Recently, I received an email from someone who had attended a webinar of mine and they asked this question.

Now these are terms that are batted around a lot in math education. We hear a lot about procedural fluency, operational sense, conceptual understanding and it’s like how do all of these mesh together? Do they mesh together or all they all different things?

Watch the video or read the transcript below:

If you want more friendly tips to help you build your students’ face fluency with addition and multiplication, make sure to click below to watch my free trainings. Just click the button below to access the recordings!

Terms That Are The Same

Let’s first start off with the ones that I feel like are the same, okay. Now you may feel different about this, so in the comments below, let me know if you think about it differently, but this is the way my brain thinks about these pieces. When I see procedural, computational, operational, those three terms to me mean the same thing. It means that I am computing, I’m doing procedures, I’m performing an operation in mathematics. So to me, those are the same thing. The procedures are the steps that I’m taking to complete that operation or that computation. So when I see those, I instantly think those go together.

Terms That Are Different

Now, what are the pieces that are different then? So if I take her two terms that she had there, let’s just look at computational and she had that twice, so computational fluency versus computational sense. So if we agree that computational just means that our kid’s ability to do the computation. Can they add, can they subtract, can they perform that mathematical thing? We’re just looking at can they compute.

Now let’s take a look at the last piece. The fluency versus sense. That’s what I want us to take a look at for a moment. Let’s look at one problem in particular. So let’s say we’ve got this 39 plus 45 here.

Now, can the kids just compute and get an answer? To me, that is computational fluency. Now there’s a bit more that goes into that, that’s one of the things that I talked about on the webinar’s was that fluency really does have three parts. It’s not just can they get the right answer but are they efficient?

A kid who’s sitting there doing 39 plus 45 and they’re making 39 tally marks and 45 tally marks and coming back and recounting it all, that’s not efficient. They could compute, they got an answer but it’s not efficient. So a piece of fluency is they have to have it right, they gotta have it be an efficient way to get it right, but the third piece of fluency is really are they flexible? Do they have other ways to think about it?

And to me, this is where there’s a little bit of overlap because as we talk about computational sense, the computational sense is really that flexibility piece. It’s when kids look at a problem, are they first looking at the problem and then using their sense of numbers to know how to do that computation? Because how I solve 39 plus 45 is actually very different than how I would solve 22 plus 45.

Now, don’t get me wrong, back in the day when I was in school, I solved those completely the same because I was very good at the procedures. I did not have any computational sense. I could just perform procedures. So computational sense is when kids look at problems, do they have different ways to think about it? And this is where that flexibility piece of fluency comes in.

So, there is a little bit of overlap but the computational sense is do they understand that those problems can be done differently? And some of you may do those exactly the same and hopefully by the end of this, you’ll have a little bit different way to look at it.

Conceptual Versus Procedural

Now, one of the pieces that I did want to talk about was something that she really didn’t ask for but I’m gonna bring it up because it works well here. And she had talked about it at the very end, her last question is what is conceptual understanding? And to really understand conceptual understanding, I want to talk about what it’s usually compared to.

There’s kind of this debate in math education about which is more important, conceptual understanding or procedural fluency? So procedural fluency, as we’ve talked about, is doing the procedures. Can they fluently do procedures? So when kids are solving that problem, are they fluently doing a procedure? Now, the other piece of math education is that conceptual understanding. Do they understand why it works? Do they understand how this works? Do they understand connections?

For me, as a kid growing up, I call myself a recovering traditionalist because back in the day, we were only taught the procedures and I was really good at that but I had no understanding. So when it was 39 plus 45, the only way, and I’m serious, the only way I knew how to do it was this procedure where you stack it and you add your ones and then you carry over the one and then you add your 10s and that gives you your answer. I seemed to have procedural fluency.

But, I had no idea what I was really doing. I was just following the steps. I didn’t even understand that that flying little 1 was really a 10, that’s how much I did not understand. And it wasn’t until I became a teacher that I realized, I have to understand that stuff. It’s not enough to just be able to do procedures.

That’s why there’s been such a push to build in that conceptual understanding. When we’re doing the same problems, still 39 + 45 but instead of just the procedure, what do we want kids to understand, what’s the conceptual basis that they need to understand about adding with multi-digits?

We want them to understand that within 39 is a 30 and a 9 and with 45, it’s really a 40 and a 5 and I could add my ones and I could add my tens and then I could put those together. And I want them to be able to make that connection to the procedure.

All of it works together. It is not enough to just build conceptual understanding and let kids have that piece. Kids do need to build their procedural fluency and the standards do talk about that. Kids need both pieces. It’s not that one is better than the other, it’s that kids need both.

And I just want to come back around and tie this all together from this procedural fluency that we were talking about and conceptual understanding that the other piece of that procedural sense is knowing when that traditional procedure may not be the best thing.

Now, with 39 + 45, I now solve that differently. Now that I have more of a sense of how numbers work, I can look at problems and decide is the traditional procedure or that traditional algorithm, the best way to go with this or using my sense of procedures, of numbers, my conceptual understanding? Or, is there may be a better way that’s more efficient but yet still gives me accuracy and shows my flexibility?

All of those pieces go together to build a true mathematical thinker. The last thing I’m going to leave you with is why it’s so important. Because of a problem like this, 17 – 8, you don’t know how many times, I’ve seen middle schoolers, even high schoolers, when they get confronted with 17 – 8, they’re either counting on their fingers or they stack it like this and then they’re doing the procedure.

They know how to do the subtraction procedure, they go over and they say, in their mind, they’re like, “well I can’t take eight from seven, so I’ve got to borrow one or regroup,” whatever terminology you’re using, but they’ll regroup it and now they’ve got a little flying one there with the seven and they don’t understand that still guys, it’s 17 – 8, nothing has changed. Just because you’ve done that procedure for subtraction, that doesn’t mean they have any conceptual understanding of what’s going on.

This is why I really am a proponent of doing both. So, conceptual understanding, procedural fluency and part of building that conceptual understanding is helping kids develop that computational sense. It’s not just about being able to perform it, to be a calculator, a computer, we want to build their math minds and that comes by building their sense.

Now, for a limited time, I do have the webinar’s up on replay. If you’re lucky enough to be reading this right when I post it, the webinar’s are up and available for you to watch. There’s one on addition and one on multiplication to help you really understand, how do we build fact fluency and also build their sense as we do it. I hope that this has helped you build your math minds so that you can go build the math minds of your students.

Pin This To Pinterest for Later