We are going to talk about how to unlock the power of math manipulatives and why they are for all students and there are just a few small changes that you can make to maximize your use of math manipulatives. We are going to start by looking at just a brief overview of why math manipulatives are for all students. There are a lot of myths around who math manipulatives are for, why math manipulatives are either effective or ineffective for different groups of students. So I want to take some time to just acknowledge why math manipulatives are so effective in the classroom. We will also talk about a few perks that come along with using manipulatives with your students. I also want to give you a quick framework for how math manipulatives fit into your math block and how they help grow students’ understanding in all phases of the learning process. And then, we’ll finish up with three small changes you can make that will get the most out of the manipulatives and it just skyrockets student learning in the classroom.
The Purpose of Math Manipulatives
Manipulatives allow students to be active participants in the learning process. Manipulatives allow students to make critical connections. They are engaged in learning when they learn with their hands first.
Myth vs. Truth #1
Many people believe that math manipulatives are for students in lower grades, not upper elementary because they have outgrown them. The argument for that is that students are just learning about numbers in lower grades and so they need those concrete experiences, which I agree with.
However, students in upper elementary are still learning new math concepts and need those concrete learning experiences to guide them through the learning process. They are still new to operations with decimals and operations with fractions. So, if we’re using that same thinking that’s applied to lower grades, we should be bringing math manipulatives into the upper grades. When students are experiencing a new concept, they need those hands-on learning experiences. One of the things that I love about math manipulatives is they are like magic, but we, as the teacher, have to know how to activate that magic. The reason I say they’re magic is that they can both challenge our advanced learners and support our struggling learners.
Myth vs. Truth #2
Most people believe that math manipulatives are only for our struggling learners, which I hate labeling students with advanced and struggling because really if we are teaching math the right way, all of our students should be engaged in productive struggle. So technically, we can say everybody is a struggling learner because we want them struggling at some point in the learning process.
Math manipulatives push or more advanced learners, those that catch on pretty quickly. Because of that, they typically perform better on standardized tests, but math manipulatives push them, beyond the formulas, beyond the algorithms, and they force them to make connections and build meaning behind the procedures that they are so good at doing.
Advanced learners, need to be able to construct meaning for themselves. A lot of times our learners catch on a bit more quickly, they’re good at following rules. They’re good at following procedures. But, can they build meaning for themselves? Can they take something that they know and apply it to a brand new concept that they’ve never been exposed to before? We want that for all learners, but math manipulatives allow these learners to accomplish that.
Myth vs. Truth #3
This myth is not as common or is not as verbalized amongst teachers, but you can see it in their teaching practice that they may have some beliefs here. This myth is that struggling learners need practice and don’t have time for math manipulatives.
The other part of our magic with math manipulatives is that they are for our learners who typically struggle as well. Now, one of my issues with the term struggling learners, especially in math is this, math is such a broad subject and so, a student cannot just struggle at math. That is like saying that a student struggles with English. Well, what part of English do they struggle with? Is it speaking? Is it understanding? Is it writing? Is it grammar? What aspect of that are they struggling with? And then, let’s pinpoint that and work on it. But in math, when you think about that, there are so many different concepts and domains. And, I have seen students struggle, labeled as struggling learners, and be amazing when it comes to geometry because that visual aspect is just strength for them.
Math manipulatives allow these learners to access the math. Typically with these, with art, these learners who are labeled as struggling learners in math, they may not catch on as quickly to the algorithms. They may struggle to memorize and follow these procedures. And so, math manipulatives allow them to access any type of problem because they have the concrete manipulative that they can use to support them in carrying it out and trying to solve it.
The second part of this myth is that it gives them the understanding to fall back on. So, when they’ve had enough experience with math manipulatives, when they’ve built those connections and they understand the why behind the algorithm. When they eventually get to a point where they make an error and they’re not sure how to fix it, they have the understanding to fall back on. And so, that is very empowering for them because it’s not like they are working on a problem and they say, “Oh, I can’t do this. I need help.” You can coach them through it by having them remember their work with manipulatives and think about how that can help them fix their mistakes.
Additional Benefits of the Power of Math Manipulatives
Math manipulatives are engaging, students love working with them, especially when we build a classroom culture that values student work with manipulatives and values true thinking. They also provide a memorable and common experience you can refer back to. These common experiences can be really powerful as you grapple with new concepts later in the year. And then, one of my favorite things about my current role in working with teachers is building their confidence. And through that, allowing them to be creative in the lessons that they plan and be creative in the experiences that they give students.
The CRA Framework to Unlock the Power of Math Manipulatives
The CRA model stands for concrete, representational, and abstract. This is a long research method for, just the process that students go through in learning math. What it means is they begin with the concrete. So, these are our manipulatives in our hands-on learning experiences. We can use structured manipulatives and unstructured math manipulatives as well. A lot of times I will use post-it notes in the classroom. Those can be considered manipulatives because it allows students to explore the math in a hands-on way.
We have our concrete learning experiences, then we have our representational learning experiences. These are ones where students are using pictures and drawings to represent the math.
Then lastly, we have the abstract. This is where students are using written methods, algorithms, numbers, and symbols to represent the work that they’re doing with their math. Essentially, students can do the same work through all phases. It is how they are interacting or representing the math that changes.
This model says that students should first learn in the concrete, then move to the representational and then move to the abstract. I agree with this. I will say one area I think many teachers, I guess kind of mess up a little bit is that we are not bridging the gap between these phases. A lot of times I will hear students or hear teachers say my students can do it with the manipulatives, but then as soon as we get to the drawings or the written methods, they all of a sudden can’t do the math. My response to that usually is, have you made those connections? Have you shown them how the written methods connect to the work that they just did with manipulatives? I think there needs to be a lot of overlap between these phases. We, as teachers, need to be very intentional to plan and focus on the bridge between each phase. The more effort and focus that we put in connecting that concrete phase to representational and also connecting the concrete phase to the abstract, the more successful students will be when they are working solely in the abstract when they are working solely with written methods and algorithms and procedures and formulas.
3 Small Changes to Unlock the Power of Math Manipulatives
Change #1
Give students context. We do not experience numbers in the real world without context. You are not walking down the street and you see a sign that says five on it. It typically says $5 burgers or $5 for a tank of gas, which hopefully it doesn’t say $5 for a tank that would be outrageous. But, there is always meaning to these numbers. We want to model this same experience in the classroom.
We want to give them context because that’s what it’s like in the real world. But then also, math becomes real difficult, unnecessarily difficult when students don’t have something to visualize when you say three plus five, which is a lower grades concept, but when you say three plus five, it does something in a student’s mind when you add apples to that, or when you add dollars to that, now students have this concrete object that they can visualize adding together. Giving students that context helps when you are talking about the math manipulatives as a class, you have a word or just meaning that you can connect back to. You’re not just throwing out numbers, you’re saying, “Oh, they had three apples, and then this person added four more to the bucket.” And, it just, it’s so much more impactful.
One of the ways that I like to include, or to give students that context is by using problem frames. So, I would use problem frames when problem-solving is not necessarily the focus of the lesson. Maybe you are trying to highlight a specific connection, and so, you just need some context. You’re not looking to solve an in-depth problem. You just want some context around the numbers, and so, a problem frame is just a numberless word problem. We’re just taking a very basic problem, taking the numbers out of it, and then, you’re just plugging in different numbers. There are times more often than not, that problem-solving should be our focus, but, when you are just wanting to use math manipulatives, throw a problem frame in there with them, so that way students have something to visualize, have something to connect back to, as they are doing the work to uncover whatever it is you’re working with to highlight with the manipulatives.
Change #2
The second quick and easy strategy is to as often as possible, provide students with some type of think sheet. What a think sheet is, it can look very, very different. It can be a worksheet where students are keeping track of their work with manipulatives, where they are maybe representing their work in a written way to just document their thinking. So that way, when you go to discuss it later on, students have, they can kind of remember what they did on their own with manipulatives. It can also be a math journal. It can be where students, as they are working with manipulatives, they are documenting things that they saw, that they were surprised by, things that confused them, connections that they made, something they discovered. So, just setting up a sheet in their math journals for that as they are working with math manipulatives, that can be a think sheet. So, it doesn’t have to be anything special. It’s just some written way for students to keep track of their thinking with manipulatives.
The bonus of this, besides the fact that it holds students accountable, is that a lot of times using that think sheet is an automatic bridge to overlapping the different phases in our CRA framework. Because if you are having students who work with concrete manipulatives, and then you’re having them draw a picture of their thinking, or using number sentences to represent their thinking, well, now, you’re connecting the concrete to the representational and the concrete to the abstract.
Students may not always document their thinking in the way that you intend for them to, but it’s still a really powerful activity and process for them to make those connections. So, as often as possible, include thinking sheets or plan for think sheets in your activities.
Change #3
The last thing that allows students to talk about what they’re doing with manipulatives. So often, we send students to a center and they work with manipulatives and they solve the answer and they move on to the next problem. Then they submit that work to the teacher, but probably the most learning is left on the table when we don’t give students a chance to debrief and to talk and to record and to showcase their thinking with manipulatives. That is one of the most powerful times in my classroom is when students are sharing their discoveries when they are sharing the different ways that they solved the same problem with manipulatives when they’re analyzing each other’s approaches. So, that is another aspect that I would always plan for when students are working with manipulatives, bring in some way for them to showcase their thinking, whether that is actually having that whole group discussion, having discussions in a small group, or if students are working individually.
Say that they are working with virtual math manipulatives, giving them, showing them some type of app that will allow them to record their screen and explain, like, do a voiceover essentially to record their thinking as they are solving that problem with the manipulatives. That is good not only for them to be explaining their thinking. We know that research tells us that students grow their learning when they are explaining their thinking to others.
The other side of that is students are also adding depth to their understanding when they are analyzing the thinking of others, when they are listening to others and having to say, “Okay, does this fit in with what I know about this concept, is this right? You know, how does this compare to my strategy?” So it works both ways. It’s really helpful to have students showcase it, showcase their learning and their understanding and their experience with manipulatives, but then also having other students kind of take part in listening and analyzing that.
So these are three really small changes that can maximize your use of math manipulatives and increase your effectiveness with students. I hope you are walking away from reading this with small changes that you can make to get the most from your math manipulatives, whether you are brand new to using math manipulatives with your students, or you’ve been using them for a while, and you just want to make the learning a little bit more meaningful and lost lasting. I hope these small changes will give you just that. While math manipulatives are really powerful math tools, they are simply tools. We have to know how to use them and we can always grow in how we use them and how we facilitate learning with them with our students. Math manipulatives are for all students, and if we are using them to build up students’ understanding of math, we’re not only creating strong math learners, but we are creating empowered learners that will continue to learn far beyond our classroom.