Math instruction can feel frustrating, not because students aren’t capable, but because traditional approaches don’t always help them see the deeper connections. Structures of Equality (SoE) emerged from the need for a better way to help students make sense of word problems. 

It started with a realization: the way students were learning to solve word problems wasn’t working. But it didn’t stop there. Through collaboration and deep mathematical thinking, SoE became a framework that helps students see mathematical relationships more clearly.

To understand how SoE came to be, you have to hear from two people: Julie, whose classroom experiences led to the first SoE models, and Valerie Faulkner, whose insights helped refine and strengthen the framework.

Julie’s Story: A Teacher’s Search for a Better Way

A Personal Struggle with Math

Math never came easily to me. In elementary school, I struggled to learn my basic facts and often felt left behind. Timed tests and memorization-based instruction created anxiety that stuck with me into high school. Despite this, I was determined to follow the college-track diploma, which meant pushing through required math courses. I passed, but I still believed I wasn’t a “math person.”

I chose elementary education because I loved helping others. And, if I’m honest, because it required little math. I excelled in my coursework and teaching career but continued to avoid math whenever I could.

The Moment Everything Changed

As my teaching career progressed, I took on different roles, including a position as a director for a private learning center. It was there that I became acutely aware of just how many students struggled with math. There were huge gaps between what students were expected to understand and what they actually knew. I was frustrated. How had we let kids fall this far behind?

Determined to make a difference, I returned to public education, teaching kindergarten and then 1st grade. That’s when I met Valerie Faulkner.

Attending her professional development changed my entire perspective. I realized my struggles weren’t my fault. The way I had been taught was broken. Valerie’s training helped me see math differently, and I noticed that many teachers carried the same math anxiety I had.

I began soaking up as much knowledge as I could about high-quality math instruction. When I had the opportunity to become a math coach, I jumped at it.

Identifying the Problem: Word Problems Weren’t Working

Working with teachers across grade levels, I saw a common struggle: students didn’t understand the relationships in word problems. Instead of analyzing the story, they pulled out numbers and picked an operation based on what they had practiced most recently.

I knew “key words” weren’t the answer, but I didn’t have a strong alternative. I experimented with different strategies, but none consistently worked.

Then, I attended Foundations of Math, where we explored Singapore math, CGI problem types, and Liping Ma’s Knowing and Teaching Elementary Mathematics. That’s when everything clicked. With Valerie’s support, I blended these ideas, creating the first iteration of SoE.

What Has Stayed the Same?

From the beginning, I knew SoE needed to be simple, easy to draw, and applicable to all problem types in the K–5 curriculum. There were a few essential components:

• A representation of equality
• Values and labels to clarify relationships
• A focus on comprehension, not just computation

These elements remain at the heart of SoE today.

What Has Evolved?

Over time, my understanding deepened. I realized that because so much of the struggle with word problems stemmed from reading comprehension, SoE needed to integrate reading strategies. I started incorporating story element questions, vocabulary discussions, and cooperative learning structures like turn-and-talk.

I also made changes to specific models. The Compare model, for instance, originally included an oval or rectangle to show the difference—something many textbooks used. But I removed it and added the line of equality instead, allowing the model to more accurately reflect the relationships in the story.

Valerie’s Reflections: Recognizing a Better Model

When I first met Julie, I saw a teacher who was deeply engaged in understanding how to teach math more effectively. She wasn’t just attending professional development. She was absorbing ideas, questioning them, and applying them in her classroom in ways that went beyond what was being presented.

At the time, we were using a simplified version of CGI models in the Foundations of Math training. It was a good system. It improved upon previous models and gave teachers a clearer structure. But Julie took it a step further.

She saw something that others hadn’t fully realized: with the right approach, all word problems could be represented using just three structures. That was a huge insight. It meant that instead of students having to memorize multiple ways of solving different types of problems, they could develop a deep understanding of mathematical relationships through a consistent, visual approach.

When I visited Julie’s classroom, I saw firsthand how she was experimenting with these ideas. She was testing her models with students, refining them based on what worked,

and adjusting based on student understanding. She wasn’t just teaching math. She was uncovering something fundamentally better.

The more I observed, the more I realized that Julie’s model was stronger than what we had in Foundations of Math. It wasn’t just simplifying word problem types. It was intentionally structured to support student learning in a way that made problem-solving accessible to all students.

At the same time, Julie recognized that there were gaps in her own mathematical understanding, and she wasn’t afraid to ask questions. She used me as a resource, and together, we worked through the finer details—ensuring that the models were mathematically sound and wouldn’t create misconceptions down the road.

That’s what made working with Julie so exciting. She had the insight to create something new and the humility to refine it. Over time, our conversations and collaboration strengthened SoE into what it is today—a tool that doesn’t just help students solve problems but transforms the way they see mathematical relationships.

What’s Next? A Closer Look at the Math

Julie knew there had to be a better way to teach word problems. Valerie helped refine that idea, ensuring it was mathematically sound and built to last.

But what exactly makes SoE so effective? What are the key mathematical ideas that set it apart?

In the next blog, we’ll continue the conversation with Valerie and break down the core concepts behind SoE—equality, decomposing numbers, and units—and explore why these ideas are essential for building strong problem solvers.