Seeing the Math You Teach – Part 1

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.

Seeing the Math You Teach Pages

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.

Seeing the Math You Teach Pages

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! 

Seeing the Math You Teach, Grades K-6

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.

 

Seeing the Math You Teach, Grades K-6

Seeing the Math You Teach – Part 2

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:

  1. Math makes sense.
  2. 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. 

Seeing the Math You Teach Pages

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.