By Jeffrey Mister
If math class feels like a never-ending list of “just do this” instructions, you’re not alone. Too often, math instruction has been built around steps and shortcuts—carry the one, move the decimal, cross-multiply and hope for the best. But there’s a big problem with that approach: students can’t transfer what they’ve learned when the problem looks even slightly different.
That’s where conceptual understanding comes in. It’s one of the three critical aspects of mathematical rigor outlined in the Common Core State Standards, alongside procedural fluency and application. Together, they create the trifecta of mathematical strength. But conceptual understanding is the foundation. Without it, even the best memorizer eventually hits a wall.
So, What Is Conceptual Understanding?
According to the National Research Council’s landmark report Adding It Up: Helping Children Learn Mathematics, conceptual understanding means grasping mathematical ideas and relationships at a deep level. Essentially, it's knowing why procedures work, not just how to perform them.
It’s the difference between knowing that ½ × ⅔ = ⅓ and understanding why multiplying fractions results in a smaller number. Or recognizing that the distributive property isn’t a trick, it’s a mathematical truth that connects arithmetic, algebra, and beyond.
In short, conceptual understanding:
- Introduces mathematical concepts before memorizing procedures
- Emphasizes sense-making over answer-getting
- Helps students organize knowledge into a meaningful framework they can build on
Why It Matters
Students with strong conceptual understanding can adapt. They can flexibly choose strategies, explain their reasoning, and tackle novel problems with confidence. Those without it? They’re often left guessing or worse, they disengage.
The National Council of Teachers of Mathematics (NCTM) puts it this way: procedural fluency should grow from conceptual understanding, not replace it. When students see how ideas connect, they remember more, reason better, and apply math more effectively.
Research backs this up.
- Students who learn procedures through conceptual contexts retain them longer and make fewer errors later (National Research Council, 2001).
- Instruction that builds fluency from conceptual understanding leads to deeper problem-solving skills and better long-term achievement (Minnesota Department of Education, 2017).
- In a 2019 study published in the International Journal of Education and Practice, researchers found that conceptual understanding was the strongest predictor of success on application and problem-solving tasks.
So, if we want students who can think—not just compute—we have to make conceptual understanding the starting point.
Teaching for Conceptual Understanding (Without Losing Your Sanity)
You don’t have to throw out your curriculum or start from scratch. Instead, think of conceptual understanding as a lens for how you teach what’s already there.
Here are some practical ways to bring it to life:
- Start with exploration, not explanation.
Give students time to notice, wonder, and make sense of patterns before introducing formal rules. For example, before teaching how to multiply fractions, have students explore with area models or number lines to see what’s happening. - Connect representations.
Move fluidly between visuals (arrays, graphs, diagrams) and symbols. When students can translate among representations, they build mental links that support understanding. - Ask “why” as often as you ask “what.”
When students justify their reasoning, they’re not only deepening understanding. They’re developing mathematical communication and reasoning skills. - Celebrate mistakes as insight.
Conceptual learning thrives on curiosity and revision. When students struggle, use those moments to clarify misconceptions and strengthen understanding.
How to Know If Students Are Truly “Getting It”
Here’s the catch: conceptual understanding isn’t always visible in traditional testing. Multiple-choice questions can’t tell you whether a student really understands why their answer works or if they just guessed correctly.
That’s where data-driven insight becomes essential. Teachers need assessments that can pinpoint conceptual gaps and guide next steps in instruction.
Enter Compass Math Assessment: Your Partner in Building Conceptual Understanding
At ANet, we’ve seen firsthand that when schools have the right data, they can close learning gaps faster and more effectively. The Compass Math Assessment is built precisely for this purpose: to help educators see the why behind student performance, not just the what.
Here’s how Compass supports conceptual learning:
- Uncovers hidden misunderstandings – Compass items are intentionally designed to reveal whether students grasp the underlying math concepts, not just the procedures.
- Balances rigor – It assesses conceptual understanding, procedural fluency, and application together, giving a holistic view of student learning.
- Guides next steps – Reports are built to help teachers make instructional moves that strengthen conceptual understanding without losing momentum on grade-level content.
- Promotes equity – By highlighting where instruction may have focused too heavily on one dimension of rigor, Compass ensures all students access deep, meaningful math learning.
Think of Compass as a compass in the truest sense: it helps teachers and leaders navigate where students are, where they need to go, and how to get there—without wandering in the dark.
The Big Picture: Sense-Making Over Memorization
Conceptual understanding is what transforms math from a list of rules into a language for reasoning. When students truly understand math, they don’t panic when the problem looks new, they get curious.
But this kind of learning doesn’t happen by accident. It takes intentional teaching, the right support, and accurate insight. With tools like Compass Math Assessment, schools can finally move from guessing to knowing, ensuring every student not only does math but also understands it.
References
- National Research Council. (2001). Adding It Up: Helping Children Learn Mathematics. National Academies Press.
- National Council of Teachers of Mathematics. (2014). Procedural Fluency in Mathematics Position Statement.
- Minnesota Department of Education. (2017). Build Procedural Fluency from Conceptual Understanding.
- International Journal of Education and Practice. (2019). Conceptual Understanding, Procedural Knowledge, and Problem Solving.