“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 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

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 Fluency

Procedural fluency deals 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

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 fluency 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 math 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 provide opportunities for students to 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.

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 mathematical knowledge to solve problems. But when math teachers find themselves crunched for time, it can be easier to sacrifice delving into the concept 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.

## Increasing Rigor in Math Classroom

Transitioning to a rigor-heavy curriculum can be challenging for districts and educators with strong backgrounds in traditional math teaching practices that emphasize rules, formulas, and memorization over deep conceptual understanding.

Here are five simple ways to increase math rigor in any classroom.

### 1. Destigmatize Mistakes in the Classroom

Students that are afraid to make mistakes are less likely to speak up in class and as a result, miss out on vital opportunities to develop greater math rigor.

Making mistakes is an unavoidable aspect of learning new concepts and skills regardless of grade, age, or experience. Teaching students that a mistake is a step towards the correct solution (rather than a conclusive failure) encourages them to approach even the most confusing mathematical concepts with perseverance and confidence.

Informative feedback is an important teaching strategy for destigmatizing math mistakes. Rather than providing binary “correct/incorrect” feedback, examine students' work with them to discover what they did right and identify where they got off track.

This process allows students to gain a deeper understanding of the topic at hand, while also providing educators with crucial insights into areas students are struggling in.

### 2. Spend More Time Discussing “Why”

While it remains important for students to understand how to practically apply skills and concepts, it’s equally important for them to develop a deep understanding of why formulas, rules, and mnemonics work.

Developing a strong foundation in the conceptual basis of rules of thumb and go-to procedures can help students connect the dots with prior learning experiences and broaden their understanding of mathematics as a whole.

Discussing why certain processes and techniques work can also enhance students' ability to think abstractly and approach math challenges from multiple angles, another important facet of learning success.

Linking algorithms and equations to their source concepts can be easily incorporated into any lesson plan to promote improved rigor and understanding.

### 3. Use Group Work to Enhance Independent Learning

While allowing students to productively struggle with challenging math lessons solo is an important part of developing rigor, working in small groups also offers opportunities for students to enhance math rigor.

Students can deepen their conceptual understanding and expand their procedural skills and fluency in small groups by:

- Discussing the steps they took to reach their respective answers
- Comparing the similarities and differences in their strategies
- Reaching a consensus regarding which methods are the most accurate, helpful, and creative.

Working in small groups also improves students' reasoning, explanation, interpretation, and understanding skills, which are the four pillars of the conceptual understanding standards put forth by Common Core.

### Encourage a Growth Mindset

The Growth Mindset model encourages teachers to frame mistakes in class as opportunities for growth.

Beyond this important shift, instilling a growth mindset also teaches students to expect a degree of confusion and struggle that requires steadfast perseverance to overcome when learning challenging or new concepts and skills.

A growth mindset is vital for developing healthy learning habits and a positive relationship with both math and learning in general.

### Celebrate Creativity and Perseverance, Not Intelligence

For many educators, it’s almost reflexive to praise a student that powers through a complex math problem and discovers the correct answer as “smart.”

And while the intention behind the praise is perfect for cultivating a healthy learning environment, choosing to praise a student’s intelligence over the hard work or creativity that went into solving the exercise can lead to poorer long-term learning habits.

That’s because hard work and abstract thinking, not intelligence, are the keys to developing strong academic rigor.

And while correct answers are important, learning how to persevere through academic exercises that are frustrating, confusing, and challenging is far more important to learning outcomes.

Similarly, teaching students to approach tough questions creatively supports the development of abstract and critical reasoning skills that are crucial to improve conceptual understanding.

## How Does Yup Support ALL Aspects of Rigor?

Yup’s on-demand math tutoring services are carefully designed to support every aspect of rigor.

Our question-based pedagogy supports deep conceptual understanding, while our expert tutors cultivate high levels of procedural skill through intensive one-on-one and small group instruction. Yup supports other aspects of rigor by:

- Explicitly seeking 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.
- After establishing conceptual understanding, the tutor connects the procedural steps to the concept as the student works through the problem, all 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? Schedule a demo to learn more about bringing Yup to your school or district.