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  1. #1

    Default Teaching Long Division

    Do you teach the 'standard' or partial quotients algorithm for long division? Or something else?

    I've demonstrated standard long division in the past and my ds8 looked at me like I was doing a magic trick. Divide, multiply, subtract, bring down, divide, multiply, one that he doesn't think he can learn. He is doing Beast Academy and just arrived at the Division chapter. It is a whole lot different that what my husband and I learned.. So I'm learning along. But then I started doubting..... Is partial quotients enough?

    -T

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  3. #2

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    I think the key is to teach division thoroughly as an idea, a REAL concrete practical idea, before teaching an ALGORITHM or procedure. In that sense I never teach either "algorithm" and yet my 10yo knows both.

    The standard method teaches a procedure for finding how many X are in Y. The partial quotients algorithm tries to replace this with just a different algorithm by having the kids guess that is smaller than the correct answer then use the standard procedure, then repeat the procedure and then finally add up all these partial answers, so X = x1+x2+.... In *principal* this is easier since they are building up their answer using easier math facts and their problem progressively gets easier. For me the problem is it is STILL an algorithm but there is no guidance for how to choose those pesky xi's. You can also spend a lot of time and do lots of little problems if you pick badly.

    I still recall being taught the partial method in 3rd grade and I was a kid that HATED it. Because I was indecisive and I didn't know how to chose a GUESS. I always rather calculate than estimate as well! I would say estimation comes from experience, not guessing, too. I preferred to play with the standard method working backwards and forwards until I understood what was happening. Other kids will benefit from breaking a problem into parts.

    My point is just use whatever works best, but the point is not to teach your child to calculate division, but to understand AND calculate. Understanding means they can walk away years and pick it up and still do it when needed.

  4. #3

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    I don't know the partial quotients method, but when my daughter learned long division the school used a unique method that had me doubting in the beginning. It's a method where the kids start by "building the problem" with blocks and then doing an operation on the blocks to solve the problem. Once they've mastered this very concrete and highly visual approach they move to the symbolic algorithm that we think of as long division (with some differences that I still don't quite get. My daughter still looks at me oddly when I do long division on the white board as a part of a solution. She gets the right answer by her method, so I don't ask about it anymore.). I had to go to a 2 hour long class on this approach just to help my kid with homework and I hated it at first. However, it worked. There was something about the process of transitioning from one method to the other that grounded the process and kept long division from seeming like a magical process for her. It answered the "whys" along the way.
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    Senior Member Arrived Avalon's Avatar
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    We used base 10 blocks to demonstrate all the steps of the standard algorithm at the same time as showing it on paper. It took a long time, but my daughter did understand it. If we just did it on paper, she had no idea what we were doing, and it was too many steps to remember, so doing it simultaneously with the blocks worked very well for her.

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    Senior Member Arrived dbmamaz's Avatar
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    First of all, I would totally trust Beast Academy. I would accept that there is value in learning what they are teaching.

    Second of all, I would wait and see if BA (or whatever you use next) covers the standard algorithm - even if its next year - before you go off road and teach him something free-hand.

    Now, to be honest, I did teach it 'off road' but only because we were not using a curriculum. And fwiw, teaching it was PAINFUL! My son catches on to most things quickly, but long division took weeks.
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  7. #6

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    I agree that the conceptual why over just the process of the algorithm is what you're aiming for. Partial quotients is sort of an algorithm, though unlike the standard algorithm, there are often multiple ways to solve the problem. I think it serves as an excellent introduction for most kids. I think it relies on experience and really seeing the number relationships.

    Just as a note, I'm pretty sure BA does the standard algorithm in the fourth grade set. We really liked the method introduced in BA, but it's similar to how Miquon introduces it so it was actually mostly review for ds.
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  8. #7

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    Teaching Long Division
    We waldorfed it:

    image.jpg

    It is the standard method, with storytelling. "Three knocks on the door of 54... how many threes fit on each floor... etc" . The stories just stick with my kid. She had the 'how' in five minutes and hasn't forgotten it yet.

    Then we had to work on the 'why'. We use the soroban abacus in place of base 10. You can line the thing up right under a long division problem, and voila!

    I can't believe how well waldorfing the soroban has worked out so far. Trippy Tiger Math.
    Home schooled two kiddos from a remote location for seven years. DD16 has transitioned to public high school. DS8 tried PS, but likes home schooling better.

  9. #8

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    I'm pretty certain that he totally gets the concept of division. If I give a simple word problem that requires division he can do it. When he is doing the problems in the workbook he just looks at it and basically estimates the answer. "It's gonna be about 43.. no 44." He's usually close, in his head he divides fine, but he doesn't care about remainders. When I prompt him to work the long division on paper he just gets stuck in the middle, throwing out random numbers for the answer. We've hit a wall.

    dbmamaz- BA has been fantastic for us ,, so I should probably just stick with it, yes.

    atomicgirl- I'm trying to wrap my brain around building the problem... and do I need blocks or not? Can it be done on paper..

    AnonyMs- He was in Montessori last year and they knock-knocked & jump-jumped for addition and subtraction, and it caused a lot of confusion

    Yesterday was terrible,, if today is the same.. I think I'm giving it a rest until January.

  10. #9

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    That Waldorf picture, while it might help kids remember the order of the steps in the algorithm, is one of the sorts of things that really concerns me about getting something conceptually because it doesn't support the concepts at all. It reminds me of in Knowing and Teaching Elementary Mathematics, the American teachers who put an apple or a little picture instead of a zero when multiplying larger numbers and told kids it was just "to hold the place" or "to help you remember to skip that space." But that's really misleading... there's a reason we put a zero there and if kids don't get it, then they're missing something about why they're doing what they're doing. It's sort of like being able to sound out the word versus having memorized the word to me.

    SunshineT, you say he gets division, but part of getting it at this stage, is getting the why behind the algorithm as well. So whether you're using the standard algorithm or the partial quotients one, then he needs to get why he's doing it that way. So why do you break up the dividend and divide each part and then add the quotients? And, actually, when you do the standard algorithm, you are still breaking up the dividend, just in a more proscribed way.

    I assume you're in 3C? I say get through the problems and then just hold on for 4B, which will be out in February anyway and which I'm like 99% sure will present the standard algorithm.
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  11. #10

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    And one more resource, if you have C-rods, Education Unboxed's long division videos are really good:

    Education Unboxed - Multiplication and Division with Large Numbers - Math Videos
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Teaching Long Division