The Math They See, the Math They Use: Demonstrating Concepts with Math Manipulatives

by Kathy Kauffman Piper Burdge

In math, we expect students to memorize steps and learn processes with the hope that they will understand the concepts as they use them. But this last step—understanding—often never happens, say Piper and Kathy. Without understanding why the rules work, students often miss the ability to use them in real life. The Math-U-See curriculum addresses this problem. Piper and Kathy demonstrate Math-U-See’s approach to multiplication and area, and offer their thoughts on using the curriculum effectively. Even if your school uses a different curriculum, consider how you can incorporate manipulatives and conceptual learning in your math classes.

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Piper: I’ve worked in food service and I’ve gotten people who don’t know how to make a recipe three times larger than it should be. I can say to them, “You need to double that recipe or you need to triple that recipe,” and what they’ll do is they’ll take the half cup and they’ll measure three times with it, rather than realizing that, “Oh, that’s a cup and a half.” And probably they could do the faction work in their math book, but it never jumped to understanding how to use it in everyday life.

Kathy: The Math-U-See curriculum that I use here, what they really want the students to understand is the “why” of math. For example, if I have a multiple digit multiplication problem, like 14 times 12. When I went to school, my teacher just told me where to put the answer. So this is 8, this is 2, 4 and 1, and we add it, and that’s my answer. But not understanding why the numbers go where they do.

Piper: For some students, their math sense is kind of innate and learning those processes will give them what they need because they’ll make the connection between the concept and the process. But most of us have students sitting in our classroom who will never make those connections or make them accurately on their own. If they don’t make the connections between the concept and the process, their ability to use that math outside of a textbook is not what we hope it will be for them. With Math-U-See, we teach students that multiplication is a block or a rectangle. When they begin learning multiplication facts, we often build those rectangles with them. So if they were doing 5 times 3 they would take their 5 blocks—and in Math-U-See each of the blocks is a different color which makes it really easy for them to distinguish where they are. 5 blocks are a light blue. And they would build a rectangle that was 5 across and 3 up.

So we talk about dimensions, we actually don’t use length and width. We teach them to stay across and up. It’s not a big difference but it’s a more natural way to talk about it for younger students.

We often have them build a rectangle, sometimes I have them do it on a marker board and then they would write the dimension five underneath and three across. That’s a visual for them to understand the multiplication, sometimes we say 5 times 3, sometimes we also talk about it as 5 counted 3 times. One way that I teach this at the beginning of third grade is to have them count, they build the block and then they count it 5,10, 15 and they move the block as they go.

Once they learn to build those basic blocks and they learned their facts—the first number of lessons are about the facts and about multiplication—you also teach area with multiplication.

Kathy: If you want to find the area of a rectangle or a square, you have to know your over and up dimension here. So that’s what this really is. This is actually showing me I have 14 over and I have 12 up.

Piper: The next step once they’ve learned all their facts—which really takes the first half of the year or so—is that they move into multiple digit multiplication. To do multiple digit multiplication, they first need to learn to build larger problems.

Kathy: We build it with our manipulatives. I will get my top number here and I know I have 14 so that means I need to have one 10 bar and a 4 bar.

Piper: If a student is doing this problem, to begin with we’re going to build the problem. And our over dimension is 13 our up dimension is 12. Sometimes we mark that: the over dimension and the up dimension. And then we build it.

Kathy: My goal here is to make a [rectangle] because multiplication is actually a [rectangle].

Piper: We start with the over dimension which is 13. So to make 13, I take a 10 block and a 3 block and that’s 13, 10 plus 3. And then we need to make it 12 up. We take a 100s block which is of course 10 high and 10 across—10 over and 10 up—and we put that on. Now I’ve got 11 and I need 12. I add another 10 block just to get my 12. Right now we’re only focusing on the up dimension so we got that—it’s 12. And then we need to fill in: 13 is right here, going across, 13, so we’ll fill that in. Put another 10 block there, and a 10 block there and a 10 block there. That fills me up right to that. Then obviously I need another 3 there.

Kathy: This is my answer to my multiplication problem. And so they think, “Great, I can just write it in.” But I have them actually break it down, tell them to take the square apart.

Piper: This takes a while, sometimes, for students to learn. I found that I needed to take a lesson—a day or two—and have them work with building blocks. For some of my students that came pretty easily and for some of them it didn’t come quite as easily, to move into these larger blocks.

One of the really neat things about this as is that when you look at this block you actually see the four problems that you do as you multiply.

Kathy: They have to put into their place value. So they’ll bring their units over here, their 10’s are right here, and this is the hundreds place.

Piper: When you work with a problem like this, there are two ways that you can work with it, and it’s good for students to learn both ways. The first way or one of the ways is to use what Math-U-See calls place value notation. It can also be called expanded form or expanded notation in different curricula. Place value notation is when you take a number like 13, and you divide it into its units and its tens and if you have more numbers, if there were 100s or 1000s, you would divide out that way.

Kathy: I have my students break this down into its place value. They know that 14 is 10 plus 4, and they know that 12 is 10 plus 2.

Piper: So to write 13 there are three 1s in 13. So we write a 3 and there’s one 10, so you write a 10. Put a plus sign in between because 10 plus 3 is 13.

Kathy: The important thing is that they understand place value. This 1 actually is 10. It’s not 1. This is actually four unita.

Piper: And students can see this with the blocks, if they make 13. You’re going to put your 1s here, here’s your 13, your 10s here, and you can split it apart into your units and your 10s.

Kathy: That is what they want them to understand here is the “why.” Why is this 10 plus 4 that means I had—it’s one 10 plus four of these.

Piper: Our place value notation for 13 is 10 plus 3, and for 12, it would be 10 plus 2. There’s a 1 in the 10s place and a 2 in the units place.

Kathy: If they still get a little confused, I have them draw a line like this. This is my units. This is my 10s place and this is my 100s place. So this is his—that his home, is what we would call it.

Piper: If we multiply this, you can see the four different problems that the block shows us. So the first problem is 3 times 2 or 2 times 3. The second problem is 2 times 10. And then the third problem is 10 times 3 and the fourth problem is 10 times 10.

If you look at the block, you see those four problems. The first one 3 times 2; the second one, 2 times 10; the third one, 10 times 3, and fourth one 10 times 10.

When we multiply we go 2 times 3, and that’s 6—that was our first step. Our second step is 2 times 10 and that’s 20. We write that in the 10s column, and go ahead and put our plus sign in between because we’re going to add those two numbers together, ultimately, in the end. Then we move over here: 10 times three is 30.

It’s really important to keep that lined up in the 10s place. It does not go in the 1s place because 30 doesn’t go in the 1s place. It’s too big for the 1s place. We would have to move it over because it would be too many numbers here. You can only have up to 9 in the 1s place.

And then we have 10 times 10; that’s 100. That has to go over here in the 100s place.

When we’re all finished, we go ahead and add, and we get 6 plus nothing is 6. 20 plus 30 is 50 and 100 plus nothing is 100. When we’re finished, you can add everything together: 100 plus 50 plus 6 for your final product, which is 156.

So when I did this problem I was thinking about it from my perspective. I’m used to doing my blocks on a board. I have a magnetic set of blocks. But if you’re doing it for students and you’re on a flat surface and they’re looking at it you need to make sure that you keep your place value lined up. So probably the block should’ve built like that because these are your 1s. You’re going to want them on the side that students would see the 1s on, which is their right, your left.

Kathy: I have them teach it back to me, and once he can teach it back to me, I know that he has mastered it and then we move on to this.

Piper: One of the really fun things that Math-U-See does I’ve never seen any other curriculum do is the way that they teach multiplication, not in expanded form. I like this because it mirrors the expanded form experience.

So if you go 2 times 3 here, we get 6. 2 times 1 is 2. Then this is 10, just like you see it here. The 1 is in the 10s place and that’s a 10. 10 times 3 is 30 and because it’s a 30, we have to put the 3 in the 10s place. If you want to you can go ahead and fill in that 0 here or sometimes I have my students make an X to show that there’s nothing in that place but the 3 from 30 has to go to the 10s place to keep in lined up with the 2 from 26. Then we move on and go 10 times 10, again, the 10 is in the 10s place here and here so its 10 times 10, and that’s 100. 100 goes in the 100s place—the one from 100. Now we’re able to add everything together. 6 plus nothing is 6, 2 plus 3 is 5 and 1 plus nothing is 1, for a final product of 156.

There’s a very strong idea that the writer—the person who designed the curriculum—has: that students are going to learn something, they’re going to master it, and then they will be able to move on to the next step. Part of the idea there is that they spend longer on individual pieces of the process. So in the first book which is Primer, they’re learning just numbers, number sense, very simple addition. In Alpha and then Beta, they work with addition and subtraction.

And by the time they get to Gamma, which is the fourth level, a lot of people would think of that as a third-grade level. They are supposed to have mastered addition and subtraction. And they’ll continue to review that, but the idea is that that foundation has been laid and they’re ready for multiplication. So, in Gamma they learn how to multiply, the whole way up through, at the end of the book multiplying multiple digit numbers together. It’s fairly high-level multiplication for a fourth grader. And then when they move on to Delta, the fourth-grade book, they learn division.

So, it’s very structured and part of that structure is that each thing builds on the next. So if you’ve learned multiplication fully, you should be able to move into division with confidence. Along with that, there’s a lot of emphasis on students understanding what they’re doing and not just learning a process. So that’s why you’re going to spend as much time as you do with place value and place value notation because understanding place value is really important to understanding how multiplication works. Otherwise, you can learn a process—”Do this process and your answer will be right,”—but you’re not really understanding why it’s right or why you have to put this number in the 10s place and not in the 100s place or the 1000s place or the 1s place.

That’s a really important piece of Math-U-See that maybe is not as strong in some other lower-elementary curriculums. The sense of students learning why they’re doing what they’re doing and understanding it. You also see it in story problems, their story problems are a lot less formulaic. The students actually have to picture what they’re doing, picture what the story probably means.

So there’s a good deal of comprehension and not just, “Oh I see this keyword, that means I multiply,” or “This keyword means I add.” Which hopefully translates into better math sense as they move on into upper grades but also into their lives outside of school.

You should be really aware of a couple of things. Math-U-See was initially designed, as I understand it, to be used more one-on-one and individually. And so, there are distinct challenge to taking a group of students and moving them through Math-U-See.

I think there’s a common misconception that Math-U-See was a remedial program and it’s not. It is on grade level. Because of the focus on individual processes—multiplication, addition, subtraction—in different grades, it doesn’t align completely with traditional math curricula. So if you are switching from a traditional math curriculum to Math-U-See, you need to be really aware of where your holes are going to be and have a plan for what you’re going to do to fill those gaps. There is that switch that happens when you move from any curriculum. When you move from a curriculum that is very traditional to one that’s less traditional, the gap is going to be larger and you have to be really careful with what holes are getting left.

The other thing I will say with Math-U-See is that their review is not as strong as many students need. So you’re going to need to supplement with lots of review, both fact practice but also concept review, vocabulary review. They do not incorporate that into their curriculum as much as in my experience you need to incorporate it in a classroom.

Even just being able to walk through the store and say, “Oh that’s 20% off,” and quickly and roughly figure out what you’re paying for an item: those are the types of connections.

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