First posted December 2006. Ponytails was nine years old.
Ponytails has been doing multiplication since first grade. Miquon Math starts teaching multiplication concepts early, since saying "three five-rods" is no harder than saying "a five-rod plus a five-rod plus a five-rod".
However, now that she's in fourth grade and has moved on to Quine's Making Math Meaningful series, we need to do some serious work on multi-digit multiplication. We worked on that a bit in the last year of Miquon Math, but Ponytails has forgotten some of it, and anyway, she's older now and can make more sense of it.
The Apprentice did this level of MMM several years ago, and I remember going through extreme frustration with it (both of us). They kept explaining and showing, explaining and showing, breaking questions apart until we weren't even sure what we were looking at anymore. Finally I told the Apprentice, conspiratorially, that I was going to teach her a shortcut, and I taught her the multiplication algorithm--the old-fashioned way, the school way. She got it. For her, that was a relief. No more explaining--just do it.
Ponytails needed a slightly different approach. We go in and out of the MMM book; we've skipped a lot of pages in it because there are things she already knows well (like place value and addition), but then there are things that she needs some extra preparation for, and the MMM teacher's book doesn't always explain them in a way that makes sense to her. So we've been working in this sequence: single digits multiplied by single digits; multiplying things that end in 0, which MMM does do a good job on (like 300 x 20); and now two digits multiplied by one or two digits. Yesterday we talked about two ways to handle those bigger numbers, and today I added a third, the one that MMM emphasizes and that the Apprentice found frustrating. What do you know--it makes sense to Ponytails.
Let's say the question is 23 x 45. The first way is to list the smaller questions you could break those down into, multiply them, and then add them all up. So, 20 x 40, 3 x 40, 20 x 5, and 3 x 5. The problem with that method is that you aren't always sure if you've gotten all the combinations.
The second is to use the "school way," the algorithm.
It's the quickest way for me because I've been doing it that way for thirty years. The problem with it for Ponytails is that she isn't sure yet of all the steps, and keeps adding where she should be multiplying or vice versa. It takes time to get familiar with this one.
This is the third way, and it's almost like the first. You draw an empty square. Across the top you write "20, 3" and down one side you write "40, 5." You divide the square into four boxes (in this case) and fill in each box, as if it were a times table chart.
The advantage over Way # 1 is that when you're done the boxes, you know you're done and you haven't missed anything. The disadvantage is that then you have to recopy all your products to add them up, unless you can do it in your head. Ponytails says she doesn't mind that, and it's easier for her right now than remembering all the steps in the algorithm. I wrote out some word problems for her to do, and she decided to do one of them with the algorithm and the rest with Way #3.
It's always nice to have choices.