In my last blog, you heard both Valerie Faulkner’s and my perspectives on how Structures of Equality (SoE) came to be. But SoE isn’t just about a way to structure instruction around word problems, it’s about changing how students understand mathematical relationships.

Since Valerie has been such a huge influence on my work, I asked our colleague, Stacy Eleczko, to catch up with her and ask a few questions about SoE and how it has evolved. Their conversation was so rich we decided to share it across multiple articles.

Today’s focus? The mathematics behind SoE—specifically, equality, decomposing numbers, and unit—three ideas that are critical for students to become strong problem solvers.

Why Decomposing Numbers Isn’t Intuitive for Students

One of the biggest misconceptions in math instruction is that students naturally understand how to break down numbers into parts. But as Valerie pointed out, that’s far from true:

“If you show a student 12 in the longer bar of a Compare model with an 8 in the other, they often have no idea what goes in the missing space. To an adult, the answer seems obvious, but for many kids, it’s not. They don’t yet see that 12 can be decomposed into 8 and another part.”

Many students rely on symbols and procedures rather than developing a deep understanding of how numbers work. Without explicit instruction in decomposing numbers, they struggle to connect the relationships. This makes (what we perceive as) simple problems feel confusing.

SoE changes that. By focusing on how numbers break apart and come back together, students build a flexible understanding of numbers that are critical to develop higher math thinking.

Teaching Equality as More Than a Symbol

Another major shift in SoE is how it teaches equality. In many classrooms, the equals sign is often interpreted as “find the answer.” But in reality, it represents a relationship, a statement that two expressions are equal in value.

This distinction might seem small, but as Valerie explained, it has a huge impact on how students think about math:

“Most models are plug-and-play. You put the numbers where they go, and the structure does the work for you. But with SoE, students have to take action. They have to think about where equality exists, what’s missing, and how to balance the equation. That’s a completely different experience.”

With SoE, students actively search for equality in every problem. Whether it’s part-whole relationships or comparisons, student’s brains are trained to identify where two things are equal, shifting them from passive answer-seekers to mathematicians analyzing relationships.

This explicit focus on equality also prevents the learning gaps that appear later in algebra. Many students struggle with equations in middle and high school because they never developed a solid understanding of what equality actually means. As Valerie pointed out, Liping Ma’s research highlights how in the U.S., students often don’t explore the difference between expressions and equations early on, making advanced math concepts feel inaccessible later. SoE changes that by making equality a core idea from the beginning.

Why “More” and “Less” Are Harder Than We Think

A concept that seems simple—understanding more and less—is actually one of the hardest things to teach in elementary math. Teachers often assume that students will intuitively grasp the difference, but in reality, many children struggle to identify the missing amount in a comparison.

“Ask most teachers in first grade. Teaching more and less is extremely hard. But when students use SoE structures, they can actually see what’s more and what’s less.”

SoE makes this concept more accessible by turning more and less into a missing part problem rather than a vague comparison. Instead of students trying to reason about “more” and “less” in an abstract way, SoE provides a visual structure where the missing part is clear.

For example, if a student sees that one quantity is 8 and the total is 12, they no longer have to guess what “less” means. They see that the missing amount must be 4. By making comparison a concrete relationship rather than an abstract idea, SoE helps students develop a much stronger number sense.

Understanding Units: A Small Detail with a Big Impact

Another critical but often overlooked idea in math instruction is the importance of units. In traditional classrooms, students are often taught to tack on the unit at the end of a problem. But SoE builds units into the problem-solving process itself, preventing common misconceptions.

“People just add five plus three, and then teachers will, as an afterthought, suggest you tack your unit on. But the way Julie has structured SoE, it’s a critical attention point for both the teacher and the student.”

When students pay attention to units early on, they build a deeper understanding of mathematical relationships. This prevents errors like treating different units as interchangeable (e.g., assuming “2G + 3R = 5T” when G stands for green, R for red, and T for total). By reinforcing that only like units can be added, SoE lays the groundwork for algebraic thinking.

This small shift in focus makes a big difference. Instead of seeing numbers in isolation, students begin to recognize the structure of mathematical expressions, making them better prepared for higher-level math.

A Tool for All Learners

One of the most powerful aspects of SoE is that it meets every student where they are. Instead of separating students into those who “get it” and those who don’t, it provides a structure that all learners can engage with, regardless of their starting point.

“It’s not about the numbers. It’s about the mathematical thinking. And when students can confidently explain and represent their thinking, that’s what’s really fun for them.”

SoE builds confidence and ensures every student has a structured way to approach problems. This supports all learners in deepening their understanding.

Up Next: The Role of Reading Comprehension in Math

If equality is one foundation of SoE, reading comprehension is the other. Together, they make math accessible to all students. In the next blog, we’ll explore how visualization and language work together to make math more accessible for every learner, and why SoE is designed as a reading comprehension tool just as much as a mathematical one.