s/o Short Division vs Long Division

[Halcyon]

My son is just finishing SM 3A and long division. All has gone fine, and long division has (knock on wood) not proved to be an obstacle. However, when I learned, I learned what now seems to be called "short division" (as mentioned on the other thread, you can find a brief video on it

).

 

My question is, why would one prefer to use long division over short? Short seems much more intuitive to me, and certainly less steps to "mess up". It also seems easier to explain using base-10 blocks.

 

Anyone?

[mammaofbean]

from looking at the two it seems that it would be easier to pinpoint a mistake in a student's work in long division easier. which doesn't really metter in a homeschooling situation because you can have time to sit and talk to them about every single problem if you want.

 

i have yet to reach this bridge, but i would say doing it both ways would be a good idea. what is perfect for one kid is not for another. though short looks like the winner to my eyes my kids have been known to surprise me with their methods.

[Dana]

In algebra, when you do polynomial division (example here), I don't see how you could do it with "short" division. I had a rough time in calculus because I'd only learned synthetic division (example here) which only works when your divisor is linear with a leading coefficient of 1. It doesn't work when you need to divide by a quadratic for instance.

 

Having a procedure that can be generalized for other topics and expanded is part of the beauty of math. The long division algorithm works for polynomials as well as integers. That's pretty nifty - and pretty powerful.

 

I also think it's better to have a procedure that will work for multiple topics rather than one that works only under certain conditions. (Thus polynomial long division and knowing when synthetic division will work instead of only synthetic division.)

[MerryAtHope]

The demo only shows division by a one digit number. I wonder if it would seem as easy to remember when dividing by 2 and 3 digit numbers and larger. How about this one:

.......______

457 / 97431

 

What do they suggest doing then? If they still have you just remember everything in your head, I think it becomes a much harder method.

 

Merry :-)

[AngieW in Texas]

My Aspie had a horrible time with long division because of her difficulties with keeping everything lined up and her difficulty with remembering the proper sequence. By the time she got to division with polynomials, she was able to do division the long way without a problem.

 

My dyslexic dd primarily had difficulties with the remembering the sequence for long division. She didn't have any problems remembering the sequence for short division. She also has visual processing issues, so the short division method was much cleaner for her to look at.

 

When they got to division with bigger numbers like 97431 divided by 457, they would actually work through the multiplication on the side. My youngest usually does it all in her head. Writing it down confuses her (visual processing), but she can carry it in her head pretty well.

[Capt_Uhura]

RS does short division for only dividing my 1-digit number. Once you hit 2-digit divisors, RS teachers long division. I believe it is the same in Math Mammoth.

[kalanamak]

In algebra, when you do polynomial division (example here), I don't see how you could do it with "short" division.

 

But, but, but, once you understood division, the long is just (really) the short with more steps, and seems like it would be easy to comprehend.

[Mallory]

We use if for all division, even problems where they are dividing by 10s or 100s. They do often do some figuring on the side but the remainder of each problem is just added into the dividend.

 

Of course they aren't to dividing polynomials yet, but it is the same thing.

 

In the example give on that site I'd write my x on the top then I'd figure out what happens when I multiply that by and subtract it. Sure I might have to figure this part on scratch paper, but then I get a "remainder" (it was -10x) and I could add that to the dividend. So under the division sign I have something that looks like this x2- 9x--10x -10. Then I'd start working on the -10x-10 part, it works just the same as long division.

 

I don't feel like this is a place where short divison is a tricky short cut for math. It isn't figuring 9 times facts by putting down that finger and reading the rest of your fingers or figuring out the 11 x 11 or more by spliting the number other then 11 and adding the two digits to the middle.

 

There are places you can do some fun math tricks and not have to know the math, but in short division you still have to know what is going on.

[Janice in NJ]

Teach it both ways.

 

Long division represents the "longest" arithmetic problem a child can do in elementary school. Teach them to be neat, quick, attentive, and careful. Start and finish the problem without daydreaming. (A speed issue - they need to learn to work quickly enough to maintain their focus but not too quickly; that leads to silly errors. That personal pacing clock matters when you approach tougher problems in mathematics. They need to learn to feel their effective personal pace.) Make sure all of the numbers are clearly written, and make sure they all line up on the page. Learn how to go back and follow your work to find errors.

 

ALL valuable skills for upper level math!

 

Easy to teach when they are little. A TON harder to teach if they have spent years developing bad habits. The good habits are needed for upper level math. Teach toward the good habits. And recognize that long division is YOUR best tool for doing that. :001_smile: (And like all things deviously mama - you don't have to tell them that this isn't really about long-division. You are the only one who needs to know that it's about so much more. It's fine if they think adults go around solving long-division problems in their "real" lives.) ;)

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

Edited October 24, 2010 by Janice in NJ

[Karenciavo]

Teach it both ways.

 

Long division represents the "longest" arithmetic problem a child can do in elementary school. Teach them to be neat, quick, attentive, and careful. Start and finish the problem without daydreaming. (A speed issue - they need to learn to work quickly enough to maintain their focus but not too quickly; that leads to silly errors. That personal pacing clock matters when you approach tougher problems in mathematics. They need to learn to feel their effective personal pace.) Make sure all of the numbers are clearly written, and make sure they all line up on the page. Learn how to go back and follow your work to find errors.

 

ALL valuable skills for upper level math!

 

 

:iagree:

 

It's never hard to agree with Janice. I'll just follow her around. :D

[Debbie in OR]

In algebra, when you do polynomial division (example here), I don't see how you could do it with "short" division. I had a rough time in calculus because I'd only learned synthetic division (example here) which only works when your divisor is linear with a leading coefficient of 1. It doesn't work when you need to divide by a quadratic for instance.

 

(Thus polynomial long division and knowing when synthetic division will work instead of only synthetic division.)

 

I just wanted to say that I have no idea what you just said here. But I think it's cool and I think I want you to teach my kids. :001_smile:

[Beth in SW WA]

:iagree:

 

It's never hard to agree with Janice. I'll just follow her around. :D

 

Me, too. :001_smile:

[Dana]

But, but, but, once you understood division, the long is just (really) the short with more steps, and seems like it would be easy to comprehend.

 

True... and I do see how the short division would work well with a single digit divisor and be much faster than long division. I just see it as dangerous to start with the shortcut. The real key being "once you understood division" and I really think that starts with the concrete. Thus, my suggestion in the other thread of base-10 blocks, then going to the standard (long) algorithm.

 

I know it's off subject, but I'm seeing for the first time this semester the results of Everyday Math in my Beginning Algebra class. It's just horrid and I have students who can't do simple arithmetic without the calculator. (Seriously... enough got 1203.34 - 407.83 wrong that I commented on it in class... they don't get place value and how borrowing works - skipped right over the 0 tens and moved 2 hundreds to give 13 ones. OUCH!) So I'm currently very much wanting the standard algorithms right now. :001_huh:

 

I just wanted to say that I have no idea what you just said here. But I think it's cool and I think I want you to teach my kids. :001_smile:

 

:D I've been teaching at the cc level for about 14 years now. I see a lot of students come in with a shaky arithmetic foundation and it's tough getting them the confidence they need and help them move along to algebra. Some days though I do think my students hear me as the teacher from Peanuts.

 

But what's really important to be aware of is that the basic arithmetic skills/procedures/understanding will be used/needed in algebra. The patterns continue when you're working with things other than numbers! (Understanding how to add, subtract, multiply and divide fractions is essential when you're doing the same thing in algebra with rational expressions - and when I have students who only can do fractions with their calculator... they just are dead in the water for algebra.)

[Mallory]

True... and I do see how the short division would work well with a single digit divisor and be much faster than long division. I just see it as dangerous to start with the shortcut. The real key being "once you understood division" and I really think that starts with the concrete. Thus, my suggestion in the other thread of base-10 blocks, then going to the standard (long) algorithm.

 

 

 

See, I don't see short division as the short cut at all, in fact I see it as the more complete way to teach division.

 

If I were dividing 78935 by 45, and I have decided that 45 fits into 78,000 1000 times. I'd take out that 45,000 and have 33,935 left. I'd put the 1 up where it will be the start of my 1000, knowing that I am still going to have some hundreds and tens and so on left to figure out. Under my division sign I'd now have written 7 8339 3 5, then I'd start working on how many times 45 will go into 33,935. I can clearly see how I am breaking down this big number.

 

I find this a much better way for me to really see what is going on. In contrast the regular long division method feels like the shortcut to me, in the sense that it isn't as clear what you are doing. It is the way that feels like moving numbers around magically.

 

Of course I also find subtraction much easier working from the largest units first! So maybe I am just wierd.

[Annabel Lee]

This is way OT, but I just wanted to say thank you, Dana, for adding another solid confirmation for my decision to pull Abe from our ps system (which uses EDM) in 1st grade. Little things like your comment help eliminate nagging self-doubts.

[Janice in NJ]

Mallory,

 

I teach both methods; we practice with both until it's knee-jerk solid. They are both useful for different reasons. :001_smile:

 

In calculus a student practices solving problems in several different ways. Each way illuminates a different side of the beauty of mathematics.

 

"Life lesson" and all that. :001_smile:

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

[Debbie in OR]

:D I've been teaching at the cc level for about 14 years now.

 

 

OK, I feel much better now :tongue_smilie: I'm still pretty sure you should teach my kids math since I don't even know what "synthetic" division is and it sounds like something I should know :lol:

[Mallory]

Mallory,

 

I teach both methods; we practice with both until it's knee-jerk solid. They are both useful for different reasons. :001_smile:

 

In calculus a student practices solving problems in several different ways. Each way illuminates a different side of the beauty of mathematics.

 

 

 

We have done both, but I don't really feel like they are even two different methods. I know I have been "arguing" for short division, but I really just feel that it is the easiest way to understand dividing.

 

If you need to write down some of the inbetween steps, then short division doesn't make that impossible, in fact we often have our products written under the dividend, but it just makes much more sense to put the remainder up into the problem.

 

So I guess our main method is kind of a mix of both, but I think it is putting the remainder into the original dividend that is so helpful in short division.