How to Differentiate Instruction in Elementary Math
From the Cool Cat Teacher Blog by Vicki Davis
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Young children have many different ability levels with math concepts. Understandably, kindergarten and elementary teachers need to reach every student where they are on their learning journey, but it isn't always easy. Dr. Carol Ann Tomlinson, an expert on the topic, gives an overview of how to differentiate math instruction with younger students.
How to Differentiate Math Instruction in the Elementary Classroom
Dr. Carol Ann Tomlinson
William Clay Parrish, Jr. Professor Emeritus at the University of Virginia’s School of Education and Human Development
Resources on Differentiating Math Instruction Mentioned in the Show
- So Each May Soar: The Principles & Practices of Learner-Centered Classrooms (2021)
- Everybody’s Classroom: Differentiating for the Shared and Unique Needs of Diverse Learners (2022)
- eSpark. – Sponsor of today's show, a differentiation tool
- How the Brain Learns by David Sousa
- Teaching Channel Math Video
Dr. Carol Ann Tomlinson - Bio as Submitted
Carol Ann Tomlinson is William Clay Parrish, Jr. Professor Emeritus at the University of Virginia’s School of Education and Human Development where she served as Chair of Educational Leadership, Foundations, and Policy, and Co-Director of the University’s Institutes on Academic Diversity. Prior to joining the faculty at UVa, she was a teacher in public schools for 21 years, during which she taught students in high school, preschool, and middle school and also administered programs for struggling and advanced learners. She was Virginia’s Teacher of the Year in 1974. She was named Outstanding Professor at UVa’s School of Education and Human Development in 2004 and received an All-University Teaching Award in 2008. In 2022, she was ranked #12 in the Education Week Edu-Scholar Public Presence Rankings of the 200 “University-based academics who are contributing most substantially to public debates about schools and schooling,” and as the #4 voice in Curriculum & Instruction.
Carol is the author of over 300 books, book chapters, articles, and other educational materials. Her two latest books are So Each May Soar: The Principles & Practices of Learner-Centered Classrooms (2021) and Everybody’s Classroom: Differentiating for the Shared and Unique Needs of Diverse Learners (2022). Her books are available in 14 languages. She works throughout the United States and internationally with educators who seek to create classrooms that are effective in reaching diverse student populations.
Transcript - Episode 793 Dr. Carol Ann Tomlinson - Differentiating Math Instruction
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John Davis – This is the ten minute teacher podcast with your host, Vicki Davis.
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Today's sponsor is eSpark. Stay tuned to learn about this amazing differentiation and intervention tool for kindergarten through fifth grade reading and math teachers.
Introduction to Dr. Carol Ann Tomlinson
Oh, I'm so excited today. We are talking with Dr. Carol Tomlinson, chair of Education Leadership Foundation and Policy at the University of Virginia. And she's also author of Everybody's Classroom: Differentiating for the Shared Unique Needs of Diverse Students.
Differentiating Math Instruction for Younger Students
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And Carol, today we're going to talk about differentiating reading instruction in the math classroom when we're focusing on elementary kindergarten. Can you actually differentiate instruction for math for younger students?
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I think you have to. And I think that's probably the game plan all the way through school. I think there are things that all of the kids need to do together in math. But I think any teacher in the classroom for 15 minutes knows that just because we thought kids could learn something in two days doesn't mean they all did.
00;01;07;28 – 00;01;31;01
And some of the majority knew it. And some of them really are going to need considerably more time than that.
Create Shared Math Experiences
So I think part of the plan is to think about shared experiences that kids can have, for example, with learning numbers or learning to count simple things and make sure that those really challenge kids to think and to try to express themselves and to show themselves in different ways.
00;01;31;05 – 00;01;40;18
Here's the way I think of it. Here's something I can do and then have time that set aside consistently so the students in that class can take their own next step.
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So you're saying spend time having shared experiences with groups where students are doing things together, so they're actually part of that differentiating process?
“Highways and Exit Ramps” in the Classroom
00;01;51;22 – 00;02;17;20
Yes, I talk about this often in terms of highways and exit ramps, highways, the time, all the kids should be together, thinking about numbers are a little higher up, trying to figure out what multiplication has to do with addition or what that might mean if you're multiplying popcorn or whatever it is you're doing. And all the kids need to be in math conversations with the whole group or trying to talk with two people, but still in the whole group.
Being Part of a Team of Learners
00;02;17;20 – 00;02;38;25
And they need to be part of the state themselves. Being mathematical, it's really important for kids to see themselves as part of the whole class. We're a team of learners and I'm part of that team and all kids need to see interesting demonstrations by the teacher about whatever they're talking about. They need to share their answers and explain how they got there.
00;02;38;28 – 00;02;59;02
They need to trial things with a partner. And so the whole class aspect of math instruction and everything else I think is critically important. We are in the US and we need to be on us a lot at the time. But then there are times consistently where some kids need to spend a little bit more time on something.
00;02;59;02 – 00;03;27;24
Some go back and patch up things or don't have the prerequisite skills they need. Then some kids learn in different ways and would do much better if they could use a different mode of learning. And some kids are ready to move ahead. So the highway needs to have periodic exit ramps and that predictable and frequent not once every two weeks, part of every day, so that kids can work on things they're interested in or things that take them to the next step or try out new ideas with peers.
Don’t Use Terms Like Bluebirds, Buzzards, and Wombats!
00;03;28;04 – 00;03;50;13
And even in those exit ramps, I think it's important for kids to collaborate so that they feel a part of a group. From my perspective to say that those ethnic groups should not be bluebirds, buzzards and wombats, that every time they get together they sit in the same place and the kids that are having trouble are always together, and the kids that are flying high are always together.
00;03;50;14 – 00;03;54;20
That's not helpful either. They need to have very flexible grouping as well.
Learning to Add with Apples and M&M’s
00;03;54;25 – 00;04;15;07
So this is very real to me. My mom was a teacher and even though I went to Georgia Tech took a lot of calculus, the hardest thing I ever really learned was actually addition. I remember first grade didn't make sense to me and mom sat down at the table and we added apples and sometimes we added M&M’s and I just remember how she worked with me.
00;04;15;08 – 00;04;29;20
One on one. It didn't make sense to me. But then when the lightbulb went on, it changed my life. So just because these basic concepts are difficult for students, it doesn't really say anything about their intelligence level, I would hope.
Don’t Use Buzzwords that Label Children!
00;04;29;24 – 00;04;50;05
And that's one reason that we don't want to have bluebirds buzzwords number one best because that does become predictive. Kids with less opportunity often end up in the buzzwords, and everybody begins to think of them as buzzwords, and they begin to think of themselves that way. And we have every reason to believe that every kid has much more capacity than we can see.
00;04;50;18 – 00;04;55;15
And we need they're smart, not plug them into a slot that's going to become the stereotype.
The Biggest Mistake with Math Instruction
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So fill in the blank for me, Carol, as teachers differentiate math instruction and the elementary grades. The biggest mistake they could make is blank.
00;05;06;15 – 00;05;27;07
I think teaching it as an algorithm, teaching it as this is what we're doing, “This is how you do it. Now do it.” versus having conversations about how that makes sense to you, why you do something and what other ways you can do it. Look how cool it is that we can get the same answer, four or five different ways that everybody can explain it.
00;05;27;07 – 00;05;49;15
Then we learn more that way. I think part of the reason you discussed your floundering– is my floundering happened because that a bad juncture for me. I had a teacher who had no patience for anybody that couldn’t see her do something on the board and reproduce it. And it made no sense to me, you know, I was terrified. And so that's not you can think your way through it and he can make his way through it a different way than you.
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One of the big ideas of mine that I really love is an infinite number of ways to get the same answer. And I think we need tell kids that in the classroom.
The Problem with the Pressure “to Cover”
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Wow. Some teachers feel obviously these days, a lot of teachers feel a lot of stress and they have a lot to “cover.” Oh, I know you probably cringe at that word like a lot of his teachers do. What's your advice to the teacher who just feels like they have so much to “cover” that they feel like they don't have the time to differentiate?
Coverage is an “Un-brain friendly” Approach to Teaching
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Well, I certainly understand why teachers feel that, the pressure on them is immense. But not only do we not have any evidence that when the teacher covers something, all the students learn it, or even that most of them learn it. We understand now that it really is teaching in a way that's totally un-brainfriendly. Howard Gardner long ago said “the enemy of understanding is coverage.”
00;06;44;15 – 00;07;11;20
And David Sousa, How the Brain Learns, talks about the fact that” coverage races are not brain friendly,” and if we are thinking that the most important thing is that we cover something and we have forgotten that the aim is that the aim is that the students learn, then that's where we need to start rethinking things.
Our Goal as Teachers
Our goal, though, I think it's twofold: to help students learn as much as they can, but also to dignify them as thinkers and as human beings and to excite them about the world around them.
00;07;12;03 – 00;07;39;22
That's a good reason to teach. Covering stuff for the test is not. And I think any time we start feeling a huge amount of pressure and we end up hearing ourselves play or think something like, “I can't teach kids that they can learn because I have to cover this myself.” Listen to that and think and regroup. Truth is that teachers are much happier and much more creative when they're teaching kids for meaning and when they can do it in creative ways and help kids be creative.
00;07;39;22 – 00;07;55;12
And we all. This is hard for us to trust, but we have good evidence that when teachers teach students, for meaning, they do better on the test than when we teach them algorithms and cover things. So I think there's powerful evidence for that, but habits are powerful, too.
A Classroom Teacher Who Understands Teaching Math with Differentiated Instruction
00;07;55;26 – 00;08;15;17
So describe for me, Carol, a classroom or a teacher that you may have met recently. You don't have to share a name. Just describe what you saw happening in that classroom that got you really excited. And you felt like that teacher understood the research and the things you espouse that need to happen for differentiated instruction in math.
00;08;15;17 – 00;08;33;28
I can think of two. One was a fourth grade class. I think the other one is young children. So I'll start with the little guys and then you can decide if we have time for another example.
Kindergarten Students Learning Numbers
But this was a class in kindergarten where kids were learning numbers and so they were learning to count from 1 to 3 and 1 to 5 and on up.
00;08;33;28 – 00;08;59;26
Of course, they were counting everything. They count the days in the week that were left, the days of the week that were gone or how many people were wearing red shirts, everything that counted all day.
But at a math center, she assigned students to go at various points of the day, and all of them were doing work to determine something about how many students were in their class and what she did at the center,
Different Tasks for Different Groups
00;08;59;26 – 00;09;28;12
I think, is probably not terribly remarkable. But what she did was to give them all a task to do that was a little ahead of where they were functionally comfortably. The coolest thing about it was that after the time at the center, she met with the kids and her conversations with them were really wonderful. So one group had to determine how many students were in their class that day and how many people are here.
00;09;28;28 – 00;09;50;06
And those little guys were adorable to watch or walking around counting people who were moving at the same time. But that was not a surprise. And then you'd see them counting as they knew the count. One, two, three, seven, nine, 11. And so the numbers were just all over the place. The conversation she had with them was wonderful.
00;09;50;06 – 00;10;11;25
She went through each of the kids. “Tell us how many people you thought were here today. How did you get that number? Do you think more people are here today than Jack does, or does Jack think they're more people than you? Oh, did you all get the same answer?” No. “Should we all get the same answer? Yeah,” but then some kid would say, not if somebody left to go to the office.
00;10;12;06 – 00;10;33;14
And then she ended up by saying, “So, since you didn't get the same answer, do you think it would be worth doing this again tomorrow and trying again and see how we're thinking about it?” They thought that was a great idea and the conversation really positioned them as mathematical thinkers.
She did the same with the second group, pretty much the same questions.
00;10;33;26 – 00;10;52;03
That group was supposed to figure out how many kids were there and how many kids were absent. And of course, it's much harder to count something that's not there. They talked about that. They talked about greater than less than how they had gotten their answer. And she did the same thing again with them. And she also after the. “How did you get your answer?”
00;10;52;05 – 00;11;11;04
I forgot what the other question was, but it had to do really with their next step in reasoning and then a third group had to count how many boys were there, how many boys, how many girls were there, and how many girls were absent. And the one question she asked them that was different than the others was, explain to us how you know, you're right.
00;11;11;19 – 00;11;36;27
The other investigation with numbers, thinking about numbers, finding different ways to do things by listening to each other. But this group, even at that age, was ready to say, I'm right because I know this and this and explain. It was a wonderful way to have all the kids together for the lead in but with different tasks, for them to do it, etc. And then conversations that accepted where they were and pushed them on a little bit further.
Treating Students as Mathematical Thinkers
00;11;37;10 – 00;11;57;05
And of course that's a cycle that they'll keep going through. But that, I think, is an example of a classroom where all of the kids are sharing good and exciting and interesting beginning instruction and asked to do work that pushes them on a bit and then are talked with as though they are really mathematical thinkers because that's what they're trying to do.
Great Conversations with Teachers Are Part of Great Classrooms (And AI Can’t Do That)
00;11;57;18 – 00;12;16;13
I love the part of the conversation because I think that truly great conversations are very much a part of excellent classrooms, and it's something that air and algorithms and robotics are never going to be able to do, is to have these meaningful conversations that truly push kids. I love that.
So let's talk about your second example.
00;12;16;28 – 00;12;38;08
This is a teacher I really can't remember the grade level, fourth grade maybe, but this is a teaching channel video. And this teacher is talking about how interested her students have become in math as a result of the kind of thinking they were doing. And she seems to be reflecting all the time when it was more algorithmic formula like. (Note from Vicki: I think the video is this one but there are several relating to this topic on the Teaching Channel.)
00;12;38;21 – 00;13;01;27
And so she's having a discussion with – I’ve forgotten exactly what the topic is– but she'll ask the kids or somebody to volunteer to do something and they come up and do a demonstration. And then before that, she'll always say to the kids, somebody and talk about what you're thinking and challenge each other's thinking. Does your thinking agree is your thinking helping each other?
00;13;02;08 – 00;13;20;29
And after they've done that, then they finally get to the place where she'll say, okay, let's do a true, false problem. And this is not exactly what they were doing, not the same problem but the same understandings. And so the kids determine whether they think the answer that she gave to this problem is correct or not, true or false.
00;13;21;17 – 00;13;48;11
And she says they love the true false thing because they love to be able to defend stuff. She's watching the kids and they keep doing the talk. And then this little gal explains why another student was wrong. And her answer is adorable. Using the idea of having M&M’s, for example, that the students know that you're going to divide and she gives this meaning to this kid, this meaning to this kid in this minute, this kid, and in the end, this girl.
00;13;48;11 – 00;14;05;18
It can't really be right, because after she's gone to the first two girls, she's already given them away. And so there isn't anything to give to the third one. And the other kids are hearing that kind of reasoning. And she says to one little boy, “So what did you think about what she said? How did that make you think?”
00;14;06;04 – 00;14;25;25
And clearly he wasn't really tuned in and didn't listen. And one thing I love about it is when he pauses and it's obvious that he didn't hear what she said. That doesn't bother the teacher, she says. “Okay, what she says was this and this. How do you feel about that?” And he thinks about it for a minute. He said “She's beginning to make me think I was wrong.”
00;14;26;12 – 00;14;52;07
“So she caused you to rethink and to understand more deeply?”
“Yes, ma'am, I think she did.”
But when that video was posted and then people who watch it can post a comment, the first one, they're not surprisingly, was surely you don't think all the kids understand this after that lesson? And her response back to it was, Oh no, this was just the whole class lesson.
00;14;52;22 – 00;14;59;20
Meaning again, that they're going to be doing things that help them go down that pathway to understanding with what comes after this.
00;15;00;07 – 00;15;25;24
Carol, thank you for coming on the show. You've really helped us understand differentiating instruction with the elementary classroom. This is a fantastic conversation.
Thank you eSpark. Thank you eSpark for sponsoring today's episode. eSpark is a differentiation and intervention tool that helps teachers of kindergarten through grade five save time by providing a ready to go standards-based reading and math activities that students love.
00;15;25;24 – 00;15;48;03
eSpark is free for teachers. Each Quest in eSpark includes a free quiz, framing videos, instructional videos, practice activities, critical thinking challenges, a pose quiz and optional student recording. I love that for activities to remain in the eSpark catalog, it must have a high in student engagement rating based on student shows and thumbs up or thumbs down.
00;15;48;04 – 00;16;27;13
You can even import NWEA or STAR data to give students a more differentiated experience from the first log on. Oh, and did I say it is free for teachers?
So, go to go.espark.com/coolcatteacher/ and sign up today for a spark you'll be glad that you did and as always email me at and Vicki at cool cat teacher dot com and let me know what you think about today’s sponsor or today's show.
00;16;27;20 – 00;16;45;13
You've been listening to the ten minute teacher podcast. If you like this program, you can find more at coolcatteacher.com. If you wish to see more content by Vicki Davis, you can find her on Facebook and Twitter under @coolcatteacher. Thank you for listening.
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