A Journey Through Everyday Math

Before I get to the next promised topic, a bit of a digression to take a dig at EDM.  For those parents who have made it to third grade (with their children, of course), the biggest complaint I hear is about the Lattice Method of Multiplication.  Never having used the Lattice Method for real calculations (although I have played with it a few times), the question I have is how can anyone say it is "very efficient and powerful", as the EDM Teacher's Reference Manual does?  So I did a little study.

If we look at the operations performed in the standard algorithm and Lattice Method, the Lattice always has more operations (defined below). If I can prove this (and I will), and it is very evident that it takes longer to draw the lattice as the problems becomes larger (moving from 3x3 digit multiplication to 4x4 to nxn), it therefore disproves the EDM Manual's assertion that "The authors have found that with practice, it is more efficient than standard multiplication for problems involving more than two digits in each factor." If there are more operations in the Lattice Method, AND you need to draw the lattice as opposed to just two lines in the standard algorithm, the Lattice Method has to take longer. I don't know how you define efficient, but I have taught my daughter that it means it takes less time (Dictionary - "producing effectively with a minimum of waste or effort "- and we all know what the "waste" is here).

This is not nitpicking, this is a direct attack on the poor quality of material in the teacher's manual, and is reflective of a cavalier attitude toward truth, accuracy and believability in EDM (more to come on that).

And so to the proof (for n x n lattice):
Operations involved:
1. multiplications
2. additions
3. carry in multiplication process
4. carry in addition process
5. writing multiplication results
6. writing addition results

There will always be equal values for multiplications (a nxn digit multiplication will always involve n-squared multiplications), carry in the addition process (there will always be 2(n-1) additions), and writing addition results (the answer will always have at most 2n digits).

For the other operations, the lattice requires 2n-squared (2 x n x n) additions, 2n-squared numbers to be written for multiplication results, and zero carrying in the multiplication process.

For the other operations, the standard algorithm requires (n(n+1)) + (n-1) + (n-2) + (n-3) + .... (n-n) additions, as well as the same for numbers to be written for multiplication results, and (n-1) squared for carrying in the multiplication process. I tried to simplify this but failed (should have paid attention in series class), but doing it by hand for 3x3, 4x4, and 5x5 showed that the lattice method used 2,3, and 4 more operations. When you go to higher n's (yes, I used a spreadsheet), you add three operations to the difference for each increase in n (so for the 6x6, the lattice uses 7 more operations; for 7x7 it uses 10, etc.),.

So, in reality, it is less efficient than the standard algorithm.  Don't you wish your children brought home a text book so you could see what the authors of EDM are telling your children about math? 

Can you believe EDM?  To alter a famous song title (Journey): Don't.  Stop Believing!

Brian BTN