There are many practices from long ago that we need to rethink as teachers. This series looks to bring up those practices and offer alternate ideas that are more relevant in today’s classroom. Today we’re discussing using alligators eating bigger numbers to teach comparison.
The time has come in my class to begin comparing numbers using > and < but guess what? We’re not using alligators or crocodiles and there are no numbers are being eaten. Instead, the symbols are being taught in a way that students can understand. We’re using language to teach the symbols and trust me, it really isn’t too hard!
Why Alligators and Crocodiles?
As teachers we love doing things that are cute. Trust me, I love using cute things with my kids. Its fun, it makes the learning fun and it engages the kids. Cute teaching resources can come in handy. When comparing numbers it can be hard for kids to understand. Especially when we are using symbols that kids haven’t seen and symbols that look very similar. So we did what we do best as teachers and we made up a cute story for kids to remember. The alligator eats the bigger number. It seems simple enough. The kids can get the answer and they can solve problems comparing numbers. But what if our cute teaching resources are undermining students’ understanding? Maybe not even at our grade level but at grade levels higher than us.
What's The Problem?
When we tell kids to draw the symbol with the alligator eating the bigger number, we are only focusing on which one is larger. Every time the child draws the symbol in their head their thinking, “12 is bigger so the alligator eats the twelve” while writing 9 < 12. You’ll notice that the symbol drawn was less than but in the child’s mind they were focused on which one was greatest. Many of my students who learned the “alligator trick” will read this as 12 is greater than 9. While not incorrect, 12 is greater than 9, this should read 9 is less than 12. This is very confusing. Yes the answers are correct because they know how to draw the symbol but there is virtually no understanding of the symbol. Most kids don’t even realize that there are two different symbols!
This year we began saying the words “is the same as” when we saw an equal symbol. Why not just use the word equal? Too often students understand the equal sign to mean the answer is coming next. That isn’t what the symbol means.
The = symbol means that what is on the left is the same as what is on the right.
Just shifting our language within the math classroom helped students gain a better understanding of the equal sign. According to the Common Core State Standards students in grade 1 should under the meaning of the equal sign. (1.OA.D.7) How many upper level students don’t realize that the symbol means that the expression on either side is equal, or the same?
We also switched up our equations so that sometimes there were answers that didn’t come after the equal sign. The equations we showed the students had the equal symbol all over from the very beginning of grade 1. Blanks were in different spots and answers were in different spots and the equal sign moved all around. This helps the kids see that equal really means the side on the left is the same as the side on the left.
This work ties in with the mathematical practice standard 6- attend to precision. Attending to precision doesn’t only mean paying attention to the problem and solving it accurately. It also is using precise language when we talk about math. I highly recommend reading this Think Math! article about this standard.
Use Language First
When it came time to compare two numbers this year we created three sentence strips. We have already been using the phrase, “is the same as,” so it was not new to students. Once they had these three sentence strips, I had them build two numbers and choose which strip went in the middle. I reminded them that just like readers, mathematicians read left to right.
To my surprise and joy they were able to start comparing numbers accurately from the very start. Some of my most struggling mathematicians found success in using the sentence strips. I was very excited to see where this was going to go. They were not only able to compare two numbers but they were also able to use the correct language while doing so! This is just what we want our kids to do!
As we finished math class on this day, one of my students asked if he could guess what was coming next. Of course! He whispered, “I know there are symbols that go with these words. Is that what the blank space at the start of our sentence strips is for? Are we going to get to use the symbols?”
Isn’t it a magical moment when kids predict where their learning is going and they are excited about it? Guess what, we were going to learn about the symbols next!
Then Teach Symbols... Relying on the language
After using the sentence strips to build our understanding of the language we use to compare two numbers we introduced the symbols to the class. You’ll notice that I put the symbols on post-it notes. This was done as a scaffold in case students need the symbols removed in order to practice longer without them.
Before putting the symbols were in place, we had an understanding that in math we use symbols to represent certain words. They already knew the symbol = went with the words “is the same as” so adding two new symbols wasn’t to tricky.
To introduce the symbols I wrote them on post its and placed them on my sentences strips. Then the class did the same. As they were drawing the symbols on to their post it notes I could hear a small buzz of observations. Many kids noticed that the symbol opens to the side that is greater and is closed on the side that is less. This isn’t a bad realization to have but it wasn’t their only knowledge of the symbols and how they work.
As we compared numbers that day I noticed most students relying on the language and then matching the symbol with the language. They would look at a problem like 82 __ 45 and say “eighty two is greater than 45. Then they would get their sentences strips and match the strip to the problem. Finally writing the symbol on the line.
What I've Learned
This is my first year as a grade 1 teacher. In the past I have taught second and third grade and my students have already come to me with the alligator story. Once you learn a trick it is hard to ever go back to reasoning. I wasn’t sure how this approach was going to go when I introduced it to my students. In fact, I waited a long time to even complete this post to see if this method of teaching symbols really even worked. Let me tell you, it did! Right now almost all of my students can compare two numbers using the symbols on their own. They now know which one means greater than and which one means less than. We still have our sentence strips to use in case we need them but the kids have eased themselves off of that scaffolding.
One Last Word About Tricks
As a former Math Curriculum Leader I have a lot to say about tricks in math. So often we teach kids tricks because we think that the math is difficult and we want to make it easier. Or we learned with a trick and we really aren’t sure about the real math behind it. Math is built on reasoning. If you are able to connect knowledge and reason through a problem you can more than likely solve it. When we teach kids tricks oftentimes it eliminates reasoning. When we take away this reasoning we are limiting our students’ math potential. Mathematicians rely on reasoning more than anything else.
Next time you go to teach a math trick in your classroom can you ask yourself these questions?
-What mathematical properties does this trick rely on? If the trick doesn’t rely on any math but more of magic then it isn’t really teaching math at all.
-Is the mathematical reasoning eliminated through this trick? Math is built around reasoning and we don’t want to take this away from students. They don’t really have the right answer if they can’t reason through a justification.
-Is this trick just getting the right answer? Math is so much more than answer getting. Please check out Phil Daro on Answer Getting.