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New Method for Teaching Division


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Joined: Oct 04, 2005
Posts: 10
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Posted Oct 04, 2005 at 10:35:49 PM
Subject: New Method for Teaching Division

Double Division - a method for doing division
http://www.clubiris.com/DoubleDivision.html

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 place that I might post.

Thanks,
Jeff

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victoria
Joined Jun 13, 2003
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Posted:Oct 05, 2005 2:34:19 AM

No thanks. Your students are going to be in a world where people who don;t use a calculator use the standard long division system; teaching them something else is setting them up for hassles the rest of their lives.

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victoria
Joined Jun 13, 2003
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Posted:Oct 05, 2005 2:35:11 AM

No thanks. Your students are going to be in a world where people who don;t use a calculator use the standard long division system; teaching them something else is setting them up for hassles the rest of their lives.

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Oct 05, 2005 11:37:00 AM

Double Division

I don't have any students, just a daught who will be learning long division soon. Teaching her double division first will reinforce the idea that division is subtracting off multiples of the divisor. It will also reinforce the distributive property.

The value of teaching manual division is to give people a method they can actually use if they have to, and to teach about mathematics.

It could be argued that if students seldom use long division after leaving school then it might be better for them to have a simpler and more intuitive method at their disposal. Would someone ten years out of school have an easier time doing long division or double division? - I'm not sure, but double division seems simpler.

About teaching math, I'm not sure that long division teaches math very well. It teaches people 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.

I don't think this is setting people up for hassles the rest of their lives. It's hard to imagine a situation where someone would be ridiculed for using a different method of division. It is a new method, but that doesn't make it worse.

I appreciate your thought none the less,
Jeff

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Sue
Joined Jun 14, 2003
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Posted:Oct 07, 2005 10:58:24 AM

So... teaching the long way of doing long division is bad 'cause students don't understnad it...

...No offence, but your "double division" doesn't seem *any* more comprehensible to my eye.

I do agree, long division is tedious and hard to grasp. But what you're doing is equally confusing, though perhaps not to its inventor, but that's 'cuase you thought of it, probably after you learned the original way?

I agree that repeated subtraction is a little bit closer to being able to be understood, but with the extra "doubling" steps it's not really clear that that's what's going on, and in my experience, once the numbers get big and complicated, the comprehension of the "repeated subtraction" gets lost in the symbols. It's a tough compromise where I would do the simultaneous "instruct and accommodate," where I'd work on understanding with the smaller numbers, with images and manipulatives and athe rest of it - and use mnemonics and survival strategies with the big stuff until the comprehension catches up.

It's a noble effort... but it owuldn't make it any easier for me...

Sue J, webmastress www.resourceroom.net

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Oct 07, 2005 1:13:55 PM

Thanks Sue, I appreciate your feedback.

The challenge is that it's hard compare the "easiness" of something new like double division compared to something you know already like long division. (I probably know double division better than long division.)

Here is the same problem done both ways:
http://members.aol.com/loydlin3/Ldivide.jpg
http://doubledivision.org/Ddivide.gif

- Long division is very dependent on lining up all the digits correctly. You have to figure out where to start writing your answer and you have to make sure to write the result of each multiplication FROM RIGHT TO LEFT starting under the particular digit of the answer. Students can learn to do this, but I don't think they know why they are doing it. And it's easy to forget these lining up rules after a few years of not doing it. (That's why I thought it might be easier to accumulate the answer on the right side rather than having to get it exactly right on the top.)

- I don't like having to guess how many times one number goes into another. How many times does 357 go into 1399 - I don't know. Having to guess and getting it wrong and having to erase makes the whole thing frustrating to me. (I guess it could be argued that this is good for people.)

- Much of long division is procedure without meaning (to me). What does bringing down the next digit really do? What does the first digit "2" in the answer mean? It means 2000, but I don't think students would know this. When you bring down the next digit and result is too small then you have to write a zero in the answer and bring down another digit -> easy to mess up.

- I think the middle and right sides of the double division example are very clear. It shows the real size of all the numbers, and it shows that each one is multiplied by the divisor and subtracted from the dividend.

- The left side (where you double) is just a place to write various multiples of the divisors. Actually you don't have to do this part - and maybe it would be taught this way first. You could just subtract off the divisor times powers of ten. You could double the number once or twice instead of three times. You could write down all the multiples from 2 to 9. I just chose doubling three times because it seemed the most efficient. It would be important that students be taught that the doubling is just a quick way to get divisor multiples to subtract with.

Am I swaying you?

Jeff

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Sue
Joined Jun 14, 2003
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Posted:Oct 07, 2005 4:56:58 PM

You still have to line the digits up your way. You have to figure out the zeroes... and believe it or not, most of those kiddos don't know why they're putting in that many zeroes.

The bottom line is, the argument that "they don't really understand the long division" applies just as well to your version. When the numbers are that big, they don't understand the reasoning.

You don't have to guess - but you do have to figure out all those doubles. That's a lot of calculating. If you didn't want to guess, you could just calculate all 9 possibilities right off and choose the best one, just as you did with the doubling. I guess you think doing all the calculatiing is "good for you" the same way you accuse the standard-method teachers of regarding erasing. Actually, you're foolish to erase - once you've got it calculated, *leave it there* so you can refer to it for the next step, just as you did with your pre-emptive calculations.

You don't have to sway me. Your arguments are all about why it works for ***you.*** So... feel free to do it that way with every long division problem life gives you :-)

However, I think one of the worst crimes in math instruction *is* this pervasive tendency to expect children to figure out math the same way we do. Again, you are coming at your method from understanding that the students, mu;ch as you don't want to believe it, generally DO NOT HAVE. You're right, most of 'em don't know the 2 'means' 2000... they also don't really know what 2000 means, so they don't know the 14,400 (or whatever) means fourteen thousand, four hundred.

Yours is just as easy to mess up. Not for you - so feel free to do it. However, I pity the poor confused young soul you try to show this to... who's already confused enough with the way the teacher shows her. She *really* won't know right from left and where to begin and end.

Anyway you look at it - yours or the 'standard' way - long division is a highly complex procedure. You understand yours better. How long did it take you to figure it out? Was it instantly, intuitively obvious?

Sue J, webmastress www.resourceroom.net

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Oct 11, 2005 12:53:02 AM

Hi Sue,

I see what you're saying, and I'm revising my own thoughts on the subject.

If you don't know what 2000 means, then both methods are just procedures without much meaning - like magic basically. But I still think the middle and right sides of the double division example are pretty clear.

Maybe the biggest difference between the two methods is that you don't have to guess and do little trial multiplications everywhere on your paper. When I said I hated to have to erase my work - it was because I write my first guess assuming it is correct - under the dividend. If I'm wrong I have to erase it.

The two things I remember about long division are one) I never know how many times one number goes into the other, and two) if I don't write everything in the right place I'll get messed up.

I will be teaching my daughter double division before long (in a month or two) and we'll see how it goes. I KNOW she has NO understanding of how long division works. (I think she has done it - but strictly as a procedure.) We'll see if double division helps her or not. I'll let you know.

Jeff

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Oct 15, 2005 12:40:12 PM

Long Division Teaching Aid, "1-2-4-8 Division"

Here is one teachers comment after teaching 1-2-4-8 Divsion (double division) to his class:
http://www.teach-nology.com/forum/showthread.php?t=1163#post5288

Jeff

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Sue
Joined Jun 14, 2003
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Posted:Dec 08, 2005 1:08:56 PM

Okay, I am curious (in some ways I'd love to be proven wrong, even anecdotally)... are you doing it? Is it working? What parts were easy/hard?

Sue J, webmastress www.resourceroom.net

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Dec 08, 2005 1:45:14 PM

Besides the 6th grade teacher above, I haven't heard from anyone after they tried teaching it. (I am not a teacher myself.)

I did get this comment yesterday:
http://www.teach-nology.com/forum/showthread.php?t=1163&page=2#20

Jeff
http://doubledivision.org

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Sue
Joined Jun 14, 2003
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Posted:Dec 09, 2005 6:18:28 PM

Quote c22b416c26="Jeff Wilson":

Hi Sue,

Quote c22b416c26="Jeff Wilson":

I
I will be teaching my daughter double division before long (in a month or two) and we'll see how it goes. I KNOW she has NO understanding of how long division works. (I think she has done it - but strictly as a procedure.) We'll see if double division helps her or not. I'll let you know.

Quote c22b416c26="Jeff Wilson":

Jeff

This is what I was referring to.

The 'guesswork' could also be solved by simply listing the ten possible multiples, instead of just some of 'em... I'd like to know if that teacher had success :) Sometimes what I'm sure won't work does... and vice versa...

Sue J, webmastress www.resourceroom.net

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Dec 09, 2005 6:24:28 PM
Subject:not yet

ahhhh,

Haven't done that yet. Actually I realized she doesn't know her basic math facts yet, so we've been working on those.

I'll let you know how it goes when I finally teach it to her.

Jeff

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Mar 09, 2006 1:17:52 AM


I created a calculator that shows the steps of double division. Double
division is a method for doing division that may help teach the concepts
of division - and may be less frustrating because there is no trial and
error guessing.

Check it out if you'd like. Just click "Step 1" then "Next Step", "Next
Step",,, It's gotten mixed reviews so far, but now many more people
are learning how it works, and some are liking it.

http://www.doubledivision.org/

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Brian
Joined Sep 22, 2003
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Posted:Mar 11, 2006 5:01:58 AM

Thanks Jeff for taking the time to develop your method and to share it with us. I haven't looked at it yet (read the thread and wanted to respond to the replies more than the method for now).

I think the following is a bit narrow-minded:

Victoria: "No thanks. Your students are going to be in a world where people who don;t use a calculator use the standard long division system; teaching them something else is setting them up for hassles the rest of their lives."

Everyone is going to be in a world where they will be better off knowing A WAY of doing long division than not knowing ANY way. Since I left elementary school I've never had ANY input from ANYONE regarding the method I use. It's been my experience that long div is a pretty private activity from teenage on, and I've certainly never been asked to hold to any "standard". Really, Victoria, we all know that all anyone cares about is that we get the right answer - not even that sometimes. What hassles might our children face through learning a different method of long div?

The bottom line in long div and any other life skill is that people find A WAY that suits their own particular learning style and one that they can use to expand their brains a little more (why else learn such a dumb thing in the calculator age?)

Sue, you didn't respond well either in my view. Perhaps, Jeff's method is more difficult for all. Perhaps for only some. Perhaps it's the easiest way for everyone except you. Who knows? What Jeff is offering is an alternative. Embrace that always. A better response would have been: "Thanks Jeff, I'd like to hear more. Please don't spare the explanatory details. We need more ways to use our brains, not less, and we don't have to grade everything and always use the easiest method to the exclusion of all others." Standard = stagnation. Alternatives = progress.

As a lifelong cat-skinner (just like everyone else), I've found that there is usually more than one way to do it, but that I normally fall in to the RUT of preparing each type of cat the same way. That's a failing on my part (not helped by my standard schooling) and one that I'm sure limits the dendrite production in my brain. I believe that once children become proficient in the "standard" method of doing anything, they should immediately be set to use whatever alternatives are available (a calculator NEVER being one of them - how fast do you have to do the 20 or so lon div problems per year the average adult faces?) So you know red, green and blue, now what other words could you use to express those "standard" concepts?

So, let's welcome innovation. Let's actively seek new ways to do and teach everything. If we don't, we'll all be receiving eulogies based on a 400-word vocabulary.

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Brian
Joined Sep 22, 2003
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Posted:Mar 11, 2006 5:48:02 AM

Hi Jeff,

I checked out the method. I like it but would reject it as a first choice for teaching a child for precisely the reason that it is TOO EASY! I believe, if possible, a child should learn to solve problems through trial and error (the "guess" you referred to, and Edison's salvation from an "addled-brain"). Your method is all add and subtract. Multiplication practice (beyond possibly 2X) is non-existent. If the brain is not to be involved much, we might as well just use calculators.

So, I'd say try to teach the more difficult ways first, then, if all else fails, or once the other ways are mastered, teach your method as an easy alternative. I think it would be really difficult getting a child to learn the method involving more multiplication if your method were to be taught first.

Interesting though. I felt like I was out of the loop as far as "controlling" the problem/solution was concerned. Would that be something to do with my personality/learning style or because I'm just used to doing long div another way?

Now, let's get rid of the pencil and paper too...

http://en.wikipedia.org/wiki/Jakow_Trachtenberg
http://mathforum.org/dr.math/faq/faq.trachten.html

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Mar 12, 2006 8:35:47 PM

Hi Brian,

Thanks for your perspective on alternate methods and your specific thoughts on double division.

I would say the three properties of double division are;

1. No guessing / estimating,

2. Deals with complete numbers rather than individual digits,

3. Give practice doubling numbers.

Numbers (2) and (3) should not be overlooked. I made it from 9 to 42 years old without realizing that when I was doing long division I was subtracting multiples of the divisor (times a power of ten) from the dividend. I didn't realize this because in long division you work with individual digits of the answer and middle portions of the dividend - instead of complete numbers. Had I learned double division at some point in my schooling then maybe I would have had a better understanding of what division is. The estimating part of long division is very valuable, but for me it dominated the process to the point that I never understood what I was doing.

My daughter is learning long division in school and I'm teaching her double division at home. She has been alternating on her homework problems between long division and double division. She likes double division because it's more "turn-the-crank," and doesn't depend on knowing her multiplication facts. I think that learning both methods will actually show her the advantages of knowing her multiplication facts and value of estimating better than if she learned only long division. And I hope that she will understand (in the end) what she is doing better than I did when I was a kid. (She doesn't yet, but we've only worked on it about three times so far.)

I'm still evaluating the whole thing myself - but my suspicion so far is that teaching double division along side long division will be positive and worth the effort. I'll know better in a year.

Jeff
http://www.doubledivision.org/

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Brian
Joined Sep 22, 2003
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Posted:Mar 13, 2006 8:23:18 PM

Hi Jeff,

It appears that you (and everyone is entitled to his/her own way of thinking) place a lot of emphasis on "understanding what you're doing and why you're doing it", in long div. I can't see how that's really important in teaching this skill to children. I've never known why the "ough's" in through and though have to have a different sound, but that didn't stop me from learning the words or using them properly.

For me, long div serves 3 purposes: Multiplication fact practice and practice in the "guessing" you alluded to earlier, i.e., in relating two large numbers in your head using experience as a guide, along with the obvious benefit of being able to divide large numbers without a machine.

Your method releases the student from the real brain work of first two. Children need estimating practice if they are to successfully compare two or more alternatives in the future. They will also multiply much more than divide in life. I must admit though, that "practice in doubling numbers" does sound better than "practice only in x2 math facts", however, I can't really see it as representing a good selling point.

Again, I believe students should be made to do the work first and only later shown shortcuts. Later still, theory may be taught - if desired.

Nobody would agree with a "new" method of long multiplication that involved adding columns of the same number.

However, if all else fails, perhaps in the case of students with learning difficulties who are falling behind, your way is a step above giving directions to the calculator store.

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Jeff Wilson
Joined Oct 04, 2005
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Posted:Mar 14, 2006 8:09:14 AM

Hello Brian,

It surprises me that you don't put a high value on "understanding what you're doing and why your doing it." We agree that multiplication facts and estimating are very important, but I also place a very high value on understanding - especially in math.

Double division may help kids understand more about division and the distributive property of multiplication. It also gives practice doubling, and may be an easier method to use 10 years out of school because it's easier. I don't think it should replace long division, but I think there is a place for double division.

It has been fun discussing this with you, and now I have to get back to my regular work for a while. Thank you for thoughts on this,

Jeff
http://www.doubledivision.org/

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Brian
Joined Sep 22, 2003
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Posted:Mar 15, 2006 6:06:39 AM

Jeff,

I didn't say that I don't place a high value on understanding what I'M doing and why I'M doing it. I said I don't see how those things are very important when teaching long div to children. I also stated that such theoretical knowledge could be taught later on.

Really, there just isn't the time to explain each and every theoretical idea behind everything that is taught in elementary school. What's important is that the child learns how to divide large numbers without a machine and gets plenty of practice in both trial-and-error estimation of solutions, and multiplication - the two items missing from your method.

Again, practice in "doubling numbers" is synonymous with practice in the 2x table.

I agree that double division IS an easier method (for those who haven't or can't master the standard method). However, for all but otherwise hopeless cases, or as an alternative method after the standard has been mastered, that is precisely the feature that I would be against.

A good example though, of how easiest isn't always best.

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