Why Making Math Visible Matters

By Kim Rimbey 

Math is the study of patterns.
Math is a way of thinking. Math helps us describe the world—quantitatively and abstractly.
Math has many expressions.

But here’s the thing: If we can see it, THEY can see it.

When it comes to teaching and learning math, getting the thinking out of our heads and into the world is essential. Visible thinking helps us communicate clearly, uncover misconceptions, and build shared understanding.

That’s exactly why we wrote Seeing the Math You Teach—to support teachers, parents, and students in making their math thinking visible. “Seeing = Understanding.” With the understanding that there are multiple solution paths to any problem, the book offers a variety of visual strategies that help learners represent and explain their ideas.

Number Lines: More Than Meets the Eye

While prepping for a recent conference, I took a deep dive into one of our most powerful tools: the number line.

We often start with number tracks—like snap cube trains—to help young learners count units. Then, we shift to number lines with tick marks. But this shift calls for a shift in focus, too. It’s the spaces between the tick marks that matter. That’s where the “jumps” happen. That’s where the math lives.

Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.

And no matter what the intervals—1s, 10s, 25s, millions, fractions—every point on a number line represents a distance from zero.

This is true even if zero isn’t shown. Even if we’re looking at just a portion of the number line, the meaning of every number still depends on how far it is from zero. And it’s true for open number lines as well. Even though the jumps may not be proportional, they still represent the distance traveled from zero. It’s all about relational thinking.

Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.

Quick Teacher Check-In: Do your students understand that every value on a number line—visible or open—is a measure of its position relative to zero?

As students start representing their thinking using number lines, we need to help them see what they’re showing—even when zero is nowhere in sight.

Letting Students SEE and BUILD the Math

Here’s the hard truth:

We can’t just tell students what to see. They need to explore, discuss, make mistakes, and revise.

And for that to happen, we need to create learning environments that encourage deep thinking and collaborative sense-making.

Enter…Vertical Non-Permanent Surfaces (VNPSs)

VNPSs—think whiteboards, whiteboard cling film, windows, chart paper, etc.—are erasable vertical spaces where students stand, collaborate, and represent their thinking.

Peter Liljedahl’s Building Thinking Classrooms popularized this approach, and I’ve seen firsthand how transformative VNPSs can be.

For Teachers 

• Easily scan every group’s work
• Step in briefly to nudge, hint, or extend
• Plant seeds and move thinking forward
• Select and sequence work to meet learning goals
• Spotlight key ideas—like number lines and distance from zero

But What If I Don’t Have Enough Whiteboards?

Good news—VNPSs aren’t about expensive equipment. Here are some ideas for where to find inexpensive VNPSs (from the Building Thinking Classrooms update posted on June 2, 2025 – used with permission – click here to subscribe):

Your Next Steps

For Teachers

Two big ideas, two next steps:

1. How might you talk about number lines in a way that conveys “distance from zero” as the central idea? 
2. How might you incorporate VNPSs (or use them more intentionally) to support visible, collaborative math thinking? 
3. Which visual do you use to support your students’ thinking? For more ideas on number lines and many other visuals, check out my newest book, Seeing the Math You Teach. 

For Leaders

Same two ideas, new lens:

1. How might you build shared understanding during PLC/planning sessions around the idea that number lines represent distances from zero? How might you facilitate doing math together? 
2. What barriers can you remove to ensure teachers have access to VNPSs and the tools to use them effectively? 

Final Thoughts

Let’s keep helping students see the math we teach.
Let’s help them own their thinking.
Let’s keep learning—together.

Interested in Workshops by Kim Rimbey?

Kim offers workshops on a variety of topics, including Building Thinking Classrooms, Ten-Frame Mathematics, Mastering Math Manipulatives, Small-Group Math Instruction, and See the Math You Teach: A New Vision for Math Teaching and Learning. View Kim’s PD Catalog to learn more.

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.