I’m Christina Tondevold, The Recovering Traditionalist, and today I’d like to talk about Alligator or No Alligator in Math Class? in our quest to Build our own Math Minds so we can Build the Math Minds of our kiddos.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog.

The Math Coach’s Corner’s post: Alligators are For Swamps

Dr. Yeap Ban Har’s website

Well, welcome, fellow Recovering Traditionalists to our monthly hot topic. Now, I know I said that this video is going to be about whether or not we should use alligators, but it’s actually so much more than that.

The difference between math conventions and math concepts

There are concepts in math that we can help our students build a very deep understanding about, and we can help them explore relationships from that concept to other mathematical concepts. Then there are also conventions in math. Now, these are things that someone has decided this is what we’re going to do or this is what we call it, and there isn’t anything deep to learn about it.

Math concepts are really big ideas that we need to help our students build a deep and connected understanding around. Math conventions are things that we just need to tell kids.

Our hot topic this week inside of the Build Math Minds Facebook Group is a prime example of how sometimes we can try to help kids to build a deep, connected understanding of a convention when instead we should be focusing on helping kids build a deep and connected understanding of the concept. 

The above image is the original post that got a little heated. It’s not like people were getting mad, but they’re very passionate about the use of crocodiles in math.

We’ve all probably seen the alligator or crocodile, whichever one you want to call it. Many of us have used that, where the alligator eats the bigger one because it’s an easy, quick way to help kids be able to know which of the symbols to use. 

The original poster, a homeschooling mom, brings up a really good point in that her son can tell which one is greater than and less than. But the problem is, he doesn’t know which symbol to use.

I know that people get really heated about the use of crocodiles or alligators, whichever one you want to call it, and in the comments, there were lots of examples and other ideas.

Ways we teach kids to use the symbols

A few people even shared the Math Coach’s Corner post. If you’re not familiar with the Math Coach’s Corner, she gives examples of ways you can teach kids to use the symbols.

One way is you put the dots on the end points and the vertex of the signs. The side that is opened up has two points and it’s next to the larger one. Whereas the side that only has one point is next to the smaller one.

Another way to help kids understand the greater than and less than symbols is showing them on a number line. As you go to the right…the numbers are getting greater on a number line and you can notice the greater than symbol over there. As you’re going to the left and the numbers are getting smaller in value, that’s the less than symbol at the end. 

Did any of you guys learn that way? I didn’t learn it that way.

The way I learned was by using my fingers. When I looked at my left hand, I’d put my fingers up in front of me in an L shape for less than. Using my right hand for greater than, I’d put my fingers up into a backwards L. 

Focus on developing the concept, the conventions will come later

All it is, is a convention. Who said that this one is the less than and this one’s the greater than? There’s no deep connection to be made to the symbols. They are just a convention that we need to help kids be able to connect that symbol to the bigger math concept. The concept that we want kids to understand is that this amount is greater than this amount or this amount is less than this amount.

The symbols are the convention, and that’s what we end up spending way too much time trying to help kids learn. We spend way too much trying to make all of these connections and build a deep understanding about the symbols when really the symbol is just a convention. There’s no reason why it’s this way. So any way we can help kids learn that this one is the less than symbol and this one’s the greater than, let’s do that.

It’s not a deep understanding. The deep understanding that we need to help the kids build is around the concept of greater than, and less than. What links to that are things like counting and place value and equality. Those are big ideas that we need to help kids understand, not the symbols. The symbols are just a convention, and there are so many other things that we could be spending our time on. 

I feel like kids don’t get enough of the actual math concept. 

We want to really focus in on the concept of comparing. The mathematical concept is comparing amounts. Whether they are greater than, less than, or equal to, that is the big mathematical concept.

The convention is how we use the symbols that we put on a piece of paper.

This is one example that I felt was the best at combining both concepts and conventions. Lori shared this image and it was funny because she was sharing something that somebody else in the Build Math Minds Group, Isabelle, had actually created. 

The reason that I really love this one is she’s just using some pins. She was using the pins and showing the kids with the quantities. That’s what I love here. They’re seeing the actual objects. They’re seeing the four things and the one thing. And when you put the pins on there, how the four things are larger. It’s a bigger quantity than the one thing, thus the mouth is open to it.

I don’t care if you call it an alligator mouth or not. I mean, we want them to see that it is opened up. I know a lot of people are like, “Don’t call it an alligator.” But you know what? It is a symbol. It’s a convention.

We need to help them know which one’s the greater than, which one’s the less than. As long as we’re connecting it to the big idea of comparing quantities, how a kid memorizes that symbol, then I couldn’t care less. 

It’s not about the symbol. It’s about the big mathematical concept we’re helping them build. And the concept is this idea of comparison: greater than, less than, or equal to. That’s why I loved your image so much, Isabel. You are tying the two together.  We’re seeing the symbol, the greater than symbol and the equal to symbol, but we’re also seeing it in comparison to the actual objects.

Why are we so concerned about helping kids learn those symbols? 

To me, the symbols are just an easy way for us to assess whether or not the kids have that understanding. If you go around and you ask each child “Is 19 greater than, less than, or equal to 24?” It’s honestly unreasonable to think that you could go around and ask every single child to tell you about comparing numbers. If you did, you would probably hear that the majority of your kids can compare those numbers.

The hard part is though they can’t put it on paper. They can’t use the symbols. The symbols give us teachers a quick way to be able to assess that because we don’t have time to go around and ask every single student about their understanding of comparing numbers.

So instead what happens is our textbook and worksheets give kids a sheet of problems and they have to put the greater than, less than, or equal to symbol into that. That’s our way of knowing whether or not they understand greater than, less than, or equal to. But really those assessments are not assessing their understanding of comparison. They are assessing whether or not a child can use that symbol.

I know that’s why we have the pressure to teach kids those symbols, but the symbols are only there to give us a way to be able to see that they can compare those numbers. It’s not that kids need to be able to use those symbols. 

This is a screenshot that I took of Dr. Ban Har’s comment on the post. When I saw that Dr. Ban Har had commented on here, I was like, “Oh!” I was totally fangirling a little bit. I love Dr. Ban Har, and I go to every training that I possibly can watch him at. If you’re not familiar with him, he does lots of trainings about the Singapore approach to math. Every time I see him present, I learn a ton. I found his comment was so informative and enlightening.

In Singapore, they don’t introduce these symbols to kids until Grade 7!!! Yet we’re doing this in 1st grade. I have seen 1st grade worksheets with the greater than, less than, or equal to symbol on them. That is not what we should be doing.

As he says here, “We need to be focusing on letting kids compare sets of objects.”

That’s the big idea, not the symbols. The symbols are just stupid. They give us a way to assess our kid’s understanding, but we can also ask them, we can also have those conversations if you’re unsure about their understanding. It is just a convention. And he even put it in there, “The symbol is a convention.” Kids will pick that up through exposure. How do kids learn that the shape of a circle is called a circle?

Because we say it all the time. When we’re showing a circle, we use the name circle. And the more that we do that with the symbols, then the kids will get it too. And he even says in here, he says, “Nothing’s wrong with various tricks to help the students identify the right symbol.” That’s why I said, I couldn’t care less if you use the alligator symbols.

There was a time when that irritated me, but what’s the big idea? Are the symbols the big thing that I need kids to understand? Heck, no. The symbols are just a symbol. 

I need kids to understand how to compare quantities. How they write that on a piece of paper is a convention that somebody made up, and I don’t care what it takes to help the kids be able to learn that convention. They will get there. But the big idea of being able to compare quantities is the piece that we need to help kids be able to do.

There are some things in math that we can really help kids discover and build their understanding around, and there are some things in math that kids will honestly never discover on their own, but they can learn about those things through us modeling it, through the exposure and by us using the terminology.

No kid is going to just come up one day and be like, “Hey, mom, I think this is the symbol for greater than.” That is not something that a kid is going to discover. But my child could come up and say, “Hey, mom, did you know that if I have 4 groups of 3, that is 12?” They’ll discover it by counting and modeling it. They’ll discover that on their own. That symbol of a greater than or less than is not something they will discover. We just have to teach it to them.

The more exposure and modeling they get, the easier it will be to recall that information. Tricks do help. I am fine with tricks for things that are conventions. Pneumonic devices help me remember things that I can’t just easily recall from making connections to other things. And that’s kind of what the tricks are for the greater than, and less than symbols.

So, the actual symbols are the convention, but the idea of greater than, and less than, and equal to is a big math concept that we need to be spending more time on helping kids develop that deep and connected understanding around. The symbols are showing the relationship between the two quantities. We need to spend more time developing that idea of comparison of quantities. 

The equal sign, the greater than, and less than signs do show that relationship from one side to the other, but the symbol is just a convention. Spend more time playing with numbers, letting kids count, looking at and comparing quantities, layer in the language and the symbols and connect it all together for the kids. 

Let’s take a look at some of the comments and see if there were any questions that came up around this topic. 

“I agree about the discovery of the symbols, but a lot of times the greater than, and less than symbols are not called by their names in the classroom. 1st grade and up. This becomes very problematic when algebraic expressions are being compared with the symbols in higher grades. Most students have no clue which expression is greater than, less than because they don’t get called what those symbols actually are.” 

— Alison

Great point. I did say, I don’t care if you’re using an alligator to help kids understand, but as the teacher, we need to be modeling how we say it. So once we get the right symbol in there, we need to always be saying exactly what the expression says. So if it’s 24 > 19, we can’t just stop at putting the correct symbol in there, we also need to layer in the terminology and say the actual expression (“Twenty-four is Greater Than 19”) just like we would if the equal sign was in there like 5 = 4 + 1 (“Five equals four plus one”).

Kids need to hear the expression that gets created when we put the greater than, and less than symbols in there. That’s one of the reasons why I think I always liked this version was because this is helping me know it’s less than. The “L” (on my fingers) is tied to the L in less than, and then I know the other one is greater than.

It is important that we do attach the correct terminology, and we don’t just leave it. I think that’s been the problem with the alligator. You’re right Alison. Is that we’re saying, “the alligator eats the bigger one,” but we’re not attaching the correct terminology to those symbols once kids get the symbols in there. That’s an important point to bring up. 

“I have the kids circle the greater number so I can tell if the concept or the symbol is what they’re struggling with.” — Beth

So I love this idea about what Beth’s saying. They might put the incorrect symbol in there, but if you also have them circle which number is greater than, then you can tell, was it the symbol that they were unsure about or are they actually not understanding which concept is bigger or greater than. 

Are there other topics in math where this would apply as far as not worrying so much about conventions? — Emily

This makes so much sense to me as I think about my first graders. So to me, I don’t know it. That would be an interesting thing to put side-by-side things that are math concepts, things that are conventions. 

If you think about a concept like when you go to teach it, the way I try to think about it is like, is there a reason for this? Is there a reason why 4 + 1 = 5? Is there a why behind it? Is there connections I can make to this thing to other mathematical concepts? If there is, it’s a math concept that is a big idea that I want to build those deep understandings, the why behind the connections to.

If I look at something and then I say, “Okay, why is this?” So, for example, when we’re doing geometry things and there’s a lesson on naming shapes, why is a circle called a circle? I’ll be honest, I don’t know. (laughs) Is there a reason why it’s called the circle? I don’t know. To me there’s no why behind it. Somebody just decided this is what it’s called. 

Now, when I look at triangle, is there a reason why it’s called triangle? Yes, because it has three (tri-) angles. Is there a reason why something’s called a quadrilateral? Yes, those are things that I could go deeper into. If there’s no why behind it, there’s nothing deeper to go.

It’s just that. It is because somebody decided that. If I’m thinking about money, right? This is a whole other subject, but money is not technically a math concept. It’s a social concept because there’s different money systems in different social systems. But here in the United States, why do we call it a dime? I don’t know. It just is. It’s just something I have to learn that this thing that looks like this is called a dime. It’s worth .10, but I can help build connections between that and up to 10 pennies, and exchange 10 pennies for a dime. There’s things that I could build connections to, but some things are just like it is. It’s just something that we have to learn. And the more repetitions, and the more times I hear it, the more it’s going to connect to me. 

There is no reason to me why a dime is called a dime, right? It just is. So, I’ve just got to learn that and I’ve got to help my kids learn that. It’s just a convention. So, that’s my kind of basis of what I look at. Is there a deeper purpose why?

I don’t know, sometimes I just don’t know the deeper purpose, and so I’ll ask other people or I’ll Google it. If there’s no why behind it to me then it is just a convention that I need to help my kids learn and know. And that happens usually from repetition, repetition, repetition. It really doesn’t matter how we help kids learn those symbols. It’s just a convention we need to help them learn.

Whatever way you do help them. They need to know the correct terms. They can’t just know that you put this one in, that the mouth opens up to the bigger number. They do need to know what those symbols are called just like they know what the equal sign is called. It’s a symbol that expresses the relationship between two sides of the expression. As they move forward, they will need that as they’re going up into what they’re doing in mathematics.

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