## Conceptual and Procedural Knowledge – Jenny Bay Williams

I would like to talk about the important
relationship between conceptual and procedural knowledge. What happens
sometimes, in our efforts to help shift away from this skill and rote practice, is
as we say things like, we need to focus on understanding. And, that is true. I’m an
advocate for understanding. But sometimes, when we say it, it sounds like we’re
pitting it against the development of skills. Which, is actually the opposite of
what we’re trying to do. We’re working on conceptual understanding because it’s
critical to students really becoming flexible thinkers and good at solving
problems. In other words, fluent in mathematics or proficient in mathematics. So, as an example I would like you to think of a story problem that you might
tell for a subtraction problem like 58 minus 49. In general, the stories that we
think of and this is true whether it’s a whole number like I just gave or whether
it’s a fraction problem, which I will share in a minute. But, the stories that
we come up with tend to almost always be takeaway stories. We forget about the
importance of telling compare stories. Now think about how you might solve that
problem, 58 minus 49. They’re so close together that we might just count up. One
to 50 and then eight more up to get our answer. A fraction example would be like
nine and one eight minus eight and a half. Like, to think about the story that would go with that. If we do a takeaway story
we’d be at 9 and 1/8 and we’d have to take away eight, oh this way. And, then
take away 1/2. That is not the way we’d want to reason through it. We would want to reason through it by picturing ourselves at 8 1/2 jumping up a half to nine and
one more 1/8. So we’d have the 4/8 jump, 1/8 jump, 5/8.
There’s a difference between those two values of five eighths. Well that means
we have to help students understand subtraction as difference. Not just
subtraction as removing something. And, that’s a critical relationship. And,
that is true for all kinds of numbers. And, we under tell stories that have to
do with compare. And, if we, if that is the case, then we limit students reasoning when it
comes to finding the difference. And, difference, as we know, all the way up
through school becomes very important as we work on the difference of two squares
and things like that. So, it’s very helpful. And, then relatedly to develop
that conceptual understanding the number line is a great tool. So, we can reason on
the number line and that in turn helps students see ways they can solve a
problem flexibly and not just use the standard algorithm when they don’t need
it. So, there’s a really strong relationship between students’
performance when they understand, when they have the conceptual understanding
with the procedural fluency. The two of them support each other. So, a student who has a weak procedural background, their conceptual understanding can help them.
But, in the same way a student with a strong conceptual unders, with a strong
procedural understanding that can help them develop concepts. So, we want to pair them up and always be toggling between the
procedures and being flexible with the procedures and the related conceptual
understanding. Why does the strategy work? So, they’re partners they’re not a choice
one or the other