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#1
10-05-2005, 01:09 AM
 Unregistered Visiting Teacher Posts: n/a
New Method for Teaching Division

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
#2
10-05-2005, 02:02 AM
 Lisa's Hotscakes Teachnology Old-timer Join Date: Jul 2005 Posts: 380

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.
#3
10-05-2005, 12:01 PM
 Unregistered Visiting Teacher Posts: n/a
double division

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

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.
#4
10-06-2005, 12:05 AM
 Lisa's Hotscakes Teachnology Old-timer Join Date: Jul 2005 Posts: 380

Quote:
 You raise some good points.
You don't even realize how many enemies you made with this statement.

Quote:
 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.

Last edited by Lisa's Hotscakes; 10-06-2005 at 12:09 AM.
#5
10-06-2005, 05:24 PM
 Unregistered Visiting Teacher Posts: n/a

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
#6
10-06-2005, 05:43 PM
 Unregistered Visiting Teacher Posts: n/a

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
#7
10-06-2005, 11:16 PM
 Lisa's Hotscakes Teachnology Old-timer Join Date: Jul 2005 Posts: 380

Quote:
 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!

Quote:
 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.

Quote:
 Do you think this could be useful to teach BEFORE long division?
Yes, if teachers can stay on their pacing guide.
#8
10-11-2005, 02:02 PM
 Unregistered Visiting Teacher Posts: n/a
A Good Alternative

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.
#9
10-11-2005, 04:13 PM
 Unregistered Visiting Teacher Posts: n/a
How did you teach it?

http://www.doubledivision.org

That's cool.

Please email me and tell me how you taught it.

wilson [at] silcom [dot] com

Thanks,
Jeff
#10
10-20-2005, 04:34 PM
 Unregistered Visiting Teacher Posts: n/a
I suggest being careful.

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.

Last edited by Unregistered; 10-20-2005 at 04:37 PM.

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