What does good math teaching look like? The answer to this will depend upon who you ask. But no matter who you ask, I bet that most people today would not say that going page by page in a math textbook is what they would consider good math teaching.

Now if you do run across someone who says that, I’d invite them to go sit in a classroom where the teacher does this and spend an hour every day sitting through that and then ask them again. In my role, I have gotten to do that lots of times.

I’ve also sat through some lessons that I come out being like, “Oh my gosh, that was amazing!!” So last week I encouraged all of you to ditch that textbook, but when you do that, what’s next?

I’m Christina Tondevold, the Recovering Traditionalist. And I hope you stick around as we delve into how to teach elementary math without a textbook. In our quest to build our math minds, so we can build the math minds of the students.

Watch the video or read the transcript below:

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Video series prior videos
#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

Developing Mathematical Proficiency: Why you shouldn’t teach math through a textbook video
IES What Works Clearinghouse
IES/WWC Practice Guide Assisting Students Struggling with Mathematics

Download your Guide to Teaching Math Without a Textbook

Now I’ve got to say I don’t like making recommendations just based off my opinion. I like to back it with research, and there’s lots of research articles and books that I could point you to. Like if I point to any of the books behind me, it would have awesome information. But I had really hoped to kind of point you to one place to find the research.

I went to my “go-to” place which is the IES What Works Clearinghouse hoping to find one of their practice guides that outlines what works for elementary math, but there wasn’t one. I will link up what they do have and one of my favorites which is all about what works for students who are struggling and in need of intervention in elementary and middle school. But what I tended to learn from that was that the recommendations they make in that document really are things that work for all of our students, not just kids who are struggling.

So some of my recommendations today come from the books behind me, but some also come from the IES What Works Clearinghouse Practice Guide Assisting Students Struggling with Mathematics which I will link to all of that right below this video.

Non-negotiable: Understand the progression of students’ learning

Now the first thing that I want to start off with is that it is so important that you understand the progression of student learning. Now we are often told that the standards are our guide. So when I talk about ditching the textbook, people will always tell me, “yeah, the textbook isn’t our guide, our standards are our guide.”

But I like to push back even on that because our standards only really tell us the end result. It’ll say things like “multiply two-digit by a two-digit number.” It doesn’t really tell us all the stuff that kids have to understand in order to be able to do that thing.

So what the progressions are is they tell us how kids progress to get to that understanding. Now they are messy, they are not normally a progression, and so there are a lot of people who don’t call them progressions. So I like to call it a progression but you’ll hear other people like Cathy Fosnot who calls it a landscape for learning, Clements and Sarama call it a trajectory. I don’t care what you call it, but you do need to understand where your students are, where we need to get them, and how do we help them progress to that endpoint from where they are.

So I have three big things that if I could design a curriculum, which I don’t want to because it’s all dependent upon your students, but if I could design a curriculum, it would be around these three things. These three things are things that if we include these three things, I don’t really feel like we need to include anything else. Let me know in the comments if you agree.

Instructional Practice #1: Focus on Number Sense, not Number Skills

So my first one that you need to be doing in the classroom is having a focus on number sense, not number skills. Let’s be a little bit clear on that because let’s be honest here, the focus in elementary school is on the concept of number. We aren’t spending months helping kids learn and identify their shapes. We are helping them add, subtract, multiply, divide. That’s the big focus.

But our textbooks take that focus and they start to focus on the wrong part of number.

They focus on skills related to number, not number sense. There’s a very big difference. So my first recommendation on what you need to be incorporating is building some sort of number sense routine into your daily practice. So I’m going to switch over into a different screen, and we’re going to talk about one of my favorite number sense routines.

This is not the only thing you should do. There are lots of other recommendations which reminds me to tell you that you need to grab the download that is linked below this video. I am putting together my guide for you. It’s not my guide, it is actually Your Guide To Teach Elementary Math Without a Textbook.

So these recommendations that I’m giving you are in there along with some extra info that we’re going to be talking about over this whole series that I’m doing about teaching elementary math without a textbook. It’s all going to be added into this whole guide that you’re doing. There’s going to be some spots for you to fill in. Grab your download so that you can have your own guide on how you plan to teach mathematics without a textbook.

Now you may do all of these, you may do some of these, but I hope you don’t do none of this, because these three things are just so darned important.

I hope throughout this video and the next ones to come through this whole series, I convince you of that, that we’ve got to stop focusing on a textbook and start doing these three things.

Okay, so let’s take a look at one of my favorite number sense routines. Reminder, this is not the only one. I will reference more inside of that guidebook that I am asking you to download. So, first thing I want you to take a look at is think of number sense routines that you might already be doing, it does not have to be this number sense routine that I am going to suggest.

So our first one that I want to recommend to you is my favorite number sense routine which is a number talk or you could do a number string.

Now really quick down and dirty here’s the difference. A number talk is when you give kids one problem, one visual, whatever it might be, and you ask them to solve it in any way that they can. That’s the piece that I want you to be doing there is that they solve it any way they can. Then you have them share the different ways that they solved it, and then talk and discuss about those different ways. Compare, contrast what was the same, what’s different, all that fun stuff.

However, if you’ve ever tried a number talk, and the kids kind of all have the same way of doing it, then a number talk isn’t the best thing. Instead what I might suggest is a number string.

A number string starts out with a simpler problem that most of your kids might know or maybe it’s a problem that some of your kids are still working on like this 5 + 9, and then it’s a whole string of problems that you show one at a time, and you’re leading them into this final problem.

Do you see where this final problem is the exact same one as the original one that I did on our number talk? But instead of just putting up that one problem there, now I’ve led them using this string of related problems to come up with maybe a different strategy that students had not ever had before.

So, number strings is a series of problems or visuals that are related in some way and you’re trying to build a relationship or strategy. Number talks are great if your students already have strategies built, they see relationships, and you’re just having them talk about the different ways they’re seeing it. So a reminder, it does not have to be just problems like bare problems, it can also be visuals.

However if you’re doing a visual and you’re doing a number talk, this is not a great visual to do a number talk, because kids need to have something to talk about. There’s not a whole lot to talk about here. It’s a five and a two. Some kids might see a four and a three. There’s just not a lot of variance that kids will see.

But if I take those same dots and put it into this random 10 frame like this,

there is so much more discussion that happens here because kids will see it in so many different ways.

So when you’re doing a number talk, it needs to be something that will bring up different ideas, different relationships, different things that kids are seeing, and then you have a discussion about that. It doesn’t matter if it’s a visual or a bare problem.

So number talk, number strings, those are some of my favorites. But again in the download, I’ll reference more and we will be doing future videos coming up about more ways to incorporate number sense and do number sense routines in your classroom.

Instructional Practice #2: Incorporate Story Problems

Our next idea that I really want you to consider incorporating in your classroom is the use of story problems. There is not enough of this. If your textbook does use story problems, it is often done in a very procedural way. It’ll give this great story problem and then it’ll tell the kids step one, do this. Step two, do this. Step three, do this. That’s not the best way to go about story problems.

There’s a lot of things that we will talk about in a future video all about the research of Cognitively Guided Instruction and how kids have their own intuitive way of solving problems. For this part of the video, I want to ensure that you all know about the problem type charts.

If you are not familiar with these problem type charts, you’ve got to go find them. Just Google problem type charts, math problem type charts, or CGI (cognitively guided instruction) story problem types. Any of that. This is a screenshot taken from the Common Core Standards. This was in the appendix.

I use this one because this might be more familiar to everybody besides the Cognitively Guided Instruction version. But this chart is not for your students.

Let me repeat that.

This chart is not for your students.

This is for you to know about the different story problem types, and there’s a whole lot to learn about why some of these are easier for kids to solve than others even when you’re using the exact same context and the exact same numbers. Some kids can solve some of these types and not others. It’s a progression, right? We’re talking about this progression again.

So kids will solve story problems using a type of progression. Some of these are easier, some are harder. And we need to have that knowledge about story problems so that we know is it the story problem that’s causing the issue or is it the mathematics, the numbers that are in there? Because sometimes kids will be able to solve and add to with a result unknown problem with bigger numbers and yet they can’t solve this bunny problem where it’s _ + 3 = 5. It basically boils down to that. But they can’t solve it because it starts out with some bunnies were on the grass, and kids don’t know how to start when it says “some” and they’re asking you how many is some, right? Have you ever had that? I have lots of times. So that’s why I’m going off on a rant about it.

The idea is that you need to understand these story problem types to ensure that your students are being exposed to these problem types, but also to understand that the problem type may be why a student is not able to solve it. It might not have anything to do with the numbers, but just the way that we write the story problem can impact whether or not a kid can solve it. Yes, we do need to have kids be able to solve all of these problem types.

There’s also a chart for the multiplication and division problem types.

When you’re going into multiplication and division, kids need to be doing it through a story problem lens. They need to have the mathematics in a context and story problems are a great way to make that happen.

Now I also love this little chart from Susan O’Connell that she kind of helps get us away from the keywords.

Keywords is kind of the death of story problems. Story problems can be solved with keywords sometimes, right? But if you really want to get into problem solving, keywords don’t do that.

If you’re having kids underline and highlight and circle, that is not problem solving. That’s just pulling out information and trying to get answers. If you want to build true mathematicians, they need to be solving story problems in context using their own understanding.

One of the big pieces to help kids understand is this key concepts, not keywords. The key concept of addition is that you’re putting things together. That’s a key concept. It’s not when it says ‘all togetherr.” We’ll talk more about all of this stuff in future trainings but this is again just a brief overview of this information.

Instructional Practice #3: Include Practice

Alright, our last one that I really want you to consider doing is practice. Now practice is often kind of a bad word I guess in my eyes in mathematics because it conjures up this image, and even when you go Google it, go Google math practice, here’s what comes up that it’s worksheets, it’s drill and kill.

So this idea of practice, we have kind of gone away from because we feel like it’s not building that thinking skill. But kids still need to practice, right? That’s how you hone your craft in anything is that once you understand how it all works, you need to practice and get better and more efficient at it. So practice needs to be a big part of what we’re doing in mathematics. But I really want you to think about ways to make it fun. It does not need to be a worksheet.

Math is not a worksheet.

And practice does not need to be a worksheet.

So my third and final recommendation is to practice via games. Make it fun for kids. Now I know that games can be scary, intimidating, a pain in the rear, however you want to say it, but games are such an important piece. Not only does it build mathematical thinking for kids, but it builds social skills that a lot of our kids do not have exposure to, because games are not being played at home as much as they were in the past. So as much as you can, bring in games.

Now the problem with games is you have to teach the games. Kids make a mess. There’s stuff everywhere.

So I want to encourage you to think about what I call evergreen games. Are there games that you can teach once and then just switch out the context or the content that’s in those games? So things like memory. You teach a kid how to play memory where you flip over all the cards and then you flip over two at a time, and if they match, you get to keep those cards. But once they learn how to play memory, all you need are different decks of cards that are paired in different ways. So here’s one where you have an equation, 10 + 28, and they are trying to figure out the answer to that, so the match is the answer.

You can have visual ones as well. Just think of ways that you can create memory games and just switch out the content, then it’s a lot easier because you don’t have to teach how to play the game over and over again.

Another popular one is Bump.

Bump’s a popular one that’s been around awhile. This is one where you roll the dice and then you add 20, and then you can put your game piece on that spot.

I like to put these in little folders and then kids can use whiteboard markers and mark on them, and if you get two marks on your spot, nobody can bump you off, but if you only have one marker and somebody rolls that same amount, they can bump you off. That’s the fun part of bump.

So games are a great way to incorporate practice. Kids are getting tons of practice without even knowing that they are practicing. It is a fun way to build this mathematical practice to help kids become more efficient. The hard part with practice is you can’t play games if you haven’t first yet built some understanding in number sense, because when you go to play games, it becomes frustrating if you can’t solve the problems quickly, because the whole part of games is that you are quick when you’re playing those games. So we need to do all of these pieces to help kids build their fluency, and then we just hone that fluency through practice using these games.

Implementing These Ideas

Okay, so how do you take these recommendations and actually implement them? A reminder of my first thing I started this with is that you need to determine the content your students need to work on based upon that progression of learning.

Then everyday start having some kind of number sense routine. Those routines should not take more than 10 minutes. When you get really good at them, you can do them within 5 minutes. One of the things I didn’t mention but I will go more in-depth with this in future videos is solving story problems is not just about solving a problem and getting an answer. The big emphasis of story problems is also then sharing their thinking and talking about the different ways that kids solved those problems.

Then the last piece is to incorporate practice using games. Again it should take 10 minutes or less. Once kids start learning evergreen games, all they do is just go pull a game and they can start playing. You don’t have to spend time teaching the rules and getting the stuff out. You will have a bunch of pre-made stuff.

Alright, so I hope that this has given you a little bit of inspiration. Like I said I hope you take on all of this, but if you don’t feel like you can take it all on, just remember I wore my shirt, you probably aren’t going to be able to see it, and it’s kinda messy here with the stripes and everything, but it’s another version of my perfectly imperfect. That is one of my big mottos is that if you are waiting to do this perfectly, it will never happen. Just try something, see how it goes, and then implement from there, right? You don’t have to do it all at once.

So don’t forget to download your guide to teaching math without a textbook. It should be right below this video. I outlined the reasons why you should move away from your textbook. I give you more detail about the three things that we talked about in this video. And there’s going to be some stuff from the future videos that we’re going to be talking about. So you’ll want that guide for all of these videos in this series. So make sure you go download it.

I hope that this video and the whole series and the guide to teaching math without a textbook helps you build your math mind so you can build the math minds of your kiddos. Have a great day.