Hands-On Math, Professional Learning, Teaching and Learning
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 Students
- Interact and record their thinking
- Erase, rework, and revise as they learn
- Represent ideas in multiple ways
- See other groups’ strategies
- Stay on their feet and activated
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:
- How might you talk about number lines in a way that conveys “distance from zero” as the central idea?
- How might you incorporate VNPSs (or use them more intentionally) to support visible, collaborative math thinking?
- 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:
- 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?
- 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.
Hands-On Math, Problem Solving, Seeing The Math You Teach, Teaching and Learning
Have You Ever Really Seen the Math?
Did the way you learned math suit you? Were the concepts connected, or were they more about following steps and rules without really understanding why they worked?
For me, dividing fractions was always a mystery. I could recite the steps: flip the second fraction and multiply. But I had no clue why it worked. And why, when I divided two fractions, did the answer sometimes get larger? Isn’t dividing supposed to make things smaller?
If you’ve ever felt this way, you’re not alone. Many of us were taught to do math, but not necessarily to see it.
And that brings me to Ted.
Have You Met Ted?
I met Ted a couple years ago at a workshop I led in his district. Ted had been teaching fourth grade for 15 years. He was confident, experienced, and deeply committed to his students. But recently, changes to his state’s standards had shaken his confidence.
One shift in particular? Multi-digit multiplication.
Instead of simply teaching students to multiply using the standard algorithm, the new standards asked Ted to teach multiplication “using strategies based on place value, properties of operations, and the relationship among operations.”
Ted was puzzled. How do you teach strategies you’ve never seen? And even if your curriculum includes those strategies, how do you use them effectively if they weren’t part of how you learned math?
This led Ted to a moment of self-reflection: his students could follow steps, sure. But did they really understand what those steps meant?
Seeing the Math You Teach
Rather than jumping into “show and tell” mode, I invited Ted and his colleagues to engage with a two-page spread from a visual math resource I was working on with my co-authors – Seeing the Math You Teach. We explored multiple strategies for multi-digit multiplication. No algorithms. No formulas. Just visuals and reasoning.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
Take a moment to imagine what Ted saw on these two pages. Think: base-ten blocks, area models, number lines, and symbolic representations all working together to build meaning.
What do you notice? What are you curious about?
As you examine visuals like these, you begin to uncover the connections:
- Place value models that show how tens and hundreds grow
- Arrays and area diagrams that highlight the structure of multiplication
- Number bonds that reveal flexible thinking about numbers and their relationships
- Symbols that begin to make sense after the concepts take shape visually
And just like that, the math starts to come alive.
Beyond Elementary: A Universal Invitation
While the examples in this post come from elementary classrooms, the ideas are universal. Whether you’re helping first graders explore base ten or guiding high schoolers through slope-intercept form, visual models support conceptual understanding.
They help students connect the why behind the how. They turn math from mystery into meaning.
So let’s explore together.
Visual Strategies in Action
Let’s look at how some common tools and representations connect concepts across K–12.
- Number Bonds. Great for developing part-part-whole reasoning and flexible number sense. Later, they laid the groundwork for decomposing expressions and functions.
- Base-Ten Manipulatives. Perfect for understanding place value, regrouping, and decimal operations. With the right scaffolding, they even support Algebra Tiles!
- Area & Array Models. Help students visualize multiplication as repeated addition and structured grouping. Also helpful for algebraic expressions and factoring in secondary grades.
- Number Lines. Useful for addition, subtraction, fractions, decimals, and integer operations. Use Linking Cubes as physical number lines in early grades. Extend into coordinate graphing and slope visualization in later grades.
- Symbols. A powerful tool, especially when paired with visuals. They offer precision and efficiency, but only after meaning is established – whether you’re connecting quantities to numerals, building understanding of comparison symbols, or establishing ways to record exponential growth.
Whatever your grade level or content area, visual representations offer a bridge to understanding.
Seeing the Math You Teach – Part 1
By Kimberly Rimbey, Ph.D., NBCT
Have You Ever Really Seen the Math?
Did the way you learned math suit you? Were the concepts connected, or were they more about following steps and rules without really understanding why they worked?
For me, dividing fractions was always a mystery. I could recite the steps: flip the second fraction and multiply. But I had no clue why it worked. And why, when I divided two fractions, did the answer sometimes get larger? Isn’t dividing supposed to make things smaller?
If you’ve ever felt this way, you’re not alone. Many of us were taught to do math, but not necessarily to see it.
And that brings me to Ted.
Have You Met Ted?
I met Ted a couple years ago at a workshop I led in his district. Ted had been teaching fourth grade for 15 years. He was confident, experienced, and deeply committed to his students. But recently, changes to his state’s standards had shaken his confidence.
One shift in particular? Multi-digit multiplication.
Instead of simply teaching students to multiply using the standard algorithm, the new standards asked Ted to teach multiplication “using strategies based on place value, properties of operations, and the relationship among operations.”
Ted was puzzled. How do you teach strategies you’ve never seen? And even if your curriculum includes those strategies, how do you use them effectively if they weren’t part of how you learned math?
This led Ted to a moment of self-reflection: his students could follow steps, sure. But did they really understand what those steps meant?
Seeing the Math You Teach
Rather than jumping into “show and tell” mode, I invited Ted and his colleagues to engage with a two-page spread from a visual math resource I was working on with my co-authors – Seeing the Math You Teach. We explored multiple strategies for multi-digit multiplication. No algorithms. No formulas. Just visuals and reasoning.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
Take a moment to imagine what Ted saw on these two pages. Think: base-ten blocks, area models, number lines, and symbolic representations all working together to build meaning.
What do you notice? What are you curious about?
As you examine visuals like these, you begin to uncover the connections:
- Place value models that show how tens and hundreds grow
- Arrays and area diagrams that highlight the structure of multiplication
- Number bonds that reveal flexible thinking about numbers and their relationships
- Symbols that begin to make sense after the concepts take shape visually
And just like that, the math starts to come alive.
Beyond Elementary: A Universal Invitation
While the examples in this post come from elementary classrooms, the ideas are universal. Whether you’re helping first graders explore base ten or guiding high schoolers through slope-intercept form, visual models support conceptual understanding.
They help students connect the why behind the how. They turn math from mystery into meaning.
So let’s explore together.
Visual Strategies in Action
Let’s look at how some common tools and representations connect concepts across K–12.
- Number Bonds. Great for developing part-part-whole reasoning and flexible number sense. Later, they laid the groundwork for decomposing expressions and functions.
- Base-Ten Manipulatives. Perfect for understanding place value, regrouping, and decimal operations. With the right scaffolding, they even support Algebra Tiles!
- Area & Array Models. Help students visualize multiplication as repeated addition and structured grouping. Also helpful for algebraic expressions and factoring in secondary grades.
- Number Lines. Useful for addition, subtraction, fractions, decimals, and integer operations. Use Linking Cubes as physical number lines in early grades. Extend into coordinate graphing and slope visualization in later grades.
- Symbols. A powerful tool, especially when paired with visuals. They offer precision and efficiency, but only after meaning is established – whether you’re connecting quantities to numerals, building understanding of comparison symbols, or establishing ways to record exponential growth.
Whatever your grade level or content area, visual representations offer a bridge to understanding.
Seeing = Understanding
Too often, in mathematics, we are in a rush to move to the abstract. In doing so, we overlook the visual.
-Peter Liljedahl
Seeing the math you teach means making ideas visible.
It means sketching. Moving things around. Building understanding.
And when we see it, our students can too.
Next Steps: What Could You See More Clearly?
Teachers: What’s one concept you teach that students struggle to grasp? Could a visual representation help uncover the why behind the steps?
Leaders: How might you use ideas like those above to spark the conversation with your teacher teams? How might you subtly use these questions to help your teachers see the math in new ways?
Everyone: The book is available now, both from Corwin and Amazon!
In the next post, we’ll explore ways to use Seeing the Math You Teach to support both teachers and students to really see the math.
Until then, keep seeing the math—and helping your students see it too.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, Problem Solving, Professional Learning, Seeing The Math You Teach, Teaching and Learning
When You See the Math, They Will See It Too
by Chryste Berda
If math has ever made you feel small, you’re not alone. Many of us have internalized the message, intentionally or not, that math is a fixed ability: you either have it or you don’t. But what if making math visible could be the very thing that transforms that feeling of smallness into strength?
In my classroom, we lived by just two rules:
- Math makes sense.
- Everyone will be good at math.
I picked up these rules at a math conference years ago, and they became my mantra from that point forward. But recently, a moment reminded me that these rules are deeply connected—and that one cannot exist without the other.
When Rule #2 Is Broken
I was screening number sense for a colleague when a 9-year-old student caught my attention. She got the problem 14-9 wrong and then sighed in frustration.
“No, that’s not right, but I don’t know why this is so hard for me! I’m just not good at math.”
In that one sentence, I heard it: Rule #2 had already been broken for her.
But here’s what I’ve learned: Everyone will be good at math only WHEN we teach it in ways that make sense. That’s Rule #1. And when we break Rule #1, we can’t keep Rule #2.
Making Math Visible Is the Key
Just the day before, I had been working with our district’s special education teaching team. The focus? Making math accessible by making it visible.
Making math visible doesn’t just mean using manipulatives or diagrams – it means illuminating the thinking, patterns, and connections behind the math. It means helping learners see what’s going on under the surface.
One teacher pulled me aside during that workshop. In a hushed voice, she said:
“I’m not a math person, but I want to be—for my students. The way you’re showing us how to make the math visible? I’m actually learning the math for myself!”
She followed up this comment by asking if we could continue to collaborate as she begins this with her students. You see, math becomes visible not just through clear explanations, but through shared exploration – with colleagues who remind us we’re not in this alone because in Rule #2, “everyone” doesn’t just mean all students, it literally means everyone – colleagues, parents, administrators, students, – everyone!
From Professional Learning to Classroom Confidence
This philosophy is at the heart of our book, Seeing the Math You Teach. We’ve seen how making math visible doesn’t just support student understanding–it builds teacher confidence, too.
At a recent 6th-grade professional learning community meeting, teachers tackled one of the most misunderstood standards: Mean Absolute Deviation (MAD). Before it appeared in standards, many educators had not even heard of it. And early resources? Practically nonexistent!
So the PLC team used page 195 from our book to understand MAD together. In those quiet moments of collaboration – around a whiteboard, over a problem, inside a question – something shifted. Confidence took root. You could feel the shift in the conversation as the process was illuminated and the math became visible—for them.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
And when they brought that same resource into their classrooms, it became an anchor chart — a living tool for students. The color coding and step-by-step clarity in the resource helped illuminate the process behind MAD and made it visible to every student, much in the same way that it had become visible for the teachers. That’s the power of making math visible. It builds what I like to call mathematical swagger—for teachers and students.
Everyone Means Everyone
Let’s stop pretending there’s such a thing as a “math person.” Confidence in math doesn’t come from talent—it comes from clarity. From ideas being made visible. From sense-making being prioritized. And it doesn’t happen alone. It happens in community, through collaboration and exploration, with people who remind us we’re not in this alone. Because when you see the math, they will see it too.
Make Math Visible for Someone This Week
Take one small step this week to make math more visible—for a student, a colleague, or even yourself. Use a model. Ask a clarifying question. Share your own learning journey. Pull out that anchor chart one more time.
And if you’re ready to dig deeper, invite a colleague to co-plan with you or explore a page from Seeing the Math You Teach together. Learning is meant to be visible—and shared.
Math makes sense. And everyone will be good at math–because of you.
Chryste Berda - Chryste is energized by sharing her passion for learning with her colleagues as the district math coordinator and as a Regional VP for the Arizona Association of Teachers of Mathematics. She is intensely curious about students’ thinking and spends much of her time listening to students explain their ideas.
Hands-On Math, Professional Learning, Teaching and Learning
Math Makes Sense
By Katie Basham
I wish I had known that when I was a young, impressionable student. Maybe some kids knew it—maybe they even believed it—but I certainly didn’t. I was the kid counting on my fingers, skip-counting under my breath, and hoping no one would notice. I thought of these methods as “less-than” strategies and felt a pit in my stomach at the thought of others noticing me using them.
Everyone Can Be Good at Math
Also true. And I wish my teachers had known that, too. Maybe then I would’ve felt proud of my own strategies, instead of ashamed. Because here’s the truth: success doesn’t belong to a select few. It’s not limited. There’s enough of it for every single student to feel it, own it, and believe in it.
These two beliefs—math makes sense and everyone can be good at math—are the foundation of Seeing the Math You Teach. Success shouldn’t be reserved for an elite few. All students can succeed when they understand that math is really about visualizing and explaining abstract thinking. Because when students can see the math, they can understand it….and, when they understand it? It makes sense.
What a privilege we have as educators.
It’s no longer our job to bestow knowledge from on high. Our mission is to create learning environments where students use concrete materials, build visual representations, and gradually progress to symbolic understanding. This book is designed to empower you—to help you see and understand the math you teach, so you can help your students do the same.
Take rounding, for example.
I recently asked a respected educator how he would explain it. Here’s how the conversation went:
Me: “What does it mean to round a number?”
Him: “Five or more, raise the score. Four or less, let it rest.”
Me: “But what does that mean?”
Him: “It means if the number ends in a five…”
As you can see, we weren’t quite getting to the heart of it.
“Rounding” isn’t really something you do, like an action—it’s more about understanding where a number falls in relation to benchmarks or friendly numbers. In other words, rounding is determining which number a given value is closest to.
- What number is nineteen closer to?
- What number is 18.34 closer to?
- THAT’S what rounding is.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
The student who can show rounding on a number line isn’t just repeating a rhyme—they’re seeing the math. They’re understanding the concept. And that, at its core, is what this book is about.
We want you—and your students—to experience success.
Because: Math Makes Sense & Everyone Can Be Good at Math!
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.
Hands-On Math
Blog post #2 in the series, Lies We Tell Our Students
We all know what a run-on sentence is. We’ve known since our early school years that run-on sentences are not okay. As this Grammarly.com quote states, “Run-on sentences are strings of complete sentences without sufficient punctuation to make them readable.” Today I’d like to coin a new term, run-on equations. And here’s the description for this new term: Run-on equations are strings of multiple unequal expressions connected with equal signs, making the entire equation untrue.
The Story
You’ve seen it…I know you have. Your students are explaining their thinking, and it goes something like this: “I added 3+4 to get 7. Then I multiplied by 2 to get 14. Then I multiplied by 5 to get 70. And then I subtracted 12….” And as they say these words, here’s what they write:
3 + 4 = 7 x 2 = 14 x 5 = 70 – 12
I’ve been in countless classrooms where students and teachers, alike, record their thoughts like this, using the equal sign as an “action” symbol rather than a “relationship” symbol. Have you witnessed this?
The (Inadvertent) Lie
We examined a similar problem last week. They’re thinking of the equal sign as a symbol of action rather than relationship. They’re using the equal sign to show the action of finding an answer and then continuing their thinking from there. The problem is that in doing so, they create a series of unequal expressions, making the entire equation false.
The Truth
Now here’s the catch. Writing equations with multiple equal signs is not problematic, as long as all expressions are equal. Strings of multiple expressions can be correctly connected with equal signs if the expressions are truly representing equal quantities. Here’s a case in point from a first-grade classroom where students have written pairs of numbers that total nine on post-it notes and then put equal signs between each:
1+8 = 2+7 = 3+6 = 4+5 = 5+4 = 6+3 = 7+2 = 8+1
As you can see, this looks a lot like the run-on equation as recorded in the story, above. However, there is a distinct difference. This string of equations is actually true! The equal sign appears between two equal expressions every single time, and every single expression is equal to every other expression. This equation uses the equal sign to represent equal relationships rather than actions, and it is, therefore, not a run-on equation.
So, how would one go about correcting the “equation” in the story, above, to truthfully describe the situation? S/he would need to separate each equation, rewriting the answer from the previous equation as the “start number” for the new equation.
3 + 4 = 7
7 x 2 = 14
14 x 5 = 70
70 – 12
In Summary
To sum up this week’s Lie We Tell Our Students:
As always, we love hearing from you! Do you think your students fully understand the meaning of the equal sign? Have you caught your students (or yourself) using run-on equations? Please share your stories in the comments box, below.
Next Steps for Teachers: First of all, model good practice for recording equations to represent thinking. Never, ever, ever, ever, ever, ever let yourself take a short-cut that leads to run-on equations. Secondly, explicitly teach the meaning of the equal sign as a symbol that represents an equal relationship between two expression. You might use the equality flashcard idea from last week’s blog post. Or, if your students are writing equations with multiple expressions and multiple equal signs, have them get out their highlighters. They can highlight each expression a different color and then determine if every single expression in the equation is equal to every other expression. If not, they probably fell into the trap (again) of using the equal sign as an action symbol rather than as a relationship symbol.
Next Steps for Leaders: During grade-level meetings, PLC meetings, or staff meetings, provide time for teachers to discuss the meaning of the equal sign. You may want to begin by replicating the story told above, recording with run-on equations, and continuing until someone in the room stops you. If no one does, then at some point, you’ll need to interrupt yourself and check in to see what they’re thinking. I cannot emphasize this enough…do not let this practice infiltrate your classrooms. Improper understanding of the equal sign will impact student learning for years to come, especially when the students get into algebra.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
