Teaching Math for Understanding

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Why it works

Research in mathematics education shows that durable mathematical competence requires both conceptual understanding (knowing why) and procedural fluency (knowing how), developed together, not procedures alone. Students taught only procedures struggle to reason, transfer, or recover when they forget a step, while those who understand the concepts can reconstruct and apply. Practices like using representations, discussing reasoning, and productive struggle build understanding.

The research: NCTM; research on conceptual understanding and procedural fluency.

The run-of-show

Choose your slot. The agenda, timings, and length update to match.

The core activity: teach one concept for understanding

Teachers leave with a plan to build understanding and fluency for one math concept.

Pick a math concept students often learn only procedurally.
Plan to build the understanding of why (representations, connections, reasoning).
Have students explain and discuss their thinking.
Develop fluency alongside, not instead of, understanding.

Understanding and fluency togetherProcedures without understanding leave students stuck when problems change. Build the why, have students reason, and develop fluency alongside it.

Facilitator notes

Teach understanding and fluency together.
Use representations, and have students explain reasoning.
Use productive struggle on worthwhile problems.

Procedures alone do not lastMemorized steps without understanding fail when problems change or steps are forgotten. Teach why the math works alongside how.

Adapt it

Elementary: concrete representations, reasoning, number sense.
Secondary: concepts behind procedures, discussion, modeling.
All settings: understanding plus fluency.

Participant handout

One page for every teacher. Print it, or save it as a PDF.

K12 Academics · Professional Learning

Teaching Math for Understanding: why and how

Both together: conceptual understanding through representations and reasoning, developed alongside fluency.

The math concept often learned only procedurally:
How I build the understanding of why:
How students explain and discuss their reasoning:
How I develop fluency alongside:
Where understanding is the real gap:

Make it stick

Why and how:

Teach understanding and fluency.
Have students reason.
Bring a concept taught for understanding to the next PLC.

Common questions

Isn't getting the right answer what matters in math?

Getting it and understanding why. Students who only memorize procedures cannot reason, transfer, or recover from a forgotten step. Build understanding and fluency together.

Doesn't conceptual teaching slow down fluency?

They reinforce each other. Understanding makes procedures meaningful and memorable; fluency frees attention for reasoning. Develop both, not one at the expense of the other.

How do I teach for understanding?

Use multiple representations, have students explain and discuss their reasoning, connect procedures to why they work, and use productive struggle on worthwhile problems.