Suppose you’re working on adding multiples of 10 with your 3rd Graders. (Suppose, further, that you should be planning that lesson instead of writing a blog post. This will only take a second though, I swear!)
Here’s the thought. A lot of kids have trouble with 67 + 40 but can handle it without trouble if you encourage them to add a bunch of 10s.
Our ability to add 67 + 40, before it’s second-nature, can come from being able to see this essentially static expression as rolling out in time. In other words, we can see “add 40” as “first add 10, then add another 10, do it again, and finally add 10.”
And I think a lot of learning is like this. There’s a process or procedure that we can do, perhaps, but it takes a lot of time. So we get good at that procedure, and then we summarize it in our minds. We can then see the summary as representing a process that rolls out in time. And eventually, the hope is that we don’t even need to remind ourselves of that process, and the summary suffices.
ANYWAY, a number string is an opportunity to work on the process, and even to summarize that process. For example:
67 + 10
67 + 10 + 10
67 + 20
67 + 10 + 10 + 10
67 + 30
67 + 40
But once kids are able to handle this comic-strip form of thinking about 67 + 40, we would want to give them a chance to mentally roll out this procedure when prompted with just a summary of it. So, for example:
67 + 10 + 10 + 10 + 10
And eventually we don’t want to offer anything other than the problem itself in its least dynamic, it’s most out-of-time form. Which is 67 + 40.
I think I’m getting closer to understanding Sfard.