Helping Students Use Fingers For Math

Add-Subtract-Place Value, Fact Fluency, Multiplication-Division, Subitizing | 0 comments

Should kids be using their fingers in math class? I’m getting ready to do some webinars all about how kids get stuck in that phase where they’re counting on their fingers for addition and skip counting on their fingers for multiplication. Every time I do trainings around this idea, I get people who will tag the article by Jo Boaler all about why kids should be using their fingers in math class. I do not disagree with that article, but what I disagree with is people just reading the headline of that article and not digging into the research and what she talks about in that article about how kids are using their fingers.

Today, we will investigate helping students use their fingers for math class effectively in our quest to build our math minds, so that we can build the math minds of our students.

Watch the video or read the transcript below.

The specific webinar mentioned in this post may have expired but you can catch Christina’s current training here.

The CGI Progression

Pretty much the basis of all of my videos is the understanding that came through the work of Thomas Carpenter and his colleagues through Cognitively Guided Instruction, also known as CGI. This is foundational to understanding how to both effectively and ineffectively use fingers. So what Thomas Carpenter and his group found, is that when we do not teach students how to solve problems, they actually go through phases. How they did it was by giving kids problems in context, and then watching to see how they solved them. As their sense of numbers grew, they would progress through stages.

direct modeling

So, the first stage is known as Direct Modeling. This is where you give a kid a problem like, “Johnny has 3 balloons, and then he gets 4 more balloons. How many balloons does he have now?” In Direct Modeling, they’ll have to model each piece of that problem, and then come back and count it all again. With fingers, they might count 1, 2, 3, on one hand, and, 1, 2, 3, 4, on the other. Then they’ll have to come back and recount every digit to figure out that it’s 7. That is totally fine if that’s where the students are at, and if that’s the phase they’re in.  Then that’s what they should be doing.

count on

As they start to progress with their understanding of numbers, they get to a phase known as Counting. This is where they will hold one of those numbers in their head, and then count on to get to the answer. Typically, they hold that first number, because they’re modeling that situation. “He had 3, and then he got 4 more,” they would hold 3 in their head and then count 4 more to figure out how much it is.

The same holds true if you’re working with multiplication. Let’s say that the problem was “Sue had 3 bags of cookies, and in each bag there were 4 cookies.” A kid who is a Direct Modeler needs to model those three groups, putting 4 in each group, and then they come back and count every single one of those. In the Counting On, or Counting phase, they are basically skip counting, so they don’t have to make all four of them. They know that there are three 4s, so they’re counting 4, 8, 12. They skip count along.

These are not bad phases, and if that’s where they need to be, it’s not horrible if your kids are there. The hard part is that we will get fifth graders, and even high schoolers, who are still counting on their fingers to solve problems like 7 + 8. There becomes a time when we need to know math quickly. Yes, I want kids to develop their conceptual understanding, but if they are not fast with their facts, it ends up hindering things as they progress through. Yes, I want kids to be fluent with their math facts, but fluent doesn’t just mean it’s memorized, that’s not what I’m talking about.

There are definitely facts we have memorized, but the problem is, if a kid doesn’t have it memorized, what they end up doing is falling all the way back down the progression to that counting phase. So, the in-between phase that Cognitively Guided Instruction found was this thing called derived facts.

derived facts

Some kids may not know it instantly, but they may know a way to derive it. So, in that 3 + 4, you might hear a kid say something like, “Well, I know 3 + 3 is 6 and one more makes 7.” If it’s multiplication and it was 3 x 4, a lot of kids might know, “Well, 2 x 4 is 8, so then just one more group of 4 gives me 12.” They’re using something they know to help them with something they don’t know.

Derived Fact, and Fact, those two phases are what is often known as fluent. When kids are up in those two phases, they are fluent. So eventually, we want to get kids up there to those two phases. If they’re first graders, they may still be Directly Modeling and Counting, and that’s okay, that’s where they’re at.  But eventually, we have to help them start developing some understanding of numbers that start moving them up this progression.

The webinars that I referenced, that’s what we will talk about because it’s a big topic. It’s not something I could do in a short 10-15 minute video. So if you want to learn more about that, there are links underneath here where you can register for one of those webinars. There’s one for PreK-2nd where we’re going to talk about why kids get stuck counting on their fingers, and then 3rd-5th grade we’re going to talk about how to move kids past that skip counting phase.

Effective Way To Use Their Fingers For Addition

In today’s lesson, we’re going to talk about how to help kids effectively use their fingers, and the difference between kids who are counting one by one and kids who are doing something called subitizing or also known as finger gnosia. Now, what in the world is that? Well, those two things are actually what Jo Boaler’s talking about in her article. If you go to her article about why kids should use fingers in math class, and then click on the research that is linked in that article, the research is all about students who are able to subitize and do finger gnosia. What is finger gnosia? Basically, it is a subset of subitizing, at least, that’s the way I’ve been taking it, from what I have read. Subitizing is when you instantly recognize without having to count.

Like, when you’re playing a dice game, and you roll it, and you see those pit marks on the dice, and you just know it’s 5. You just subitized. When I hold up six digits, are you able to tell how many there are without counting? That’s subitizing, but it’s also known as finger gnosia. It’s being able to recognize the spatial patterns of your fingers to help you know how many, without having to count. So to me, it’s like a subset of subitizing, and if I’m wrong in my understanding, let me know in the comments.

All of the research that Jo links to is all about kids who are able to subitize and have finger gnosia, those students are performing great in math class. It is not kids who are counting one, by one, to show you 5. It’s the kids who, when you hold up your hand, they know it’s 5. That type of understanding of fingers is what’s predictive of a kids ability to perform better in mathematics. So, what does this look like when they’re adding? Let’s start there. So, let’s say use the problem 6 + 7. I could sit there and count out 6, and then count out 7, but I don’t have enough fingers to show that. That’s why kids will move into that counting phase. We move them into that counting phase because you don’t have enough fingers to show it when you get to larger amounts if you’re counting one by one. But if you have finger gnosia, and you have that ability to subitize using your fingers. Can you picture 6? Can you picture 7? Now, put those pictures together. Did you count every single one of your fingers? Kids with finger gnosia don’t. They see the groupings.

When they have 6 and 7, they will put the two 5s together to make 10, and then put the 2 and the 1 together to make the 3, then they will combine those to make 13. That is a very powerful derived fact: 6 + 7 is the same thing as having (5 + 5) + (2 + 1). That is a huge way to think about those problems, and it’s a way that, without that visual, without finger gnosia, becomes very difficult for our kids.

That’s why students with finger gnosia can perform better in mathematics because they’re able to use those visuals and then solve the problem. So if you do read through Jo’s work, it talks about when kids were thinking about the problem, the parts of their brain that stimulate the fingers were being used, and they start to visualize that subitizing or finger gnosia, as they’re solving problems. So, we want to help kids develop subitizing and finger gnosia, so that they can use those visuals, and they’re not reliant upon counting one, by one, with their fingers. So, finger gnosia is not just counting one, by one.

Effective Way To Use Their Fingers For Multiplication

What does this look like when it comes to multiplication? All right, so let’s do a little bit of a bigger problem. Last time I said 3 x 4, but let’s say the problem was something like “Sue had 7 bags of cookies, and in each bag she had 4 cookies.” Now, in this case, the context matters because, when it’s like that, and it’s 7 groups of 4, kids who need to count, they follow the problem. They’re going to count by 4s seven times. Yes, it would be way easier if they just did 4 x 7, and thought about it as 4 groups of 7. It’s way less counting, way less addition if they’re doing repeated addition, but the problem does not say that. The problem says that they have seven groups of 4, so guess what they’re going to count? 7 groups of 4. If they’re in that skip counting phase, they have to go 4, 8, 12, and then some of them are like, what’s 12 plus 4? Then they’re off to the side counting one-by-one-by-one, right? Then they get to 16, then 20, then 24, then 28, and do they know when to stop?

That’s the other problem. If they don’t have finger gnosia, they don’t know when they’ve done 7 groups of 4. If they have finger gnosia and they think in their mind, “Okay, I need 7.” Remember, 7 can look many different ways when using fingers, but typically, kids will see 7 as 5 on one hand, and then 2 on the other. The cool part about this is that kids will develop that very powerful derived fact for sevens. Sevens suck! When you’re multiplying by sevens, it is not friendly for a lot of our kids, but guess what is friendly? Fives and twos. Kids learn fives and twos quickly when it comes to multiplication. So, if what they have in their mind is 7, and then they could think, “Okay, well, “5 groups of 4 is 20, and 2 groups of 4 is 8, 20 and 8 gives me 28.”

So, they just took that seven groups of 4 and broke it into five groups of 4 and two groups of 4, which is an amazing derived fact. Their fingers help them out. So yes, fingers are very powerful to use in math class, but we’ve got to help kids use them effectively, and help them use those finger patterns to move through the CGI progression.

Now, if you want to learn more about how to help your kids move through those progressions like I said, I’ve got webinars coming up.  The links are below to get signed up. There’s one that’s focused for PreK-2nd grade about how to develop these understandings and help kids move past counting on their fingers.  

There’s also one for 3rd-5th grade, all focused on how to help build this understanding and help kids move past that skip counting phase. Now, depending upon when you’re watching this video, the webinars may be over, but that link might take you to a waitlist, and you may be able to sign up for the next time that I do the webinar. So, I hope that this video has helped you build your math mind, so that you can go build the math minds of your students.