Undoubtedly, it’s true that students would benefit from seeing a 3 as a composite object.
But it seems to me a bit like cheating to say that this has to do with seeing the equals sign as the “do something signal.” The issue isn’t so much the equals sign as a signal, the issue is that The reason why kids read the equality as “do something” is, I think, plausibly explained by their reading of “a 3” as “a mystery number plus 3” instead of “the composite expression a 3,” which would indicate the more sophisticated understanding And what of Carpenter?
This passage summarizes two possibilities from Kieran and Carpenter.
If students had a relational view of equality, this would be better. Then these students would see a 3 = 10 as saying that “a 3” has the same value as “10.” They would treat “a 3” as a complex algebraic object, not as a variable with an operation.
as describing a process (“you get 97 when you multiply some number by 5 and add 32 to it”) it’s hard to take a step back and say, OK, how would this process have gone different if you didn’t add that 32? The conception of equality is made to stand for much bigger and more complex shifts in understanding.
That step back is a meta moment; it’s when you’re able to start talking But is that change all about the equal sign? (Update, 12/20: It seems to me that I might have not properly described the “do something” signal.
You should subtract 32 from both sides, then, because if you In other words, I see no reason why “backtracking” (or “undoing the steps”) wouldn’t be available to a “do something” student.
What about seeing the equals sign as “do something” would get in the way of backtracking?
In sum, if you think about the equal sign in a shallow way then you probably weren’t solving an equation using algebra in 8th Grade.
And you were less likely to correctly solve equations across all middle school grades.