I wanted to start this year’s vlog off with a challenge. It’s a challenge to look at how you are teaching mathematics this year. I want you to specifically be paying attention to how you are developing mathematical proficiency. Now, it’s a challenge that will be difficult for some of you. Some of you may be already doing this. It’s a challenge that will encourage you to do something that might be a bit uncomfortable. It’s to teach math without a textbook.
I’m Christina Tondevold, the Recovering Traditionalist and I hope you’ll stick around for this as we investigate Developing Mathematical Proficiency: Why You Shouldn’t Be Teaching Math Through a Textbook, 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:
Here are links to products/activities mentioned in this vlog. (Some may be affiliate links which just means that if you do purchase using my link, the company you purchased from sends me some money. Find more info HERE about that.)
Videos in this series:
#1: Why you shouldn’t teach math through a textbook
#2: How to Teach Elementary Math Without a Textbook
#3: Creating School Change When Others Don’t Want To
#4: Components of Number Sense in PreK-2
#5: Components of Number Sense in 3rd-5th
#6: Teaching Math through Story Problems
#7: Math Practice: Building Math Fluency through Games
- Adding it Up book
- Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction by Carpenter, Fennema, and Franke.
- Children’s Mathematics: Cognitively Guided Instruction by Carpenter, Fennema, Franke, Levi, Empson
Adding It Up book
There are two resources that have totally influenced my belief that we can teach mathematics without textbooks. The first one I want to talk to you about is the book, Adding It Up. Now it was first published back in 2001, but it still oh, so relevant. One of the big pieces that came out of Adding It Up was this graphic about strands of proficiency. Now these are interwoven strands of proficiency.
This is what I wanted to start off with is what does it actually mean to develop mathematical proficiency? Because all too often, our idea of developing mathematical proficiency is we get through the textbook and then we give them the test at the end of the year. If they perform well on that test, then we think that they are mathematically proficient. Even our tests will say that they’re proficient or not proficient. Like those are terminology that we use a lot in standardized tests. But really what makes up that proficiency? This is an image that should be still very much out there, but there are a lot of people who have never ever seen this. So let’s take a look at what this means.
So one of the first ones is Conceptual Understanding. You guys have probably heard me talk about this a lot, that conceptual understanding is not just knowing how to do something, but it’s understanding the why behind it.
Then Procedural Fluency is that being able to do it. We do need kids to be able to get the right answer and to have proficient and efficient ways to get to that answer, right? A lot of kids will develop this conceptual understanding, but still not be very fluent with procedures. So these go hand in hand as I talked about. These are interwoven. You cannot have just one and feel like your kids are being proficient. They need all five of these.
So conceptual understanding is knowing how it all works, why does it work. Procedural fluency is being able to do the work, knowing how to do it. Then the next one is Strategic Competency. This is basically problem-solving. It’s seeing things in a context and being able to solve the problem. But here’s the deal. All too often our textbooks have this view of problem-solving: we give you the problem, you have to pull out the information and solve the problem.
But in real life, all too often, the mathematical problems we encounter, we don’t even realize what the problem is. Like we have to take all of this extraneous information and then figure out what the problem really is. So even though our textbooks will line out problem-solving, it’s not a true reflection of what real problem-solving is actually like for our students out in the real world.
Now the next one is Adaptive Reasoning. This is where kids can be able to reason and explain and justify. They can look and reflect about their understanding versus someone else’s understanding and make connections between those.
The last piece, which is often something that is forgotten, is Productive Disposition. This is where kids actually see math as sensible, useful and worthwhile. It’s also about one’s own ability to do the math and believing that we can do the math. Now this is kind of, this is the one that hits home for me because I feel like my own kids, my four kids, have the first four of these things. But even today, when I’m recording this, it is the first day back to school for my kids. One of my sons said it right in the car on the way to school. ” I hate math.” Like that right there tells you that we are not developing mathematical proficiency. If their view of mathematics is that it’s boring, it’s dumb, why do I have to do this, we are not developing true mathematical proficiency.
I’ve got to tell you that one of the things that I feel like does this to kids is our textbooks. It like sucks the life out of mathematics. I want you to bring the life back to mathematics for your students, for yourself. All too often, textbooks are just so boring to teach out of, right?
I know that there are good parts of textbooks. Don’t get me wrong, there are great parts about having a textbook, following it and knowing that you are “getting through the material.” That is comforting to know, but it also doesn’t build this true mathematical proficiency for a lot of our students.
Now this brings me to another piece of what’s in Adding It Up. Because it’s not just about these concepts of mathematics, these strands of proficiency. What it also talks about is how do we actually do that in the classroom? They talk a lot about something known as the Instructional Triangle. It’s the interaction among teachers, students, and the mathematics, all three of those inside of a context. So I’ll bring up the visual that they use in that book.
Yes, you can use your textbook as the context that you are doing this interaction between, but all too often, the textbook is seen as the one and only resource and guide that you must follow. It’s often a unidirectional interaction, just between the textbook and you. Like you start with textbook. The teacher reads the information. They take that information, then they give it to the students. That’s the interaction that happens.
But instead, it needs to be this triangle where they’re all working together. The teacher’s interacting with the mathematics, helping the students interact. Students are interacting together. They’re interacting with the mathematics without the teacher involvement. There are times the teachers get involved and interact, like it’s all going back and forth. All the while, the context is on the outside, giving us a way to think about and view the mathematics. One of the big parts of this triangle that is so important for students to understand is helping build that disposition and understanding that they are in control of their learning.
One of the reasons kids have a bad disposition about mathematics is that is one of those things where it’s top-down. The teacher says, “We’re learning this today.” And even if you already know this stuff or you’re not ready for this stuff, you’re still learning it. Right!?!? So, the idea of having the kids help be in control of their learning is a big part of this triangle. The students must interact with the mathematics, the teachers and the other students. Teachers should use those interactions to help shape the learning of mathematics. Now how do we actually do that?
This is what I wanted to start off with is what does it actually mean to develop mathematical proficiency? Because all too often, our idea of developing mathematical proficiency is we get through the textbook and then we give them the test at the end of the year. If they perform well on that test, then we think that they are mathematically proficient. Even our tests will say that they’re proficient or not proficient. Like those are terminology that we use a lot in standardized tests. But really what makes up that proficiency? This is an image that should be still very much out there, but there are a lot of people who have never ever seen this. So let’s take a look at what this means.
So one of the first ones is Conceptual Understanding. You guys have probably heard me talk about this a lot, that conceptual understanding is not just knowing how to do something, but it’s understanding the why behind it.
Then Procedural Fluency is that being able to do it. We do need kids to be able to get the right answer and to have proficient and efficient ways to get to that answer, right? A lot of kids will develop this conceptual understanding, but still not be very fluent with procedures. So these go hand in hand as I talked about. These are interwoven. You cannot have just one and feel like your kids are being proficient. They need all five of these.
So conceptual understanding is knowing how it all works, why does it work. Procedural fluency is being able to do the work, knowing how to do it. Then the next one is Strategic Competency. This is basically problem-solving. It’s seeing things in a context and being able to solve the problem. But here’s the deal. All too often our textbooks have this view of problem-solving: we give you the problem, you have to pull out the information and solve the problem.
But in real life, all too often, the mathematical problems we encounter, we don’t even realize what the problem is. Like we have to take all of this extraneous information and then figure out what the problem really is. So even though our textbooks will line out problem-solving, it’s not a true reflection of what real problem-solving is actually like for our students out in the real world.
Now the next one is Adaptive Reasoning. This is where kids can be able to reason and explain and justify. They can look and reflect about their understanding versus someone else’s understanding and make connections between those.
The last piece, which is often something that is forgotten, is Productive Disposition. This is where kids actually see math as sensible, useful and worthwhile. It’s also about one’s own ability to do the math and believing that we can do the math. Now this is kind of, this is the one that hits home for me because I feel like my own kids, my four kids, have the first four of these things. But even today, when I’m recording this, it is the first day back to school for my kids. One of my sons said it right in the car on the way to school. ” I hate math.” Like that right there tells you that we are not developing mathematical proficiency. If their view of mathematics is that it’s boring, it’s dumb, why do I have to do this, we are not developing true mathematical proficiency.
I’ve got to tell you that one of the things that I feel like does this to kids is our textbooks. It like sucks the life out of mathematics. I want you to bring the life back to mathematics for your students, for yourself. All too often, textbooks are just so boring to teach out of, right?
I know that there are good parts of textbooks. Don’t get me wrong, there are great parts about having a textbook, following it and knowing that you are “getting through the material.” That is comforting to know, but it also doesn’t build this true mathematical proficiency for a lot of our students.
Now this brings me to another piece of what’s in Adding It Up. Because it’s not just about these concepts of mathematics, these strands of proficiency. What it also talks about is how do we actually do that in the classroom? They talk a lot about something known as the Instructional Triangle. It’s the interaction among teachers, students, and the mathematics, all three of those inside of a context. So I’ll bring up the visual that they use in that book.
Yes, you can use your textbook as the context that you are doing this interaction between, but all too often, the textbook is seen as the one and only resource and guide that you must follow. It’s often a unidirectional interaction, just between the textbook and you. Like you start with textbook. The teacher reads the information. They take that information, then they give it to the students. That’s the interaction that happens.
But instead, it needs to be this triangle where they’re all working together. The teacher’s interacting with the mathematics, helping the students interact. Students are interacting together. They’re interacting with the mathematics without the teacher involvement. There are times the teachers get involved and interact, like it’s all going back and forth. All the while, the context is on the outside, giving us a way to think about and view the mathematics. One of the big parts of this triangle that is so important for students to understand is helping build that disposition and understanding that they are in control of their learning.
One of the reasons kids have a bad disposition about mathematics is that is one of those things where it’s top-down. The teacher says, “We’re learning this today.” And even if you already know this stuff or you’re not ready for this stuff, you’re still learning it. Right!?!? So, the idea of having the kids help be in control of their learning is a big part of this triangle. The students must interact with the mathematics, the teachers and the other students. Teachers should use those interactions to help shape the learning of mathematics. Now how do we actually do that?
Cognitively Guided Instruction
Well that brings me to my second resource that made a huge difference in my understanding of how we should teach mathematics. It’s Cognitively Guided Instruction. Now I first learned of Cognitively Guided Instruction when the research came out about cognitively guided instruction back in the mid ’90s. It has been around a long time. It originally was a professional development program, but has turned into a massive movement. It’s not something I can do justice in these little videos, so I’ll definitely link to some of the major research and of course, the book, Children’s Mathematics: Cognitively Guided Instruction. So whichever way, if you want to read an article or read the book, I’ll link to that underneath this video.
But basically, the underlying idea was to help teachers understand the development of children’s thinking and then use that to drive their instruction. In particular, they focused on how children intuitively solve word problems. They started with how children would solve these without any instruction. Then they trained teachers to understand how to use students’ thinking to pick the next thing they were going to do to guide their instruction.
Now one of the interesting pieces that I want to bring up with you about Cognitively Guided Instruction is the research that they discovered from this. They compared the group of teachers’ students who were part of Cognitively Guided Instruction with students who were in more traditional-like classrooms where they focus on computation and worksheets and that kind of stuff. In that research, what they discovered with the kids who were in CGI-type classrooms performed just as well as the other kids on basic computation stuff. But when it came to problems and tests that involved problem-solving, guess what? The CGI kids scored way better than the compare group of students who were from the more traditional style of teaching. So even though the emphasis in CGI classrooms was not on number skills and computation, they still did just as well, but were way better at problem-solving and thinking. Cognitively Guided Instruction is just such a powerful way to approach the teaching and learning of mathematics.
How to teach math without a textbook
So what does this all mean? And why am I telling you this stuff? Well what I wanted to do in this video is to encourage you to teach math in a way that focuses on those strands of proficiency, as well as using your students’ thinking to guide your instruction, not the textbook. Too often, we are just going lesson by lesson without considering if it’s the best thing for our students. As I like to say, fidelity to your students is greater than fidelity to your textbook.
Now if your textbook is great and those strands of proficiency are in there and it also has you paying attention to student thinking and using that to guide the instruction, then by all means, keep with it. But my guess is, it doesn’t. So I want to encourage you, that if it isn’t, it’s time to start thinking about changing that. Now how do I suggest you do that? Well that’s to come. I can’t fit it all in one video.
So next week, I’m going to give you my top suggestions on how to make those changes. Then I’m going to have future videos as well that go a little bit deeper into my recommendations. But for right now, I just want you to consider that the teaching of mathematics can and should be different for our students.
The way that we do that is by focusing on those strands of proficiency and starting to use our students’ thinking to guide the instruction, not the textbook. So I hope that this video has helped you build your math mind so that you are ready to go build those math minds of your students. Have a great day.
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Welcome! I’m a Recovering Traditionalist elementary teacher and now, I help teachers and children learn to love math. Explore how I can help provide you PD at your fingertips! BuildMathMinds.com