Funny arithmetic samples
February 12, 2008 am29 8:26 am
See yesterday’s post and comments. Click photos for larger images. Is that “scaffold division?” Notice that 36 goes into 6552 first 100, then 30, then 20, then 20 then 2 more times…
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JD2718 
Your “greenleaf” is morally equivalent to what I called “scaffold”; in scaffolding, the accumulated quotient (the 100, 30, 30, 20, 2) is stacked above the solidus, then added above that. This is becoming a rather popular variant on traditional long division.
Some may hate it because it looks foriegn, but it has some HUGE advantages in the classroom. The connection to a repeated subtraction model for division is explicit; the memorized but not understood “divide, multiply, subtract, bring down” mantra is banished; and it is nearly impossible to muck up. Not sure whether the next digit should be a six or a seven? Try a six, and if it turns out later you could have been greedier, go ahead and do one more stage of subtraction (no need to erase and feel stupid). (Or do as this student did; subtract 30 copies of the divisor if that is easiest; then see what can work at the next stage.)
There’s also a nice emphasis on the meaning of place value. Notice in the image that the student has written “100”, not “1”. They are explicitly subtracting 100 copies of 36 at that first step.
It’s not funny; it’s fabulous.
I think it’s a great way to teach long division, but I can’t understand not then (after weeks or months) transferring to the standard algorithm.
It is precisely what you identified, the retention of place value, that I think makes this a powerful teaching tool.
I think those who teach this (and other repeated subtraction methods) are impressed that you don’t have to get the ‘right’ number on the first go. I think it’s great. At the beginning. But with practice it gets easier. Let’s be careful not to celebrate the inability to estimate a product.
I don’t celebrate student inability. But I rejoice in having a robust algorithm, one that students can learn, understand, and make progress with, even as they continue toward fluency with their multiplication skills.
If the primary goal of instruction in long division is to produce pupils who are fluent in the so-called traditional algorithm, then naturally it is critical to shift to that method at some point in the course of instruction.
If the primary goal of instruction in long division is to produce pupils with greater fluency in arithmetic, who are capable of computing quotients of multidigit numbers by hand, with a deeper understanding of division and multiplication, and who will be able to apply their understanding to polynomial arithmetic, then I honestly don’t see any imperative to move beyond the scaffold or greenleaf algorithm.
Clearly, I’m not sold on standardization as a goal, or even as desireable. (Cue the Pink Floyd music…)
I don’t think I’ve every *seen* this division algorithm before, but (unlike lattice) it was immediately obvious to me what she had done, and that it was reasonable. I think it’s a wonderful algorithm for division, actually. I don’t see any reason to require or favor the “traditional” method over this one.
Thanks for posting this. Like mathmom, I’ve never seen it either but was able to follow it the student’s work. Interesting.
This is how I learned long division, and I think it gave me a leg up. I use a form of it, still, in checking divisibility mentally for some numbers.
But I think there is value in having a shared algorithm, and ours looks different from this.
Our standard algorithm knocks the zeros off the “special numbers” leaving place value represented by physical place within the space of the written work. Shorter? Yes. Better? No. Accepted? Yes.
If we decided that all kids should learn this type instead, I think that could be ok. But in the meantime, it would be good if we had a shared type.
The thing I like about this is that it doesn’t require you to know, or maybe write out in the margin, your 36x table before you start. I suspect it’s in many ways more efficient than the standard algorithm for kids who don’t have good mental multiplication-by-one-digit skills, which these days seems to be most of them.
While we’re talking standard methods, what’s your subtraction method? Borrows or carries?
My method of subtraction? For teaching? or for myself?
I don’t teach subtraction. I am of two minds for polynomial subtraction. (Different topic. More, maybe, later).
But for myself, I neither borrow nor carry. I add to the subtrahend until it is a “specialer” number, and then add that to the distance to the minuend.
Hmm.
3247 – 1984
2000 – 1984 = 16
16 + 1247 = 1263
Fairly automatic, I don’t even think about it. I do have to stop before helping anyone, though.
JD, as we’ve discussed in the past, this is how my 6yo likes to subtract also. He did also learn how to “borrow” at some point last year, though. He finds the “trick” about borrowing across multiple zeros quite amusing, anyhow. But he still subtracts (and does other mental math) in his head using techniques like this.
I totally forgot the discussion, but it’s so cool that he discovered this on his own.
Constructed knowledge, right? I love it, even though I’m no ‘constructivist.’
That’s how I first learnt to do division, as a step towards the standard long division algorithm.
If we decided that all kids should learn this type instead, I think that could be ok. But in the meantime, it would be good if we had a shared type.
My daughter is pretty good in math, but struggled with long division — or at least found it annoying enough to complain endlessly about. And I think a lot of that had to do with jumping straight to the standard algorithm. The scaffold step may be a crutch, but crutches are useful sometimes, particularly when they’re not just mnemonics, but have a really connecting to the process behind the algorithm.
I sometimes finds myself blanking a bit on the quadratic formula (I don’t use it on a day-to-day basis…), but I know I can always go back to completing the square. And I think scaffold division is similar — the standard approach is quicker and more familiar to most people, but the other gets you to the answer in an obvious, if slightly clunky, way.
I’m high enough on the education chain that what I really want it some algorithm where students will have a basis for figuring out polynomial division. It really helped 20 years ago when you could count on students knowing long division via the standard algorithm. The scaffolded division works fine for polynomials, though, so I’d be happy with it too. I’m under the impression that a lot of my freshmen, however, don’t know how to divide at all without a calculator, and that’s not so good. So if there are any middle school math teachers listening, please make sure they know how to do it one way or the other.