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.
