fit does not work, for every child!
For the teachers that are open to this and other alternatives, I say Thank you, as a parent for caring enought to help children learn.
You are the true hero's in our culture today!
THANK YOU, once again!

[Unregistered]

It's a shame that some people and teachers are not open to other alternative methods to teach our children concepts.

And it's a shame that some suffer from a lack of reading comprehension. The problem with alternative methods is that future math topics, such as long division of polynomials, are based on the traditional method.

The importance of division does not rest so much on the answer - a calculator can provide that. Instead, it's the process of dividing that is important. Therefore, I urged caution whenever replacing a traditional method that one THINKS is easier. As a supplement for those who struggle, fine; but the traditional method needs to be taught.

[Unregistered]

And it's a shame that some suffer from a lack of reading comprehension. The problem with alternative methods is that future math topics, such as long division of polynomials, are based on the traditional method.

The importance of division does not rest so much on the answer - a calculator can provide that. Instead, it's the process of dividing that is important. Therefore, I urged caution whenever replacing a traditional method that one THINKS is easier. As a supplement for those who struggle, fine; but the traditional method needs to be taught.

To attack me and feel that I am not comprehending what you and others are saying, as you are attacking this alternative method is showing your own ignorance. Please, realize that what I stated is, "that traditional methods do not always work for every child, everytime". I am not knocking traditional methods. Just the need for some children to to be shown alternative methods for a greater understanding of the concept. Again, I repeat myself in stating, that "one method does not work for every child"!
I exampled my son who was a child who struggled with the traditional method of long division for 3 long years. This is not the easy way out, this method provided an new approach for a difficult struggle.

I am thanking the author, Jeff for creating a place that we can go and learn this method and teach it to our students. Do not mistake my gratefulness with your ignorance of trying "new" approaches.

Here lies the real issue of "some" teacher's without forward thinking.

Teachers, I sure hope that you do not view this and other alternative methods of teaching as a threat your traditional methods, just view it as another learning tool to help your students. Please, please.... do not let your ego take over, and distract from the real goal of teaching children, so that can better understand and comprehend.

[Unregistered]

To attack me and feel that I am not comprehending what you and others are saying, as you are attacking this alternative method is showing your own ignorance.

First of all, you already insulted many of us in here by calling us closeminded simply because we pointed out some criticisms of an alternative method. Now you get upset because you are, in turn, "attacked." I did not appreciate being called closeminded, and I bet others didn't appreciate it either. You should be more mindful of your own tone when posting.

Please, realize that what I stated is, "that traditional methods do not always work for every child, everytime". I am not knocking traditional methods. Just the need for some children to to be shown alternative methods for a greater understanding of the concept. Again, I repeat myself in stating, that "one method does not work for every child"!

And let me repeat myself: It is the process of division, not the answer, that is important. Yes, there are many methods of arriving at an answer when dividing, but a calculator will provide the answer just as readily. However, the traditional method of division contains a process that is identical to that one used when dividing by polynomials. So replacing the traditional method with one that is "easier" is not necessarily the best way to go.

Now I have explained this numerous times in this thread, and so far you haven't even bothered to acknowledge it. If you disagree my analysis, fine. Let's hear it.

Teachers, I sure hope that you do not view this and other alternative methods of teaching as a threat your traditional methods, just view it as another learning tool to help your students.

I don't have any emotional attachment to mathematical methods. However, I do understand why these traditional methods are so widely adopted and have stood the test of time. Non-traditional methods need careful examination, because their ease of use may subvert a crucial process that is needed at a later age.

The easiest way to "get the answer" in mathematics is to use a calculator. And just about every student in the country will find it easier than the "hard way." In many instances a calculator is necessary; but, before using it, ask yourself, "Is there a mathematical process that is not being learned when relying on this non-traditional method?" In many instances, the answer is "yes."

Please, please.... do not let your ego take over, and distract from the real goal of teaching children, so that can better understand and comprehend.

First you call us closeminded, now you are raising the issue to the personal level by calling our motives into question. Sorry, but I don't appreciate the psychoanalysis -- that there must be something wrong with us because we found fault with someone's method of dividing numbers.

I thought we were all entitled to an opinion on the matter.

[Unregistered]

i was looking around the web to try and help a kid i know do some long devision but couldn't think of a way to explain how to do larger problems. I was reading the post and couldn't understand what you were supose to do with this double devision. I went and looked at the calculator that got posted and i think that i might start using this system. I think that whoever came up with it should push it more and see if they can get it into some cirriculums. Even if kids have to do other steps to get the algorithims and things that people say they get from long devision fine. It is better to have to teach two things and have the kids eccel than to teach them one and have them get discuraged. i love the doulble devision thing. Thanks a lot.

[Unregistered-CKL]

I'm a special education teacher, and I believe that double division has merit ( as an alternative algorithm) for several reasons:

First, it takes away the guess and check and repetition of the DMSB steps, which can be frustrating for students who have working memory deficits or who have trouble estimating or rounding. These students get estimation practice without having to deal with the consequences of picking the wrong number, even if it is close. Also, there is an actual record of what has occurred on the student's page( with long division, you can't see the division or multiplication steps-they're meant to be done in your head).

Secondly, students with learning difficulties especially need to feel sucessful. This has worked when long division hasn't. Think of it as an available stepping stone instead of a crutch. I believe that feeling sucessful with one method boosts confidence to learn the traditional method.

[Unregistered]

"And let me repeat myself: It is the process of division, not the answer, that is important."

I agree it is not the answer that is important, but I disagree that it isthe PROCESS that is important. I am a secondary math teacher and I believe the UNDERSTANDING of the process is most important. I currently work with elementary teachers in our district as they work to help students understand the mathematics while constantly pushing them to become efficient in "doing" mathematics.

We encourage students to first understand what it means to divide. When looking at a problem like 547 divided by 26 and to ask how many times does 26 go into 5 or into 54 is WRONG! That 5 represents 500 and the 4 represents 40. A student needs to be considering "how many groups of 26 can be subtracted from 547?". To say that 26 goes into the dividend 2 times is equally WRONG. It goes into there 20 times.

We begin by asking, "what is the most groups of 26 you can think of to subtract from 547?" Any number of groups that the student is comfortable with should be acceptable. Most students will start with 10 if it is not too big. It is our job as teachers to accept that in the beginning but to push the students to use larger multiples of 10 whenever possible. The method demonstrated at http://mb.msdpt.k12.in.us/Math/PartialQuotients.wmv is what I am talking about.

We have found that not only do our students understand division but they come to the point where they can do it much quicker that those using the traditional long division algorithm used in the US. Because the understand division they do quite well when they reach formal algebra and certainly do not suffer for having used this strategy.

[Unregistered]

Okay, I admit that this method appears better than the double division method and the traditional method, in my opinion. And it appears that it would work for polynomial long division and most methods down the road that involve division, which was my main concern. I think this one is a winner. -- Lisa H

[Unregistered]

Before learning of double division, I had been using the method of creating multiples of the divisor and subtracting the closest from the dividend. This worked very well with my 10-12 year old students in Trinidad. moreso, the cocept of divison was well reinforced. I reccommend this method to any teacher.
e.g. 1560 divided by 24
(make multiples of 24 by repeated addition) 24 48 72 96 120 144 168...
Subtract closest multiple to 156 (144) 24 x 6...(Write 6 above dividend)
Remainder to divide is 120...subtract closest multiple to 120 (120) 24 x 5
Write 5 next to 6 above dividend..........Answer is 65......(65 groups of 24 in 1560.)
This method has proven doable by even the weakest Mathematics students. give it a try

[Unregistered]

Its a fantastic way to teach division and is based on the notation that division is about grouping. students who have grown up using this method as part of the national numeracy strategy in England are confident mathematicians. The problem is when they have not been taught this and have to remember a set of rules or procedures ie formal long divisionwhich not based around understadning of the number system. the'new' way doers confuse them. But as with all methods I must say that some children just prefer being taught a method rather than a strategy

[Unregistered]

students who have grown up using this method as part of the national numeracy strategy in England are confident mathematicians.

What exactly does that mean? Do you have any links?

[Unregistered]

The importance of division does not rest so much on the answer - a calculator can provide that. Instead, it's the process of dividing that is important.

I absolutley agree with you. I also agree with Jeff that his method(if he really invented it) very easy; it's like using calculator. However, I prefer the traditional method to teach our kids, at least below highschool. That way they can use the their own brains to solve math problems.That's the purpsoe of doing math, to excercise your brain, isn't? Why do we have to make everything so easy?

[Unregistered]

...they use the method presented on this website.

They even went to separate schools in separate towns and they both learned this method. His father is an engineer for Boeing and had no trouble with the higher maths. In fact, I would say his son and I probably struggled more with higher math than he did.

[Unregistered]

Interesting that everyone here seems to think double division is new. I learned it when I was in grade school 35 years ago. It remains, for me, a way that makes it clear how the number positions work on division and it made it easier for my math-resistant daughter to understand the division she hates.

[Unregistered]

At 29 years old I distinctly remember giving up on long division back in school. Now I am taking C&G maths level 2 exams and long division cropped up in the brush up course. After spending a while and still not understanding I decided to look on the net for a better explanation of long division. This method makes it easy and accurate and I have no problems with it.

I can safely say, today I learned something new.

Thank you.

Sam

[Unregistered]

I am discalculate (most people have never heard of this)-my brain literally has a problem comprehending, rememebering, and manipulating numbers. I did not learn this until I was an adult however, and I struggled with mathematical concepts, and functions through out my entire life. I was a staright A student in all areas except for math, from the time I was in second grade. I could never get the hang of things. I would beg my teachers and my parents to explain it to me in a way I could understand, but was always told something to the effect of "This is the way you do it, if you work harder/study longer you'll get it." Well I never did "get it" and even to this day I count on my fingers, I struggle to make change in my head, long division causes anxiety...

My oldest son is in third grade, and we are now working at long division, and though he has been identified as "gifted", LD has brought him to tears; he is not "getting it". So I was surfing the net to see what I could possibly do to get him to grasp the process, and found Double Division. In approximately 2 hours he and I were able to figure out ANY division problem set before us, not just having the correct answer but with a basic understanding of the process itself.

I agree with a previous poster that said, something like, "If a child cannot learn to get the answers to basic problems they aren't going to go on to higher mathematical operations anyway." If children cannot "get" the basics they will never have the desire for more difficult concepts, but if they meet with success now they may have the ambition to try something else later, even if it means learning a different process. The DD process is no more difficult than LD (they are both foreign to anyone just learning them) it is merely different, and a bit longer. I think that it is a great way to understand LD in the long run. I understand LD uses many mathematical concepts all at once and is necessary for higher mathematical concepts, but for the mathimatically challenged individual and for those just learning (i.e. grade schoolers) all those concepts rolled into one can be very confusing and frustrating. I think the DD method is fabulous. We are having great results.

Just a note side thought: My husband does ALL of his calculations in multiples of ten, he is very fast and efficient with mathematical operations and concepts; always obtaining the correct answer, never requiring paper and pencil or even a calculator. So their are different ways to do the same thing, we need to find what fits the individual, children specifically; but there's the rub when thinking about Public School, it is a cookie-cutter education, if you don't fit the mold you're are S-O-L. Just my 2 cents.

[Unregistered]

From what I've seen of the UofChicago partial-quotient division method--it serves to help understanding but sadly it is not helpful in a practical sense (higher levels of mathematics). Some students who are taught this method and prefer it over standard division are totally incapable of doing "regular" division and continue to revert back to partial-quotient, simply because they have not been taught to estimate with large numbers. Similar to the lattice method in multiplication, this method serves to delay learning of fundamental operations.

I have to disagree about delaying learning fundamental operations. I think it depends on how it is taught. I have had much success with the lattice method because I take is from the basis of place value and to me it makes more sense than the traditional algorithym. Just because we have always done something a particular way doesn't make it the best way. What I have found is that my students come away with a better grasp of number sense and then can apply and understand application to the "traditional algorithym" world. One this that I like to ask myself and my students is "How many ways are there to get out of a box"??

Have a great Sunday!!

M~

[Unregistered]

I was very confused with the method at first. I did not understand it (probably because of the way it was presented). I put it aside for a while but then I looked at it again recently. It makes sense now.
I disagree with what some people have said though (that it shows students the real meaning of division). It doesn't. The only method that shows students the meaning of division is "repeated subtraction".
The "double division" method is NOT showing students the meaning of division. Neither does the traditional "long division" either. The double division method takes advantage of some advanced concepts number theory.

[Unregistered]

Double Division - a method for doing division
http://www.DoubleDivision.org

I found a new method for doing manual division that ?may? be better than long division in a number of ways. It might be useful to teach this method BEFORE teaching long division, or maybe INSTEAD of long division.

I think it's simpler and more intuitive - but I may be wrong. Please check this out and give me feedback. Feel free to forward this to anyone who may be interested - or suggest other places that I might post.

Thanks,
Jeff

i tried this method and found it very confusing and when i calculated with a calculator, it wasn't the same answer.

[Unregistered]

To the person that posted the following, show us the problem you were doing and the steps you did... (the method is not explained very well on the site):
i tried this method and found it very confusing and when i calculated with a calculator, it wasn't the same answer.

[Unregistered]

I must say that I was stunned to see the passionate/hateful exchanges about a topic as dry as long division! Are you teachers, or children in need of social supervision? Petty, petty, petty. And, as a parent, a bit scary, as well.

IMO if another manner of explaining a topic allows a particular individual to grasp the concept, it is worth pursuing. I have read many books in my life under the assumption that different voices were worth hearing, filtering, examining. I have not found all of them touching or valuable, but for another person in a different place they might be fantastic. Look no further than a best seller list to realize we speak in different languages all the time. If this method yields a correct answer, so be it! Offer it on a platter of options for kids who are struggling.

[Unregistered]

it's wrong

[JennyC]

While trying to work out long division to help my 14yr old, I came across this site. It's 26 years since I last tried to do it and I never quite got it then - seemed a matter of blind faith. I really like this method. It makes a lot of sense to me and I have had a fun hour playing with ever larger numbers - very cool. Thank you!

[Unregistered]

I have actually seen this method for several years. The problem that I see being a 5th grade teacher myself is that the grade levels before me must also introduce things this way to create a better flow. If the 4th grade teacher below me introduces division using the old method then by me introducing the new method I have only baffled half of my class and it takes me twice as long to teach the concept. Which loses all the LD or ADHD kids along the way in the process. They typcially need short and to the point instructions not long winded steps. Long division is already tooooooo long for the severely ADHD child not on medication. It has to be short and to the point. I think the method could be good if all the teachers buy into it and use it. Of course we can always fall back on the old standard but looking at this method it seems to me to be much easier to start with this one. If you start with the old method you might as well stay with the old method and keep plugging away at it and try some alternate helpful tips, but I would never change the whole method mid swing.

Sheri

[Unregistered]

OMG
THiS iS SOOO HARD
SOMEONE HELP ME
iGOT SOME NEXT HWK YEH
AND
iCANT DO iT
iM iN YEAR 9 AND SO DUMBB =[[

[Tigger]

I felt that way about a new multiplication method, but the kids learned from it. I think sometimes we have to step outside the box. I would introduce it and teach the other way. Some kids need different options. They do not all learn the same!

[Unregistered]

my daughter came home and told me that her 4th grade teacher don't want her to use the old way and if she does, she'll mark it wrong. Is this right for a teacher to do? I've tought my daughter the old way and she understand it fully and are very good at it.

[Unregistered]

52 divided by 135

[Unregistered]

After readidng this I still do not understand why our early school math teaching methods are not beginning with the true math of capitalism that is taught daily at home at everywhere but in school - the dollar. From this single fundmental part of our lives from earliest birthdays we deal iwth it. From adding pennies, nickels, etc to dividing a dollar with your brother and sisters it appears the basic skills in kindergarden to third grade should focus from this basic concept first. Even negative numbers can be found here. And all else can grow from it even the double divison you discussed.

I got here trying to figure way to help grandson who is having difficulty per my daughter. I suggested dividing the dollar to begin with coins such as quarters to confirm the numbers on paper. It gives meaning to the sumbers that kids understand, certainly did when I helped grandson with subration. Then she can move to other coins and finally to any number on paper as the concept has real meaning.

We need these dollar skills from the beginning, but I like most of you it appars never got this from our schools, but at home and the local store selling pop and candy.

Again, I ask why is this not the basis in the early grades for counting and learning math? Everything in America re-enforces it daily. Yet I have found the same old exercises on the web I used 50 years ago still being used for second and third grade and almost none base with or dollar.

Wonder how great we will be when reallyeducate our kids and they can move on to do real things when they get out of school vice larning how to use a checkbook, budget, and figure out their bill vice income for the first "x' years?

Bob in VA

[Unregistered]

To an adult who already knows division this method might make sense but to elementary grades students this would be very complicated and difficult and inappropriate to teach before they understand the concept of division. Students need to explore division through activities that encourage and develop student inquiry, and critical thinking skills. The use of manipulatives greatly enhances student comprehension. Mathematics is so much more than just doing problems on a piece of paper. Students need to understand the why of things, not just "do this trick". As an elementary educator, I would never introduce this "trick" to my students. Perhaps in high school, but never elementary school.

Double Division - a method for doing division
http://www.DoubleDivision.org

I found a new method for doing manual division that ?may? be better than long division in a number of ways. It might be useful to teach this method BEFORE teaching long division, or maybe INSTEAD of long division.

I think it's simpler and more intuitive - but I may be wrong. Please check this out and give me feedback. Feel free to forward this to anyone who may be interested - or suggest other places that I might post.

Thanks,
Jeff

[Unregistered]

my daughter came home and told me that her 4th grade teacher don't want her to use the old way and if she does, she'll mark it wrong. Is this right for a teacher to do? I've tought my daughter the old way and she understand it fully and are very good at it.

NO, your daughter should not be punished for using multiple methods to solve mathematical problems. There are many paths that can lead to an answer, and as many as possible should be explored. The only reason I can think of that the teacher would have for insisting that your daughter use her method is that it addresses a particular skill that the teacher is trying to teach the students. It is good to know more than one method to solve problems. Perhaps if this was explained to your daughter, the why behind learning the teachers method, she would be more open to learning it. There is no reason not to explain the why of things to students. Adults like to know why they have to do something, why should children be any different. As an elementary educator, this understanding is arrived at early in my classroom so that children are more open to other methods.

[Unregistered]

Some people really like double division and others don't. One problem is that the extension to decimals is not very clear. I've been asked several times about this and haven't had a good answer.

But I might have a better answer now...

Basically if you want to extend your answer by say two decimal places, then you start again with just your remainder times 100. You can do this as many times as you'd like, or until you find that one of your remainders repeats itself.

Here is an example:
http://www.doubledivision.org/images/remainder-to-decimal.html

- Notice that the 1x, 2x, 4x, and 8x numbers are the same each time.

- Also notice that the original problem does not get bigger and bigger and bigger. Instead you just do additional smaller problems.

Jeff
http://www.doubledivision.org

[Unregistered]

I began teaching a Pre-GED class recently and I'm finding that several students have problems with long division and I'm wondering if this method of "Double Division" has been tried in Adult Ed situations. If so, has it helped or hindered students in the long run as they prepare for the GED exam? (I was too lazy to read ALL the posts on this.)

[Unregistered]

I am a fifth grade teacher, and we just started long division in the book that we currently use. That is not to say that my fifth graders have never done long division before, but the first nine weeks (almost) has been great review time. I came across this method when I was looking for a few worksheets to reinforce the concept for those few student who didn't get it. I am someone who can find the answer in math, and not be able to tell you how I got it. I know how I got the answer, but explaining it to another person is tough for me sometimes. While I have to say that I am not sure how well this would go over at my school with administration, I find it to be a fascinating concept. My students (most of them) have already learned division the other way, so I won't be teaching this in my classes, but I am going to show the advanced students this method, because I think they will be thrilled with it. I am amazed at the mind that was able to get this method on paper, and out into the world. I have to say that as teachers go, many of the posters on this site fear change, while I see it as necessary. What we are doing in the public school system is certainly not working, so why not try something new. Being close-minded won't help to advance the students. When you are set in your ways, and refuse to try something, I think that might be the sign that you should hang up your abacus and pointer stick. Just my thoughts. By the way, Jeff, I think your method is great, and it intrigues me. Two thumbs up...

[Unregistered]

I learned to divide large numbers in 3rd grade and always thought it was a REVERSE of what my brain was doing. In after finally (painfully) grasping long division it made sense but was a roundabout and somewhat backward method of the way I just basically taught myself. I basically would do it from right to left, and you can figure how doing this takes less steps. I saw some site about it a couple years ago and it showed it taking less steps, and was looking for it again to find the method, which is a little more refined than the way I do it because I have to store numbers in my head. If your child/students are doing division in their own way please encourage them and try to see if you can figure out what they are doing. My teachers couldn't figure out how I was doing it faster than them and basically forced traditional long division on me. I reverted back after they stopped forcing me, and it took a while to regain my ability. If it had been encouraged things would have been much easier (then I sort of forgot after calculators but when I don't have one I still do it in reverse.)
Help the way the kids do their math, don't discourage it and say "Mine is Better." The methods that are taught are basically taught because you can teach someone steps that they don't have to think about much. Then when they get to higher algebra, geometry, trig, and calculus they have to think and they are way behind.

-One of those students the teachers hated or loved because they couldn't figure me out.

[Unregistered]

I realize now I basically used the double division method and included the odd numbers as well (and all in my head, the teachers hated,) but if you can find it (still looking) someone has refined the method even further and it takes minimal writing without remembering numbers. If I find it I'll post, and YES IT HURT to learn it the traditional way.

[BegBorrowShare]

We use the Does McD's Sell Cheese Burgers Raw method. I created a template of boxes which scaffolds how a division problem looks which they use to learn basic 2 digit by 1 digit division without remainders. (I made templates for larger problems too.) Then I have them write the DMSCBR in a column next to the problem (divide, multiply, subtract, check that difference is smaller than divisor, bring down, repeat) and create a box around each step/letter as they perform it. (i.e. they underline each letter the first time through, then build the right side of each box the second time through, then the left side, etc) They don't usually go through the steps more than 4 times since we only go as high as 3 digit by 1 digit division so this works for learning the steps. [After the first couple of days, the template has only boxes for the divisor/dividend/first subtraction problem. Then I remove them altogether. This seems to help immensely.]

I have my students make multiplication charts every night because we use them for all kinds of things: equivalent fractions, simplification, conversion, etc. They can do it in fifteen minutes now but we use prepared charts for long division.

I mention this because when we do long division, I have them find the divisor at the top of the column in the chart and then ask them to get as close to the dividend as possible in the column without going over it. Then they are taught to trace from that number over to the row number to find the number of DIVISOR GROUPS that can be created within the dividends "house". It isn't long before they make the connection between multiplication and "backwards multiplication", as they call division.

I suppose using the multiplication chart is quite a bit like making a divisor multiplication t-chart. But since they are required to make the chart on all benchmark tests before they begin, we make use of it!


** COMMON SNAG & How I Fix It: As an aside, the second time through the steps they will try to use the "first" dividend again instead of dividing the new number created when you bring down the next number. To fix this, I have them bring down the number and then draw a line through the "first" dividend ("kick 'em out of the house") so they won't try to use it again. Then I have them move the "house" (the symbol that houses the original dividend...sorry, don't know what it's called) to the "new" dividend---the one created by bringing down the next number---since the old dividend is "gone" and "doesn't live there anymore".

Anyway, it cured all the problems, even with my LD students.

[Unregistered]

Long division calculator available on www.dol88.com (http://www.dol88.com)
Also able to do Additions, Substractions, Multiplications

http://dol88.free.fr/www/chgr/jeu-en-operations.jpg

Step by step explanations
User manual on the site