Double Trouble: When Doubles and Near Doubles Meet Number Talks
Double Trouble: When Doubles and Near Doubles Meet Number Talks
If you have been following my work, you already know that I love a good Number Talk. Give me a problem, some interesting student strategies, and a room full of people willing to talk about mathematics, and I am happy!
This time, however, my preservice teachers and I were dealing with Doubles and Near Doubles
.
Now, before anyone says, “Dr. Massey, aren't doubles for elementary students?”
Yes.
And that is exactly the point!
Sometimes the mathematics we expect children to understand deeply is the same mathematics that makes adults stop, stare at the board, and suddenly become very interested in their shoes.
So, onto the board went our first problem:
125 + 125
No pencils.
No paper.
No calculators.
And definitely no phones!
Just think.
Of course, 125 + 125 = 250.
Easy enough, right?
But Number Talks are never just about getting the answer. I want to know how you know.
Did you think:
100 + 100 = 200
20 + 20 = 40
5 + 5 = 10
Therefore, 200 + 40 + 10 = 250?
Or did you simply know that double 125 is 250?
Either way, we had established our friendly double.
And I was not finished with them yet.
Now, Let Me Move Some Numbers Around
Next up:
124 + 126
At first, this looks like a completely different problem.
But is it?
Look at our first problem again:
125 + 125 = 250
Now look at:
124 + 126
One 125 decreased by one, while the other increased by one.
We simply moved a one!
124 + 126 = 250
Same sum.
At this point, I want my preservice teachers to recognize that we are not just adding numbers. We are examining relationships between numbers.
That is where number sense lives.
A student who understands compensation does not have to immediately reach for the traditional algorithm. Instead, the student can reason:
"I know 125 + 125 is 250. If I take one from one addend and give it to the other, my total doesn't change."
Now THAT is the kind of mathematical thinking I want to hear!
Then Came 126 + 127
I couldn't let them get too comfortable.
Our final problem was:
126 + 127
Here is where our near double entered the conversation.
If I know:
126 + 126 = 252
then:
126 + 127 = 253
I only need one more.
But someone else might think:
127 + 127 = 254
and then reason:
"I need one less."
So:
254 - 1 = 253
Same answer.
Different thinking.
And this is one of my favorite parts of Number Talks.
There isn't just one way to think.
As teachers, sometimes we become so focused on getting students to the correct answer that we miss the mathematical brilliance happening on the way to that answer.
Wait...We're Not Finished Because You Got the Answer!
My preservice teachers know—or they are quickly learning—that giving me the answer during a Number Talk is only the beginning.
You said 253?
Wonderful!
Now comes my favorite question:
How did you see it?
And then:
Who solved it differently?
And, because I just can't leave well enough alone:
How are those two strategies connected?
I want my preservice teachers to become comfortable listening to mathematical thinking—not just evaluating whether an answer is right or wrong.
That requires a shift.
Instead of asking only, “Did the student get 253?” we begin asking:
What does this strategy tell me about how this child understands numbers?
That is a much more powerful question for a teacher.
It's Bigger Than Doubles
Yes, we were practicing Doubles and Near Doubles.
But the real lesson was much bigger.
The progression:
125 + 125 → 124 + 126 → 126 + 127
helps learners see numbers as connected rather than isolated.
The first problem becomes an anchor. The next problem invites compensation. The final problem encourages the use of a near double.
That is flexible mathematical thinking.
And that is what I want my preservice teachers to eventually nurture in their own classrooms.
Children should learn that mathematics is not simply a collection of rules, procedures, and algorithms handed down by the teacher.
Mathematics is about patterns, relationships, reasoning, connections, and making sense of numbers.
Our job as teachers is to create opportunities for children to make those connections—and then listen carefully enough to hear the mathematics in their thinking.
So yes, sometimes I give my preservice teachers what looks like a simple elementary mathematics problem.
And yes, sometimes they probably wonder where I am going with it.
But there is always a method to my mathematical madness!
Because before we can teach children to think flexibly about numbers, we have to learn to recognize, value, and encourage flexible mathematical thinking ourselves.
And that is why...
We Number Talk!
My biggest takeaway from this is that we have to not only teach children how to do things but as teachers we have to understand how they think as well. I think these number talks not only sharpen mathematical skills, but they also make you a better teacher as well.
ReplyDeleteYour point really highlights how teaching is just as much about understanding students’ thinking as it is about the math itself. I agree that participating in Number Talks strengthens our own teaching because it helps us recognize and appreciate the different ways people make sense of problems.
DeleteCourtney, I agree with your point that teachers have to understand how children think, not only teach them procedures. Number Talks do both, because we practice mental math and learn to listen for the reasoning behind a strategy. That will help us respond to students more effectively.
DeleteAlthough I wasn’t there in person, I was able to watch my classmates participate in the Number Talk, and it was really helpful to see how everyone approached the problems differently. Number Talks are important for students because they reveal how mathematical thinking develops, but having preservice teachers participate is just as valuable. It allows us to see how our peers process math in ways we may not have considered, giving us new strategies to use in our future classrooms.
ReplyDeleteYou missed a good number talk.
DeleteThe Number Talk showed me that there can be different ways to solve the same problem. I liked seeing how doubles and near doubles can help students solve problems mentally and build their number sense.
ReplyDeleteI agree with you! Number Talks are a great way to help students understand that there is more than one way to solve a math problem.
DeleteI agree with your takeaways. Showing that there is more than one way to solve a problem is important.
DeleteI agree with your comment. I think doubles and near doubles are easy to recognize; without even thinking, I knew the answer to 125 + 125 = 250 because 25 + 25 is 50 and 1 + 1 is 2.
DeleteThis Number Talk showed me how you can use different numbers and at the end of the problem you can get the same answer. It also shows that students can have different strategies to find the sum of the problem.
ReplyDeleteI agree! The Number Talk showed that students can use different strategies and still get the same correct answer. I think this is important because every student may not think about or solve a problem in the same way.
DeleteDuring our Number Talk, I got to share my thinking out loud with the class. When we got to 124 + 126 or 126 + 127, I solved it by stacking the numbers and regrouping. Explaining it to my classmates made me slow down and think about why my strategy worked, not just what answer I got. Then, hearing my classmates show me another way to see the same problem, I realized the strategies were connected. As a future teacher, I now see how it feels to be the student who has to explain her thinking, and how much it helps when the teacher asks, "How did you see it?" I want my students to feel comfortable sharing their ideas the same way.
ReplyDeleteDuring the number talk, I realized that there are many different ways to solve addition problems. Some of my classmates stacked the numbers, while others broke the numbers apart and then added them. I enjoyed listening to everyone explain their strategies because it showed me that there is not always just one way to solve a math problem.
ReplyDeleteI agree with your observation, the different ways could also work with subtraction as well!
DeleteI think we will do a subtraction one Monday
DeleteI agree, I just wish I knew how to solve other math problems in other ways. Math is confusing as is.
DeleteI agree with your observation on their being many ways to solve different problems.
DeleteI really enjoyed this number talk. My biggest takeaway is that there is more than one way to solve a math problem. Just because you solve a problem differently from someone else, it doesn't make your way wrong. The biggest thing is making sure you have the correct answer and understanding how you got there.
ReplyDeleteThis number talk was very insightful. I enjoyed understanding that there is more than one way to solve a math problem. I also believe that it is important to fact check your work by solving the problem in other ways.
ReplyDeleteI agree with your answer. Doing this in our future classroom would also show what the students know! A good bellringer!
DeleteThis number talk video was very interesting; I get how a child can be easily confused with math. Like me, I come from 7 different high schools while in foster care, and every school taught math differently, and no matter how hard I tried, I never got to understand it fully. I know the basics of math, but all that other stuff looks foreign to me, like anything that's not adding, subtracting, or simple math. This is what I will be looking out for in my class, it's important to let students know it's more than one way a person can learn to solve a problem.
ReplyDeleteI like how you understand the different ways students might need to learn math to understand it.
DeleteMy biggest takeaway from the number talk is seeing that there is more than one way to solve a problem. Everyone thinks differently so the way they come up with the same answer can differ. As a few teacher this showed me how it important is to be able connect to different learners.
ReplyDeleteMy biggest takeaway from the number talk was seeing a creative way for future bellringers. This is a nice activity to see what the students know and/or understand. And it's great because every student gets to engage with one another; it's a great bonding activity. This number talk shows the different ways that a problem can be solved. See all the different outcomes that a problem can have.
ReplyDeleteDr. Massey is indeed a mathematical wiz!
ReplyDeleteThe progression from 125 + 125 to 124 + 126 to 126 + 127 made the reasoning visible for me. I knew the double, but seeing one addend decrease while the other increased — and realizing the sum doesn't budge — gave me a real reason for why compensation works. Then 126 + 127 showed me the same near double can be reached from either 126 + 126 + 1 or 127 + 127 − 1. What stuck with me most is that the point wasn't the answer, it was being able to explain how I got there. That's the kind of classroom I want to run.
ReplyDelete