The many ways of arithmetic
… are coming around to haunt me.
I rediscover Greenleaf Division – in a student’s scrap work
I just added two algebra classes for the second term, helping cover the wonderful teacher who is moving to Alaska, until we hire from the open market (I hate that name) in April for next September. Comp time lost… It’s ok. Teaching is harder, but it feels better to be in the classroom fulltime. Exhausting, but better. I’m going to write more about that…
Anyhow, I am getting to meet my new students. Some games, some quick quizzes. So here it is, Sunday night, grading, and what do I find?
Who will correct mistakes in non-standard algorithms?
Q: 180 gallons of a 40% mixture, dilute it to 36% (original was wordier). One kid sets it up perfectly, just like we’ve been talking about over at Vlorbik’s:
and then everything went wrong. I stared for maybe 30 seconds before I understood (and I am usually very very fast).
(non-standard multiplication and division beneath the fold–>)
Lattice multiplication. Twice. With the decimals guessed at the end. Wrong both times. I do not know how to teach that. How will I help her correct it?
The division looked okay, but wait, where was the dividend? Off on the side! Was it Greenleaf division, with the standard division bracket, instead of the Greenleaf one? Yup. Greenleaf division folks! Carried out to a decimal place. (maybe I’ll scan the paper?)
Damn straight I know Greenleaf, learned it myself as a tot, and worked with it for a month? two months? before transferring the nice place value knowledge to the standard algorithm. So I can help her (fix the Greenleaf, or transfer to standard). But what if it wasn’t me? What if a kid learns something non-standard, but not to mastery, and gets a new teacher? Honest, I can’t help with the lattice.
If we deny kids standard algorithms, we may well be keeping them from receiving help later on.
Of course the real hope is that she solved the proportion with common denominators, not because it is better (maybe it is), but because we taught it, which means she is open to shifting methods.
Now, go over to Dave Marain’s site, read the poll in the upper right, and remember the handicap you give kids if you never teach them the standard algorithm. And vote!
JD2718 
Can you tell me what a Greenleaf division is? A quick, really quick, search didn’t come up with anything but your post.
The question may sound strange, but I can’t think of how else to ask it, so I just will: Why can’t you help with the lattice?
And funny that you would mention the mixture problem, because I was thinking about those few days ago. Do you have any particular reason why these types of problems are important for students to be able to do?
I hope I didn’t get the name wrong – I wrote it like 5 times. It’s how I learned, at a strange school, in the early 70’s. And I will either scan the kid’s paper, or I may just do some and post. I mean to talk about division and divisibility, so I’ll consider this a call to get started (finally)
“Lattice multiplication. Twice. With the decimals guessed at the end. Wrong both times. I do not know how to teach that. How will I help her correct it?”
In any algorithm, the key to getting the decimal place when multiplying is keeping track of where your unit is (where is the product of ones in the final output?) If you keep track of the decimal point on the two multiplicands, and find the row and column where they intersect in the lattice, that diagonal will correspond to the decimal point of the product (since ones x ones = ones, and like place values align on the diagonals in the lattice).
Is “greenleaf” similar to scaffold division?
I agree with TwoPi: the essentials are the same here, and the only thing the lattice algorithm does is change the alignment of place value from vertical to diagonal. Besides, I wouldn’t repeat your woes to either peers or college faculty who have to read essays. Oh, were we to have the same easy identification (and correction) of errors as you claim would happen with a single set of algorithms!
Algorithms easy? Sherman, are you sure you don’t teach math? Because that’s not what any of the English or Social Studies teachers I know say.
“Lattice” is taught just to be different. There is nothing to commend it. No greater understanding. No great advantage in ease of calculation. And as a reward, something that looks essentially foreign. A lifetime of not being able to show someone that a quick multiplication is correct (because she would first need to explain)…
This is, I admit, the first time I encountered a kid using lattice multiplication (I’ve taught 11 years, last 6 in a school where we get kids from a large number of different middle schools, from an assortment of districts, and some out of parochial or private school.)
She told me today that the teacher who taught her long division also taught her multiplication. Not a big surprise.
Scaffold divsion? Produces the same number of hits as Greenleaf… we could be discussing the same technique. I’ll scan the page, or maybe my own work, later this week.
Jonathan
I’m with e here – I have no clue what greenleaf division is. I’d love to see more. Just in case you need another vote in favor of scanning that paper.
Of course, if you’ve only JUST met a student using lattice, it’s hard to see the generalizability. Personally, I don’t see the attraction, but my daughter swears by it, and she’s doing fine. 7ut I won’t generalize from her experience, either.
No, I’m not a math teacher, but I’m a GREAT student of algorithms. Probably one of the few with a history Ph.D. who took real analysis in college.
I think what Jonathan is perhaps trying to say is that knowing the “usual” algorithms may be a form of cultural literacy.
If we just care that a child be able to “understand” multiplication, and “get the right answer” (maybe quickly), then it doesn’t matter which algorithm a child prefers and becomes proficient with.
But if we want children to be able to “show their work” and have it be more or less universally understood, then we need to settle on a common algorithm that everyone can understand. It doesn’t have to be the one they usually use to do computations in their head, but they should be able to “translate” to it if necessary. (Sort of how sometimes people use English as a lingua franca to communicate, even though it may not be the first language of either of them.)
“There is nothing to commend it. No greater understanding. No great advantage in ease of calculation.”
I could not disagree more with this statement.
Greater understanding: Lattice is linked directly to the area model for multidigit multiplication
Advantages in ease of calculation: lattice separates into completely isolated processes the various stages in the calculation: remembering single digit multiplication facts, adding partial products, carrying digits as needed. (Compare that to the “traditional” method, where one often finds oneself multiplying and carrying in one’s head simultaneously, with greater risk of making a silly mistake.)
Example: Suppose you are multiplying 6728 x 394, and it turns out that you made a silly mistake and thought that 7×9 was 65. If you did that in the traditional algorithm, all you would see would be 60752 (or 607520) in the second row, and finding (much less fixing) the error is virtually impossible. In the lattice method, the incorrect “6/5” is visible in the grid, replacing it with “6/3” is simple, and tracing the collateral changes in the overall calculation is straightforward.
Lattice also motivates a particularly elegant and simple way to multiply polynomials.
True story: I was on a flight to a math conference (giving a talk on Briggs’ construction of logarithm tables). One of the first slides in my talk had his construction of a table of successive square roots of 10. I had no idea how long it would take to extract 40 digits of the square root of 10 by hand, so I did it on my flight: it took me 90 minutes. (I suspect someone practiced at it could do this significantly faster.) So here I was, with a 40 digit square root of 10. How do I check my answer? (I actually wimped out and only squared the first 15 digits of the root, but the choice between using the “traditional” algorithm or the lattice algorithm was a no-brainer; lattice mult is definitely the way to go with something like this.)
I like your true story. But it sounds exceptional.
However, ‘ease of computation’ arguments catch my attention. I should play some with it. I will try it out for a while.
The area stuff though, doesn’t look right.
The key to the area deal is to think in terms of partial products.
Compute 6728 x 394 by building a rectangle whose width is 6000, plus 700, plus 20, plus 8 units, and whose height is 300, plus 90, plus 4 units. Draw a diagram of such a thing (4 columns, 3 rows, with columns getting smaller as you move left to right (thousands to ones), rows getting smaller moving top to bottom (hundreds to ones).
Compute the area of each of the 12 sub-rectangles.
In each subrectangle, all you’re really doing is computing a single digit product, and keeping track of an appropriate place value.
Notice that corresponding place values line up diagonally, and the trailing digits are more of a hindrance than an aid.
More can be said here in how this evolves to the final algorithm, but I suspect at this point the rest of the details are superfluous.
I see how to do lattice multiplication but how do you do lattice division?