khan academy has been getting lauds for being a way anybody can get good instruction.
So, let’s see what it has to say about multiplication!
I really like that it confronts those visual things students do that those of us who attribute meaning to the symbols don’t do. He notes that when we add 2 + 3, and he draws some circles to show it, that we can see the 2 and we can see the 3. WIth multiplication, we *don’t* see 2 and three… we have to count the groups of either two or three.
I also like that he shows adding first … uses the “start with what you know” principle. I *don’t* like that for adding two plus 3, he adds 2 magenta cherries and 3 blueberries… to get 5 pieces. And now, you’re going to tell me later that I need a common denominator when adding fractions because we need to be adding the same thing? I’m supposed to have gleaned that from between the lines, perhaps?
He then goes on to talk about multiplication as repeated addition, with a rather strong implication that “that’s all it is.” The pictures… oops, they’re gone, and now we’re just using the symbols, with a few arrows between them for emphasis that 2 x 3 is 2 + 2 + 2. I’m pretty sure he’ll get to some arrays, but to me it doesn’t make sense to start with the abstract symbols.
He does introduce the very important idea that you can sometimes take different mathematical paths that are both correct and get to the right answer. “Regardles of the path you take as long as you take a correct path you get the same answer.”
Now, what’s the use? Groups of three lemons… something kinda visual. There’s an easier and a faster way… (I’m waiting with bated breath — how is knowing this really *faster* than counting?) He does it by adding. 3 + 3 is 6, 6+3 is 9, 9+3 is 12. He introduces the idea of “four threes” meaning four times three, which I like ’cause it’s a good preliminary to 4x.
I like the adding of the explanation of 3 x 1 and why it’s 3, not 4.
I think we can do better, though.