November 11, 2020
The Power of Conceptual Understanding

Posted in: Aha! Blog > Eureka Math Blog > Implementation Support > The Power of Conceptual Understanding

A lot of my success as a math student was due to rote memorization. I remember learning the FOIL method to multiply binomials: first, outer, inner, last. Although I was able to remember the mnemonic, I didn’t understand why it worked. When we went beyond binomials, FOIL failed me, and I had to memorize a new procedure. As a teacher, I discovered the powerful connection between conceptual understanding and my students’ learning. When students can access concepts from multiple perspectives and see math as a set of connected topics, they’re better able to retain, extend, and apply learning.

By using Eureka Math®, my students gradually achieved conceptual understanding. Let’s consider a few things you can do to help students develop that conceptual understanding as they use Eureka Math.

Use math talk to promote conceptual understanding.

When students talk about their reasoning and math choices, they strengthen their conceptual understanding. Early in my teaching career, I was apprehensive about student discourse and didn’t use it as a key component in my classroom. I wasn’t sure how to get students to talk about math or what they should be talking about. Now student discourse is a key component of every lesson. I always plan for students to explain their processes, results, and mathematical approaches to a problem. Let’s explore some strategies to promote math talk.

Ask questions to build understanding.
  • How is this problem or lesson like the previous problem or lesson
  • How is it different?
  • How do you know your answer is correct?
  • What would happen if we changed _____?
  • What mistakes did you make? How did you correct them?
Have students turn and talk in pairs or small groups.
  • Talk to your partner about what you notice in the problem. Decide on a strategy to solve the problem.
  • How is your work like your partner’s work? How is it different?
  • Talk to your partner about how you solved the problem.
Provide sentence stems to promote peer-to-peer discourse.
  • I agree with _____ because …
  • I disagree with _____ because
  • I heard _____ say _____. I want to add …
  • I was confused when …
  • I wonder …
Debrief lessons.
  • What was new about our work today?
  • What do you know now that you didn’t know yesterday?
  • From which mistake did you learn the most?
  • What do you think is most important from our work today?


Promoting math talk helps your students strengthen their understanding and gives you valuable information to adjust your teaching based on what students know.

Deepen conceptual understanding with multiple representations.

Teach concepts by using multiple representations. The connections students make between representations deepen their conceptual understanding, make them more flexible problem solvers, and reinforce mathematics as a set of connected topics.

Imagine that students are working on equivalent ratios. Below you can see four different representations of the same proportional relationship, modeled by using a tape diagram, double number line, ratio table, and coordinate plane. Each representation contributes to a deep understanding of ratios.

Help students make connections between representations by asking questions. In this case, consider asking the following questions:

  • How are the quantities for sugar and flour shown in each representation?
  • Where do you see the ratio in each representation?
  • How can you use each representation to find equivalent ratios?

This image shows four representations of a comparison of sugar to flour using a tape diagram, double number line, ratio table, and coordinate plane.   The tape diagram representation shows a pair of tape diagrams. The tape diagram representing sugar is 6 units long with annotations showing 2-unit sections. The tape diagram representing flour is 9 units long with annotations showing 3-unit sections. The ratios, two to three, four to six, and six to nine are written next to the diagram.  The double number line diagram shows a horizontal line that represents sugar with intervals of 2 starting at 0 and ending at 6. Below is a horizontal line that represents flour with intervals scale of 3 starting at 0 and ending at 9. Intervals align to show the ratios of two to three, four to six, and six to nine.  The ratio table image shows a table with two columns and three rows of values. Column one has the header, Sugar, with values of two, four, and six below in that order. Column two has the header, Flour, with values of three, six, and nine below in that order.  The coordinate plane image shows a coordinate plane with the x-axis labeled sugar and the y-axis labeled flour. The x-axis has an interval scale of two, starting at zero and ending at eight. The y-axis has an interval scale of 3, starting at zero and ending at 12. The line on the graph passes through the origin and the points, two three, four six, and six nine.

As students work on problems, ask them why they chose the representations they use and to think of another way they can represent the problem. For word problems, ask students how each part of the problem is modeled in their representation.

When you teach with multiple representations, students get different access points into a lesson. It helps them scaffold to new ideas such as moving from ratios to the concept of slope. More generally, the practice provides opportunities for students to make sense of problems and to use their judgment when they select appropriate representations and strategies to do math.

Use patterns and relationships to build concepts.

Provide opportunities for students to identify, define, and create patterns and relationships. Students build conceptual understanding when they look for patterns, make use of structure, and make generalizations.

This image shows an eight by eight multiplication table. The first row shows facts from one times one to eight times one and the resulting product. The second row shows facts from one times two to eight times two and the resulting products. Subsequent rows show facts and products in the same pattern with the second factor increasing by one in each row.  All table cells with an even product are shaded.

So what does that look like? Consider this idea from Grade 3. Students identify patterns in multiplication and division facts by using the multiplication table. In the activity, students shade the even numbers and then look for patterns. Pose questions such as the following to help students build conceptual understanding about patterns and relationships:

  • What do you notice in each row?
  • What do you notice in each column?
  • What do you notice in each diagonal?
  • Why do you think those patterns happen?

When they look for patterns and think about why the patterns occur and what they mean, students build deeper conceptual understanding.

Throughout Eureka Math, you will find many opportunities to engage students with conceptual development. Math talk, representations, and patterns are just a few great ways to help your students build a deep understanding of mathematics.


Topics: Implementation Support