Rigor is what moves students into creative and effortful problem solving. It’s the type of learning that challenges, and occasionally frustrates as well. Effective instructional rigor has the power to expand students’ capacity for deeper understanding and intrinsic motivation. However, some common beliefs may be preventing teachers from developing consistently rigorous learning environments.

In order to make way for the positive effects of rigor, educators should steer clear of the following assumptions.

## 1. Direct instruction is enough

Students need more than the traditional “I do, we do, you do” model of instruction. Hands-on, visual experiences can provide all students, regardless of math or language proficiency, access to more rigorous mathematical problem solving.

Consider incorporating manipulatives (both physical and digital) into lesson plans to remove barriers and provide all students with an access point to the concept being taught.

Explore manipulatives resources:

- Digital Manipulatives and the Mind-Body-Math Connection
- 7 Best Practices For Using Digital Manipulatives in the Classroom
**Clever Ways to Use Math Manipulatives with EdTech to Facilitate Deeper Learning**

## 2. **Mistakes indicate lack of understanding**

It may be common to interpret a student’s incorrect answer as he or she not grasping the concept being taught. But what if educators turned those mistakes into opportunities for the whole class to learn from? Research shows that making mistakes may lead students to better understand math concepts.

Try encouraging a mindset in the classroom that celebrates mistakes as the perfect opportunity for learning. Allow students to collaborate and give feedback to each other in order to unpack incorrect answers. As much as possible, refrain from telling students what to do too quickly, so they learn to persevere in problem solving.

“Every time a student makes a mistake in math, they grow a synapse.”

- Jo Boaler, from Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching

## 3. There’s only one way to find the answer

In math, there are often multiple ways to solve a problem. In fact, researchers are finding it beneficial for educators to explore problems with multiple solutions and facilitate students in comparing problem-solving strategies. When students discuss different ways of solving problems, they learn that with a little creativity, no problem is out of reach.

Inspire students to think outside the box with problems that allow many different pathways to the solution. Once a problem is solved, challenge students to come up with another way to solve it. Work backwards, if need be, by providing the solution and having students find different ways to get there.

## 4. Memorization is better than reasoning

It’s often the quick calculators and good memorizers who are praised the most in math class. However, Stanford professor, Jo Boaler, warns that instruction based solely on memorization and arithmetic can lead students to misunderstand and dislike math. Test results show that the highest achievers are those who can see the bigger picture and make connections between different mathematical concepts.

Memorization will only take us so far in math, explains Boaler. “We don’t need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms.” (Boaler 2015)

As much as possible, incorporate opportunities for students to learn *why* the formulas and procedures work. For example, if we know the area of a square is base x height, how can we find the area of a triangle with the same base and height? You might find that this approach actually requires less memorization in the end.

## 5. Perfect scores equate to concept mastery

The perfect time to compound a student’s knowledge is when they start showing signs of understanding. Moving on as soon as students can answer a few problems correctly on a test may forfeit a crucial opportunity to facilitate transference and real world application. As soon as students start to show signs of mastery, change the context of the problem, reverse the question, and extend their thinking!

Truly rigorous instruction empowers students to anticipate and overcome challenges. Download the posters below to help keep guiding principles of rigorous instruction top of mind.

For educators: Guiding Principles for Increasing Rigor [pdf]

For students: The Cycle of Rigorous Learning [pdf]