Say it often enough, and people will believe it
I just read this piece about math education. On the whole it wasn’t too bad, but this one quote caught my attention, because the logic (or illogic) of it is something I see in many arguments about the “failings” of traditional math education:
"It is possible to learn to follow algorithms or enact procedures without having a full grasp of the concepts that underpin the procedures. But when you learn math that way, as arbitrary abstract rules without conceptual understanding, it doesn’t stick. And that’s one reason why somebody can graduate from high school with the scores on the test they need to get into college. But then by the time they get to college, which might be months or years later, computational rules without conceptual understanding hasn’t stuck with them.”
What is left undefined in this and similar writings I’ve seen, is the term "conceptual understanding". One can learn the algorithm for fractional division without understanding why it works in all cases. But that doesn't mean one lacks the "conceptual understanding"--i.e., what fractional division represents. Similarly, in calculus, one doesn't need to understand the rigorous definitions of limit and continuity in order to know what derivatives are or how to compute them. One can operate with an intuitive understanding of what derivatives represent as well as limits and continuity. The intuitive and contextual understanding is a form of the holy grail of "conceptual understanding".
But this business of "it doesn't stick because they don't know the 'why' of it" has been repeated so often that many take it as the truth. It would have some truth in those cases in which only the algorithm is taught without the context. I haven’t seen that done, but such statements lead many to believe that that's the prevalent way math is or was taught.
The straw man in these arguments is that our kids can do the algorithms but can't understand the concepts, when in fact many kids can't even do the algorithms. When I taught sixth-grade math to a homeschool pod, the kids (from high income, educated backgrounds) couldn't do long division. They understood what the concept of division was, but they simply hadn't been taught the algorithm.
Agreed. And the flip side does not get properly examined: I.e. the constructivist camp talks as if, if students have the concept they will remember everything for years. Which, in my experience, is not the case. Forgetting is inevitable and requires its own set of strategies to deal with.