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.”
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.
I agree I’m at 3rd and 4th grade and this belief has led to teaching all kinds of different methods to conceptualize basic math at the cost of kids never atctually getting a solid grasp of any of them. It also has convinced people at this level to ignore basic number sense and math facts with this idea that they will learn it once they get the concept down.
One aspect of this illogic is the claim if you learn it without conceptual understanding it won’t stick. This begs to have some data to back it up. If it didn’t stick in the next exam then it would be trivial to claim teaching with conceptual understanding leads to better exam results.
Perhaps it doesn’t stick a year later? But as your previous example of fractional division hints at lots of people remember that method but not a solid proof or even a good explanation.
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.
I agree I’m at 3rd and 4th grade and this belief has led to teaching all kinds of different methods to conceptualize basic math at the cost of kids never atctually getting a solid grasp of any of them. It also has convinced people at this level to ignore basic number sense and math facts with this idea that they will learn it once they get the concept down.
One aspect of this illogic is the claim if you learn it without conceptual understanding it won’t stick. This begs to have some data to back it up. If it didn’t stick in the next exam then it would be trivial to claim teaching with conceptual understanding leads to better exam results.
Perhaps it doesn’t stick a year later? But as your previous example of fractional division hints at lots of people remember that method but not a solid proof or even a good explanation.