Rigor: Bring it on in Math

Making Math Count in WHPS

There is a lot said about bringing the rigor to math, but what does this really look like?

I recently had a teacher say to me, "I am happy to bring the rigor, just tell me what it is." In education we often feel that way when a new "buzz phrase" such as rigor is abundant in professional learning and professional reading, but not so defined in the curriculum. Below we are going to define rigor, share practical advice on how to include it in your lesson plans, and give an example of rigorous instruction.

What is rigor?

According to The Common Core Standards, rigor means to pursue conceptual understanding, procedural skills and fluency, and application with equal intensity. We often talk about rigor as though it is an adjective or a noun. We might say it was a rigorous activity, meaning it was challenging and asking students to think or push beyond their comfort zone. We might say it is a rigorous activity if a student has to apply understanding to solve a problem. We will sometimes say that lesson had rigor, as though rigor is a thing to be had. Usually this means there is challenge to the lesson.

Rigor is only achieved when we focus on how we are asking students to make sense of math and develop knowledge. This means the actions our students take during learning should equally pursue understanding of concepts, procedural skills, and application. It's like a three-legged stool. All three legs need to be strong, or the math instruction falls short.

For many years, math education was focused on procedural skills. The goal of math instruction was to make math easier for students by teaching them the procedural steps to find solutions. Instructional time was focused on showing students ways to remember, so they could get the correct answer to an equation. The students that are "good at math" might be able to answer a word problem at the bottom of the page, but all students struggled to connect concepts and see patterns in math. Students sought right answers rather than developing an understanding of mathematics that leads to fluency and application.

Rigorous instruction moves beyond right answers. Teachers focus on using rich tasks and deliberate questions to promote understanding of math concepts. They make instructional decisions that move students to new mathematical understanding. These understandings are the underpinnings for students to replicate strategies and solve problems using similar procedures. With continued practice, students develop procedural fluency that supports future understanding and conceptual development.

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How to Plan for Rigorous Instruction

Add a copy of the guide below to your plan book for easy access. Rigor should always be on your mind as you plan math lessons. This guide was adapted from ANet.
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Resources for Tasks that Promote Rigorous Teaching

An Example of Rigorous Instruction

Here is a problem presented to a third grade class.

The Band Concert Task

The third-grade class is responsible for setting up the chairs for the spring band concert.In preparation, the class needs to determine the total number of chairs that will be needed and ask the school’s engineer to retrieve that many chairs from the central storage area. The class needs to set up 7 rows of chairs with 20 chairs in each row, leaving space for a center aisle. How many chairs does the school’s engineer need to retrieve from the central storage area?

Mr. Harris asks his students think about how they might represent the problem. Students then pair and share their ideas. Mr. Harris notices students using pictures and symbolic representations such as repeated addition and partial product. While students work, Mr. Harris monitors their work and asks students about their representations and thinking.

In planning for the lesson, Mr. Harris prepared key questions that he could use to press students to consider critical features of their representations related to the structure of multiplication. As the students work, he often asks: “How does your drawing show the seven rows?” “How does your drawing show that there are 20 chairs in each row?” “Why are you adding all those twenties?” “How many twenties are you adding and why?”

Before for the whole group discussion he has students pair with another student that has a different representation to take turns sharing. He asks students to identify similarities and differences with their partner.

While students are working, Mr. Harris chooses 3 students to share their work. His choice of representations is purposeful and he asks them to explain their representations and answer questions from classmates. The first student used skip counting by 20. Mr. Harris asks the class, "How does this representation relate back to the story?" He invites students to retell in their own words and verifies if the class agrees and why. The second and third student representations considered the aisle and students worked with 10 rather than 20. Students discuss how the approaches are different. Mr. Harris asks students to respond in their journals to the question, "Is it okay to represent this problem with both of these approaches? Justify your answer."

Mr. Harris concludes the lesson for the day by writing the equation 7 x 20 = 140. He asks students, "How does this equation relate to the strategies we discuss thus far?" He purposefully connects repeated addition and skip counting to multiplication.

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The Lesson Continues on Day 2

On the next day, Mr. Harris begis the class by having more students share their work. He asked Tyrell, "How many tens did you have?" As he responds, Mr. Harris recorded 14 tens = 140. Next the class examines the work of Ananda. She explains that she counted by tens to 70 and then just knew she has 70 more chairs and adds them up. Mr. Harris again emphasizes how she was combining tens and wrote on the board, “7 tens + 7 tens = 14 tens = 140 chairs.” He then asks the students to work with a partner to write an equation that represents they statement using multiplication. Most students wrote, “(7 × 10) + (7 × 10) = 140 chairs."

Mr. Harris used representations and a single problem to help students develop concepts that will lead to fluency in multiplication. Students are developing the why that supports an understanding of distributive property, multiple of tens, and decomposing arrays.

This lesson started with application. The discovery of concepts were grounded in a real-world problem. Students had an opportunity to work on the task independently, share ideas with partners while the teacher monitored student ideas and engagement, asked questions to move student thinking forward toward understanding, and students engaged in multiple practice standards and content standards. Finally, Mr. Harris explicitly connected the concepts to the equation that can be solved using procedural skills and fluency.

Adapted from Taking Action: Implementing Effective Math Teaching Practices (2017) NCTM