What Does It Mean to Be Good at Math?

Every math classroom has a range of learners. Some students can compute accurately and efficiently. Others reason through complex problems and generate multiple solutions. Some persevere through tasks, trying one strategy after another. But which students are the best mathematicians?

According to Adding It Up: Helping Children Learn Mathematics, the strongest mathematicians are well-rounded, able to demonstrate proficiency across five interwoven aspects of math learning called the Strands of Mathematical Proficiency.

Our students bring a wide range of skills to the classroom. That’s why we need to create spaces where every learner is seen as a capable mathematician. When we lead with the belief that math belongs to everyone, we build a classroom rooted in equity. Here are some ways I use the Five Strands of Mathematical Proficiency to ensure that no one is left on the sidelines.

1. Conceptual Understanding

As teachers, we focus our instruction on building conceptual understanding. We help students use vocabulary and symbols correctly. We make sure they are able to explain why procedures work. Curriculums are designed with this in mind.

One strategy I use in my classroom to support conceptual understanding is worked examples, models that show what thinking looks like, not just the answer. These examples go beyond numbers on a page. They include visual models, written explanations, and symbolic representations so that students can see how different forms of thinking connect. In practice, a worked example for a problem like 6 x 4 could include a visual model of an array with 6 rows and 4 columns, a written explanation of “six groups of four,” and a symbolic representation such as repeated addition 4 + 4 + 4 + 4 + 4 + 4.

2. Procedural Fluency

Procedural fluency is more than getting the right answer. Adding It Up defines it as computing “flexibly, accurately, efficiently, and appropriately.” Fluency doesn’t come from rote memorization and speed drills, but from balancing deep understanding with opportunities to practice efficient strategies.

Two strategies that support fluency in my classroom are math games and mistake analysis. Games offer low-pressure, engaging practice, while mistake analysis gives students the chance to reflect on why something works, not just how, deepening their conceptual understanding. For example, my fourth-grade students play a card game where they earn points by adding fractions with unlike denominators, using manipulatives fraction strips to help them visualize their thinking. Later, we analyze the error ½ + ⅓ = ⅖, using the same tools, and explain how to fix it.

3. Strategic Competence

If we think of the solution to a problem as “the what,” strategic competence is “the when” and “the why.” It involves posing, representing, and solving problems. Students decide how to approach tasks, what tools to use, and how to adjust when something doesn’t work.

One way I build strategic competence is with open-ended problems with multiple solutions. My first- and second-grade students loved YouCubed’s Foot Parade problem. After listing animals and the number of feet each has, students were asked to create a parade with a total of 12 feet. It’s open-ended and playful, and it invites students to make choices. They spent a surprising amount of time debating whether a snail should be considered to have one foot or no feet at all. As they worked, they weren’t just adding. They were reasoning, representing, and revising their ideas in rich discussion. This kind of problem helps students see themselves as capable problem-solvers, which is exactly what strategic competence is all about.

4. Adaptive Reasoning

Adaptive reasoning is the ability to think logically about concepts and their relationships. It’s important to create opportunities for students to reflect on how their ideas connect to each other, adapt their thinking when faced with new information, and justify their reasoning when they disagree.

For example, when I ask students to turn and talk, I will provide sentence stems such as “Our answers were the same, but our steps were different because…” or “This connects to what we learned about…” These prompts help students slow down, organize their thinking, and listen actively to their peers. During one particular exchange, a student declared, “I thought the answer was 4, but now my thinking has changed because I realize we were taking away, not adding.” That moment of reflection is adaptive reasoning in action. It’s not just about getting it right. It’s about making sense of why something works and being open to change.

5. Productive Disposition

Adding It Up describes productive disposition as a student’s ability to see math “as both useful and worthwhile” and “to see oneself as an effective learner and doer of mathematics.” This can be one of the most challenging strands to nurture because it requires a mindset shift, which starts with us.

Many adults carry negative experiences from their own math education that leave them feeling like they aren’t “a math person,” or they are just not smart enough. But productive disposition asks us to model the opposite: Math is for everyone, and that effort matters.

In my classroom I talk openly about times I have found a math concept confusing, by saying things such as, “This part of the problem really stumped me at first.” When a lesson doesn’t go as planned, I reflect out loud with my students about what I will try differently next time. I also celebrate their perseverance—when someone sticks with a tough problem, revises their thinking, or abandons an inefficient strategy for something different, that’s a win. Over time, we build a classroom culture where math feels safe, effort is noticed, and growth is expected.

If we truly believe that being good at math is about more than quick answers, we must make space for all kinds of thinkers. That means noticing and celebrating the diverse ways students approach problems.

When we only highlight one kind of math thinker, we risk sending the wrong message about who belongs. But every student deserves to see themselves as capable and creative in math. Building a classroom where all learners are valued isn’t just good teaching—it’s equity in action.