Conceptual understanding: what is it, and how do we teach it?

An ideal picture of conceptual understanding in math class:

Students exploring the patterns they see. Students making sense and reasoning about what they know. Students predicting and solving based on ideas that they can explain. Students proving answers with models. Students teaching each other. It is a beautiful thing.

**How do we teach conceptual understanding? **

One thing I have found is that I need to stop *teaching* (in the traditional sense). Teaching often translates to telling, which will not result in the picture above.

**Here are four ways to build students’ conceptual understanding in your math class:**

**Models** allow students to see why formulas work and how to solve problems.

Students struggling with basic operations? Use physical models to build reasoning and number sense.

Students struggling with fractions? Use fraction tiles to model comparisons and operations.

Students struggling with area of triangles, circles, parallelograms or composite figures? Cut paper models to prove the formulas.

Students struggling with volume? Use cubes to build prisms and fill containers.

Students struggling with polynomials? Use algebra tiles to model multiplication and factoring.

**Resources:**

- Free: Area Exploration
- Decimal Multiplication
- Fraction Addition and Subtraction with Common Denominators
- Rectangular Prism Volume with Linking Cubes

## Instead of giving students a rule or formula to use, show examples and let students figure out the rule. Not only will this build conceptual understanding, but it will also boost students’ confidence as mathematicians.

Students struggling with adding and subtracting integers? Talk about examples using temperature or depth before talking about any rules to follow.

Students struggling to remember laws of exponents? Work out examples, and ask students to develop their own rules based on solutions.

Students struggling to remember angle relationship rules? Have students measure examples and tell you what they notice.

Students struggling with transformations on the coordinate plane? Have students transform examples and find out the rules for coordinates.

Students struggling with graphing quadratics in vertex form? Show students examples and ask them to find shortcuts on their own.

Students struggling with parent function transformations? Graph changes to parent functions and look for patterns.

**Resources:**

- Angle Relationships
- Transformations Exploration
- Parent Function Transformations
- Graphing Quadratics in Vertex Form
- Arc and Angle Relationships

## This simple question has a powerful impact. If students are forced to explain why, they are pushed into conceptual understanding.

Students may know to move the decimal when dividing, but do they know why?

Students may know to cross multiply to solve a proportion, but do they know why?

Students may know a zero exponent results in 1, but do they know why?

Students may know the formula for circumference is C=πd, but do they know why?

## Experiments are not just for science class. Experiments allow students to explore, think, discover, predict, and develop conceptual understanding in math.

Want to design your own experiment? The scientific method is a great guide to get started:

- Ask students a question.
- Students make a hypothesis.
- Students test their hypothesis and collect data.
- Students analyze results and draw conclusions.

Experiments are engaging and lead to deep, lasting understanding.

Resources:

## Conceptual understanding is true understanding. For many of us, it is not how math class was presented when we were in school. Therefore, we have to think outside the box and even push our own understanding to promote it in our own classrooms.

Thanks for reading!

*-Rachael*