“Rigor” has become a buzzword in math education, and an often misunderstood one if we stop at the dictionary definition of rigor as simply “demanding” or “difficult.” Rigor in math education actually refers to a deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades.

At first glance, the idea of maintaining rigor for students performing below-grade level seems counterintuitive--but true rigor is not about setting needless obstacles in students’ paths. To promote rigor, math learning must focus with equal intensity on three aspects: conceptual understanding, procedural skills and fluency, and application.

## What Does Rigor Look Like in Math Learning?

The Common Core calls for each of these three aspects to be applied with equal intensity. Here is what that looks like:

*Conceptual Understanding* refers to building a functional understanding of key mathematical ideas (i.e. place value ). This goes beyond memorizing procedures or individual facts to a focus on sense-making. Teaching conceptual understanding enables students to understand a concept from multiple perspectives and look for patterns that can help them understand future problems.

*Procedural skills and fluency* deal with students’ speed and accuracy when performing mathematical calculations. Students need to perform core mathematical functions efficiently (mentally or with tools) to lay the groundwork for more complex skills. Fluency comes from practice, practice, practice--both inside and outside of the classroom.

*Application* calls students to draw on their mathematical knowledge in situations that require it--for example, knowing and performing the calculations needed to make a scaled drawing of furniture in a room. Students must have a solid foundation of both conceptual understanding and procedural skills to accurately and efficiently apply their knowledge.

## The Pitfall of Too Much Procedure

Over-emphasizing the procedural at the expense of the conceptual is a misstep math instructors sometimes make, especially when feeling pressured to address unfinished learning. While teachers might create their original lesson plans with all aspects of rigor in mind, the conceptual can take a backseat to the procedural in interventions or other supports, which creates a disconnect for students. For example, an approach to two-digit subtraction that focuses on the “borrowing” method can limit students’ ability to catch mistakes or apply their subtraction knowledge to other mathematical situations (not to mention, they might simply forget the procedure since it is less “sticky” than the concept!).

## A Case for Conceptual Understanding

Rebalancing the scales to focus equally on conceptual understanding can have many benefits--especially for accelerated learning. Conceptual understanding…

**Supports knowledge retention.**When students learn concepts, they organize their understanding into a schema of interconnected ideas. These hierarchical clusters of information (i.e. “properties of multiplication”) can be more easily accessed and unpacked than a set of isolated facts.**Saves instructional time.**While providing students with tips and tricks might feel like an efficiency in the moment, establishing strong conceptual understanding can actually cut down on the volume of information students need to learn. By training students to look for deeper similarities in seemingly unrelated mathematical situations, for example, they can access conceptual frameworks to tackle a wider range of problems than they would be able to grasp with just a procedural skill.**Fosters creativity and cultural relevance.**A focus on conceptual understanding can also be a move towards a more culturally relevant approach to math. While there might be one right answer to some problems, there might be multiple ways to arrive at that answer, and a focus on conceptual understanding can invite more creative problem-solving (for example, adding fractions by drawing a picture, using manipulatives, visualizing a number line, or even telling a story). Asking students to explain their approach and discussing the advantages and disadvantages of different approaches can build a culture of equity that allows students to lead their own learning.

Students need both conceptual and procedural math knowledge to solve problems, but when teachers find themselves crunched for time, it can be easier to sacrifice delving into the conceptual in favor of a quick trick or more practice--this is a missed opportunity to set students up with the critical thinking skills they will need to take on the next math challenge.

**How Does Yup Support ALL Aspects of Rigor?**

Yup’s instructional rubric explicitly seeks to ensure that student understanding is rooted in conceptual frameworks. When a student working with a Yup tutor has a misconception, the tutor concisely explains or visualizes the new concept. The tutor then connects the procedural steps to the concept as the student works through the problem, all the while checking for understanding so that students can apply their knowledge to future math problems.

Read more about how Yup is readying students for modern classroom rigor. Are you a teacher or administrator looking to build authentic rigor in math classrooms? Contact partnerships@yup.com to learn more about bringing Yup to your school or district.