|Source: Math is Fun|
In the last couple weeks she has developed her own strategy for finding answers to equations like these. She does it by adding it all up on her fingers. We did the roll-the-dice grouping game a bunch of times over the course of a month (January, I think?) and she seemed to catch on to what it was all about. But now, it seems, counting on her fingers works just fine except that when she gets to numbers over six or seven it can take a long, long time to find the answer. She's not noticeably perturbed by the effort, and from my point of view it's awesome that she understands that she's counting nine, eight times to get the answer. That's the main point of what multiplication is at this stage, right? .
Although, there she was, still trying to figure out the answer. So, I offered, "How about I go get the multiplication table?" She agreed but when I brought it back she said, "But I don't know how to use it!"
Visually tracking columns and rows is difficult, even for an adult sometimes, so I hit upon two strategies, which I showed her:
The first is to put your pencil in the first column, in this case the 0, and leave the tip just above the nine row, then move the pencil column by column until it's in the right one for the equation, in this case the eight. The tip points directly at your answer.
The other strategy is to just find the nine in the shaded area on the left and, using your finger, point to each box in that row and say "zero, one, two, three..." until you get to eight and there's your answer. Easy, right? But the girl had a different reaction:
"THAT'S CHEATING! If I do it this way, how am I going to really understand it????!!!"
I must admit I was a tiny bit impressed with this statement but I was also more than a bit flummoxed about how to respond. I mean, what's there to really understand? You learn what multiplication means (which I think she's got), you learn your facts and then you use them when you need them. I finally said:
"Well, addition is the most important thing to know how to do in your head. When you add in your head you are learning how numbers combine and recombine to make other numbers. It's an important skill to have and that's why they don't have a facts chart for it. It's not quite the same for multiplication -- what people usually do with the times table is memorize it. There are all sorts of fun number patterns to find in this chart..."
At this point she was still absolutely convinced that using the chart to find any answer constituted some kind of unlawful activity. Trying a different approach, I said, "Okay, I have an idea. If you really want to understand how and why multiplication works, you can skip count. You already know how to skip count zero, ones, twos, fives and tens. So all you need to do is learn how to do that with threes, fours, sixes, sevens, eights and nines. It's like adding..."
That was the best I could do in the heat of the moment. My daughter is prone to ginormous reactions and sometimes it's hard for me to think clearly under duress. But, the question still remains -- is it 'cheating' to use the multiplication chart? I should add that she seems to have no interest in memorizing the facts either.
After writing this all out I'm starting to think that that maybe all this (her finger counting strategy, eschewing the chart, etc.) is just her way of mastering the content. All the same, I am wondering what kind of understanding she wants. Maybe she's looking for familiarity? Maybe at some point she'll finally figure out on her own that memorizing the times tables facts is actually pretty helpful? Am I missing anything here?
Sigh. If you have any thoughts, including helpfully pointing out any errors you find in my explanations and reasoning, I'd love to hear them...