Beyond the Answer: Two Simple Strategies for Deeper Math Understanding

Getting the right answer is important. But does getting the right answer always mean a student understands the mathematics behind it?

Not necessarily.

Students can memorize formulas, follow procedures, and solve problems correctly without fully understanding why those procedures work.

Developing strong mathematical thinkers requires us to move beyond simply teaching students how to solve problems. Two powerful instructional practices can help make that happen: intentionally using academic language and providing frequent opportunities for students to justify their thinking.

1. Make Academic Language Part of Everyday Math Instruction

Academic language is much more than a vocabulary list at the beginning of a lesson.

Mathematical terms like sum, difference, equivalent, factor, quotient, and product help students communicate mathematical ideas with precision. But simply memorizing their definitions doesn't mean students understand the concepts they represent.

Students need to hear, see, use, and apply mathematical language throughout the learning process.

Consider a lesson on equivalent fractions. Rather than simply teaching students that equivalent means equal in value, teachers can consistently use the term while modeling with fraction strips, comparing visual representations, and discussing why two fractions represent the same quantity.

As students begin working independently, encourage them to use that same language to explain their thinking.

Ways to build academic language into instruction:

• Model it consistently. Use precise mathematical language when introducing concepts, thinking aloud, and demonstrating strategies.

• Connect words to meaning. Pair vocabulary with visual models, manipulatives, examples, and real mathematical situations.

• Encourage students to use it. Provide opportunities for students to incorporate mathematical language into their discussions, explanations, and written responses.

• Revisit it often. Continue using previously introduced terms as students encounter new concepts and make connections across mathematical ideas.

The goal isn't for students to simply know mathematical words. It's for them to understand the ideas those words represent and use them naturally when communicating their thinking.

2. Ask Students to Justify Their Thinking

We've all seen the instruction at the bottom of a math problem: "Justify your answer."

But mathematical justification should be more than an occasional written response. It should be a regular part of how students learn, practice, and discuss mathematics.

When students explain their strategies, defend their reasoning, or compare approaches with classmates, they are doing more than demonstrating what they know. They are actively making sense of mathematical relationships.

Consider a student who solves 8 × 6 = 48.

Rather than moving immediately to the next problem, ask:

• How did you figure that out?

• Why does your strategy work?

• Can you represent your thinking another way?

• Would a different strategy give you the same result? Why?

• Do you agree or disagree with your classmate's approach? Explain.

One student might use repeated addition. Another might break apart 8 × 6 into (5 × 6) + (3 × 6). A third might use an array. Discussing these approaches helps students recognize the mathematical relationships connecting different strategies.

Just as importantly, these conversations provide teachers with valuable insight into student understanding, including misconceptions that might otherwise be hidden behind a correct answer. And this is where academic language comes back into play.

Encourage students to use precise mathematical language as they explain and justify their reasoning. Over time, students can move from informal descriptions toward increasingly clear and mathematically accurate explanations.

Building Understanding, Not Just Correct Answers

These two strategies work best together.

Academic language gives students the tools to communicate mathematical ideas. Justification gives them meaningful opportunities to use those tools while developing and refining their understanding.

Neither requires an entirely new lesson or instructional program. Both can be intentionally incorporated into teacher modeling, guided practice, partner discussions, independent work, and everyday classroom conversations.

These strategies also align with the research-informed teaching practices outlined in NCTM’s Principles to Actions, which emphasizes meaningful mathematical discourse, purposeful questioning, and building procedural fluency from conceptual understanding.

Ultimately, we want students who can do more than follow the steps.

We want students who understand why the steps work, can communicate their reasoning, and can apply that understanding to new mathematical situations.

One Idea to Try This Week

During your next math lesson, choose one important mathematical term and use it intentionally throughout instruction.

Then, when students solve a problem, ask them to explain why their strategy works using that mathematical language.

Sometimes a simple question can reveal far more about student understanding than an entire page of correct answers.