Here’s something I often hear from parents of young children who are struggling with math: “They’re teaching them some new way of multiplying that is so slow and confusing; I think if my kid were taught the way I was taught, everything would be fine.” The primary reason for this complaint, of course, is that when kids are taught in a way that’s unfamiliar to their parents, it’s harder for those parents to help.
A friend of mine suggested a way of thinking about this kind of thing that might be helpful for the frustrated parent. Before I pass that along, just a quick bit about the method of multiplication at issue here.
The “new” way is in fact an extended version of the way that is familiar to many present-day parents. The reason for using it with young children (or at least the reason I often use it first with young children) is that it is more transparent than the old way and thus often easier to understand and remember. Here’s a little visual demonstration of the unfamiliar method known as partial product(s):
(via Mrs. Kent’s blog)
The “old” (familiar) way involves a few shortcuts that conceal the place values of the digits. In order to understand and keep track of the shuffling around of places, a person needs a little more experience with place value than many children yet have when they are first asked to multiply multi-digit numbers. They can still get the answers, if they can remember the steps, but they’re often arriving at those answers without conceptual understanding. That can be very disconcerting – kids often sense that they’re just going through some motions which they don’t understand. We think our way it should be easier for them because it’s familiar to us, but when kids don’t understand why it works, it often isn’t easier.
The “new” way is a smaller step from where kids just were (multiplying single digit numbers by each other), toward multiplying multi-digit numbers. It keeps the steps distinct, and when they’re distinct, the reason for doing each one is clearer. (It’s also often easier for a child to move on to the old/familiar method, which can be ultimately less time-consuming, once he or she has mastered the extended version.)
I’ve cloaked the words new and old in quotation marks because it’s my guess that this “new” way actually came before the “old” way. I spent a few minutes reading up on it, but that was a bit of a rabbit hole with all sorts of interesting tributaries for a person interested in such things, but for now let’s just leave it that logic would suggest that in order to create the method most of us are more familiar with, someone would first have to have built this extended method. Both methods are actually just means for keeping track of all the smaller multiplications involved in finding the answer to a bigger multiplication. They’re shorthand. The “new” way is longer shorthand than the “old” way.
So, back to my friend’s suggestion. I told him about this frequent complaint, about the unfamiliar multiplication. And I said “Why is it that when something’s unfamiliar, we’re so quick to blame it for any related confusion or failing?”
He thought for a minute and said “Well, it kind of sounds like the way we are with most new technology.”
This made so much sense I was a little irritated at myself for not having thought of it already. New stuff is easy to blame for the discomfort that comes with disruption. But it isn’t always worse than what we have, and sometimes it’s better. And then sometimes it isn’t better, which is all the more reason to look into it carefully when it comes along.
So now when I talk to parents about how their children are or aren’t being taught multiplication, I try to remember to bring up the context of new technology. Methods for performing computation might not seem like technologies, but they are. And it might seem silly to spend several paragraphs proposing the analogy. But I see so many children lose their footing in math right around the time that multiplication comes to town, I think that anything we can do to increase the chances that they have every available tool and every available mode of support available to them that might help them keep their confidence about them is worth it.