Math Manipulatives That Count: The Right Tool for Thinking

Walk into almost any elementary classroom, and you will see how intentionally the space has been designed to support literacy. Classroom libraries are inviting and accessible. Books are displayed at students’ eye level. Students can book-shop until the cows come home. Math manipulatives often tell a different story.

The connecting cubes, counters, ten frames, base-ten blocks, fraction strips, and other tools may be in the classroom, but they are frequently stored on a high shelf or tucked away in a closet. Students technically have manipulatives available, yet they cannot independently access or choose them when they need them.

If we want students to engage in Standard for Mathematical Practice 5: Use appropriate tools strategically (SMP 5), the tools have to be within reach. I’ve written before about thoughtful systems that help math centers run smoothly; those same principles should make it easy for students to find, use, and return manipulatives independently. Students also need enough experience with different tools to make informed choices about which one might help.

Students can choose tools strategically only when those tools are visible, accessible, and familiar.

But access and student choice are only part of the equation. Students also need tools that make the particular mathematical relationships they are studying visible.

As educators, we also have to be strategic about the manipulatives we introduce, the connections we help students make, and the moments when we encourage students to return to a concrete model. Manipulatives are not just for students in kindergarten through second grade. They are not all created equal. And they should not be used only in lockstep, with every student using the same tool at the same time—or no one using one at all.

“My Teacher Said We Didn’t Need Them Anymore”

A few years ago, I was working in a second-grade classroom when I noticed a student solving a subtraction problem similar to 71 – 56.

She was making a common error. In each column, she subtracted the smaller digit from the larger digit so that the subtraction would “work.” Instead of regrouping, she subtracted down in one column and up in the other.

I asked her where the base-ten pieces were.

She explained that her teacher had told the class they no longer needed manipulatives. 

I asked again where they were, and she pointed toward a shelf by the window. I retrieved them and asked her to model the problem.

Using the base-ten blocks, she represented 71, exchanged one ten for 10 ones, and successfully subtracted 56. The model allowed her to see why regrouping was necessary. It made visible something the written algorithm had hidden from her.

Manipulatives give students access to mathematical relationships that may be difficult to see in numbers and symbols alone.

Next, I asked her to draw a picture of what she had done with the blocks. Then I gave her another problem and asked her to begin with a drawing. She solved that problem successfully, too.

If she had not been successful, I would have returned to the manipulatives.

CSA Is Not a One-Way Staircase

That is an important part of the concrete–semi-concrete–abstract, or CSA, progression. One of the greatest benefits of manipulatives in math is that students can move between physical models, drawings, and notation as their understanding develops. CSA should not be treated as a one-way path students complete before leaving manipulatives behind forever.

Students can—and should—move back to a previous representation when the mathematics becomes more complex or their understanding begins to break down. The student in that second-grade classroom did not need more practice following the subtraction procedure incorrectly. She needed a representation that would help her understand the mathematics behind the procedure.

Using the same problem across materials, quick drawings, and equations helps students connect representations instead of experiencing them as separate methods.

Moving back to a concrete or pictorial representation is not regression. It is strategic sense-making.

Why Math Manipulatives Aren’t Just for Beginners

Too often, manipulatives are treated like training wheels. Once students can draw a model or write an equation, we pack the tools away. As students get older, we may assume they should no longer need concrete materials.

In reality, students may need manipulatives even more as they encounter increasingly complex ideas.

A student who no longer relies on manipulatives for whole-number place value may benefit from base-ten blocks when exploring decimal place value. A student who can identify simple fractions symbolically may need fraction strips to reason about equivalence. Older students may benefit from tools and representations such as two-color counters, number lines, area models, and hanger diagrams as they explore integers, expressions, and equations.

Moving to more sophisticated mathematics does not eliminate the need for representations. It changes the representations students need.

The goal is not to keep every student using manipulatives forever. Nor is it to remove manipulatives as quickly as possible. The goal is for students to develop a flexible collection of representations they can use strategically—and for educators to recognize when a particular tool will illuminate the mathematics.

Choosing Math Manipulatives Requires Strategic Teaching

SMP 5 is about students choosing tools strategically, but educators shape the conditions that make that choice possible. We decide which tools are accessible, which models students learn to use, and how explicitly we connect concrete materials to drawings, language, and notation.

Building that shared understanding across classrooms is part of creating instructional coherence. The Math Pact by Karen S. Karp, Barbara J. Dougherty, and Sarah B. Bush specifically identifies the use of manipulatives and visual representations as a shared instructional practice. With elementary, middle school, and high school versions, the series emphasizes creating consistent expectations for how students use tools and connect concrete models to drawings, language, and notation across classrooms and grade levels.

So, what are math manipulatives? They are representations of mathematical ideas. Each tool highlights certain relationships, places others in the background, and asks students to make particular connections. Hands-on does not automatically mean minds-on. The question is not only, “Do students have a manipulative?” It is, “What does this manipulative help students understand?”

Introducing Manipulatives That Count

That question is the starting point for a new Coaching That Counts blog series: Math Manipulatives That Count: The Right Tool for Thinking.

Over the course of the series, we will examine familiar tools, compare models that are often treated as interchangeable, and consider when each one best supports the mathematics students are learning. We will keep returning to what each representation makes visible, what it might hide, and which understandings it is best positioned to support. Together, the posts can also provide a starting point for PLC conversations about tool selection and consistent instructional practices across classrooms and grade levels.

Because the most important question is not simply, “Did students use a manipulative?”

It is: What mathematical thinking did this tool make possible?

This series will help educators explore that question in their classrooms and PLCs.

Ready to put these ideas into practice? Coaching That Counts workshops help educators turn these principles into meaningful classroom experiences. In May 2027, Beyond Busywork K–2 and Beyond Busywork 3–5 will focus specifically on using manipulatives and centers to support mathematical thinking. View the available workshops and register.