The image at left shows the lattice or gelosi method for mutipying 35 x 76. You put the numbers to multiply along the top and the right sides, with sufficient rows and columns for each digit. Each box is divided diagonally. You multiply (in any order) and the result goes in the appropriate box, with the tens above the diagonal and the units below. Once all the multiplications are done, you add along the diagonals (the red arrows in my pic), starting from the bottom right corner, working to the left and carrying where necessary.As a method for long multiplication, it works very nicely, and is probably a little simpler to learn than the more usual columnar approach. It is certainly worth showing to students - some will no doubt prefer the standard columnar approach, but others will like the lattice (at least until they get calculators).
But the other day I found another use for the lattice.
I was introducing a Year 10 class to multiplying polynomials. I used a lattice (minus the drawn diagonals) as shown at left. We collected the like terms (which conveniently lie diagonally). My students found this approach very accessible. While I did not suggest it on the day, an obvious way to speed up this approach is to omit the variables and work with only the coefficients.Then I happened to look at a blog I've been occasionally reading, and the lattice was mentioned there as well. But in this case, it was about someone arguing against using the lattice method in the classroom. Say what? Sadly, it is true - there are those who would have teachers confine themselves to teaching only one way (in this case, the standard columnar approach), and who disparage any other approach and denigrate anyone who would teach any other approach.
I can't even begin to fathom the mental bankruptcy necessary to say to someone else "there is only one way that students should do it". For me, part of the beauty of Mathematics is that there is always more than one way. Some ways will be faster or more elegant. Some ways will be slow wandering paths, but you may learn other things along the way.
The person who insists that students do things only one way is no educator.
2 comments:
Yes, my blog, I suspect, contained the essay on lattice multiplication and the brick-headed resistance to it out there. Since my post, there have been comments from at least a couple of such concrete-skulled individuals, showing that even when you try to enlighten people, no amount of reasoning can convince most bigots. And bigotry is very much a factor in the Math Wars here in the US. I've seen it pop up elsewhere, too, including in Oz, sad to say.
What's so pathetic is that if you did nothing more with lattice multiplication than presented it to kids in elementary school and asked them to figure out why it works, and then re-presented it to kids in middle school precisely for doing polynomial multiplication as you've done, you'd be doing them a great service. Ditto for would-be teachers of K-8 mathematics (high school, too, but you can't teach THOSE people ANYTHING).
You say, perhaps with tongue in cheek, that like terms "conveniently" show up in the same diagonal. Of course, it's precisely the fact that it's no coincidence that they do that lies at the heart of why the method works for various applications of multiplication. It made me borderline nuts (some would say I have been well OVER that border for a long time) to find in-service upper elementary school teachers who taught lattice multiplication in grades 4 and 5 (only because the textbook they had to use covered it), but who not only didn't know WHY the method worked, but exhibited not the smallest curiosity towards figuring it out when I asked them if they knew. And when I have presented multiplication to in-service teachers, I'm amazed that when I ground everything in expanded notation and the distributive property (e.g., 23 * 34 = (20 + 3)(30 + 4) = 20*30 + 20*4 + 3*30 + 3*4, and show that this is what's at the heart of most algorithms for doing multi-digit multiplication, including the lattice method, and when I model this physically through things like decimal graph paper or drawings, I'm told that this is all WAY too hard for THEIR kids; hence, they'll stick to what is "tried and true." That the standard algorithm is also grounded in all this, that it hides what's going on in ways that confuse a lot of kids (probably a lot of the teachers, too, if they had to explain it), and finally that these potentially more transparent approaches REALLY prepare kids for, yes, polynomial multiplication (and without the idiocy of FOIL), seems to make very little impact on some teachers. I've learned that the phrase, "My kids couldn't do it this way" is often code for "I don't get this; I don't like it; and I'll be jiggered if you'll make me teach it, short of putting a .44 Magnum to my head." And frankly, in some cases, that last idea is probably not such a bad one.
Set rant to [OFF]. :^)
Yes, it was your blog, sorry, I should've linked back to it.
I think you hit the nail on the head with the phrase "tried and true". Too often this (and another phrase "if it ain't broke, don't fix it") become mantras for "I don't want to have to think about what I'm doing, I just want to do it, and move on". But when we inflict that type of (non)thinking onto students, we do them a major disservice.
It seems to me that many of these "traditionalists" don't understand at all what they are doing when they apply the algorithms they learned as children - the whole matter is like a series of dark tunnels to them; they have learned to follow one set path to come safely out the other side, and the thought of taking a different path scares the pants off them. (Probably not a great analogy, but you get the idea.)
Brett
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