Formative Assessment

How to Use Students’ Prior Knowledge to Build Math Skills

Instruction that’s informed by what students already know, and thoughtfully supported by intentional questions, can help ensure retention of math concepts.

April 13, 2023
Nataliia Tymofieieva / iStock

Visitors to my fifth-grade classroom sometimes comment that it’s not exactly a quiet place. I smile every time, though during one recent math lesson I did wonder whether we might get a call from the teacher next door. 

My students were so excited and engaged as they debated fractional math strategies. Yep, they were excited about fractional math strategies. Specifically, they were discussing cans of cat food.

The kids are very familiar with my two cats, Spencer and Cali. They hear stories about their antics and love to see photos of them. On that particular noisy day, I posed a problem that involved figuring out how many cans of cat food I needed for an upcoming trip. I explained that each cat gets a different fraction of food in the morning and evening, and my students had to work out how many cans of food I needed to purchase. The catch: This was their first introduction to mixed numbers. 

Figure Out What Students Know About A Concept 

I use real-world math problems when introducing each new concept. This gives students the chance to develop their own strategies for making sense of and solving problems. I want them to make meaning of the math first, and I want to understand what the kids know before diving into instruction.

This student-centered approach, inspired by training I’ve received in Cognitively Guided Instruction, or CGI, makes sense. CGI is an asset-based model of instruction that builds on student knowledge and children’s natural ability to problem-solve. Teachers pose real-world problems and build on students’ mathematical understanding through questioning. 

Students engage in mathematical problem-solving from a young age. When you tell kids to share their blocks, they divide. When you say they took too many cookies and have to give some back, they subtract. Tapping into that student knowledge at the start of a unit helps me identify and address gaps and think about how to build on what kids know. 

In the case of the cat food problem, many students didn’t know how to add fractions with different denominators, and most struggled to write mixed numbers. But when they encountered the problem, they devised their own strategies using math models, manipulatives, and drawings. As students explained their thinking, I learned more about what they understood and used questioning techniques to identify misconceptions and deepen learning. My students were highly motivated to build on their knowledge. After all, they wanted to be sure that Spencer and Cali were properly fed. 

Collaboration and Hands-On Learning are Key

So, about those noise levels—after I present the class with a problem, students discuss and debate how they might solve the problem, often working in pairs and small groups. They share ideas and build connections. Then, they discuss strategies with the whole class. Meanwhile, I continue to observe and ask questions to learn about their understanding, so I know where I need to go with the lesson.

CGI-inspired lessons can help bridge the gaps that students are facing post-pandemic. During remote learning, students didn’t have the opportunity to engage in this kind of collaborative mathematical problem-solving. They were taught over Zoom with limited access to manipulatives, hands-on activities, and mathematical dialogue with peers. As a result, many of my fifth graders don’t have the core mathematical understanding or number sense needed to solve problems. Most of them can perform basic calculations, but when presented with a real-world problem to solve, they struggle to apply concepts.

Academic Recovery is in Progress

Nationally, kids are really struggling with math. The most recent Nation’s Report Card showed that students experienced the largest declines in math in three decades. Twenty-five percent of fourth graders are working below the Basic level on the Nation’s Report Card, as are 38 percent of eighth graders. 

My students are still working on their academic recovery, but we’re seeing progress. On a recent assessment involving fraction-based problem-solving, 86 percent of my students scored near or above standard. They’ve grown more comfortable with problem-solving, and scratch paper reveals that they’re using a variety of effective strategies.

Developing conceptual understanding in math is personal. I was a good math student. I was even named calculus student of the year during my senior year of high school. Yet, as I walked onstage to receive the award, I felt like a fraud. I could recite formulas and solve equations, but I had no idea what any of the formulas or equations meant or how to apply them in the real world. The focus was on memorization and calculation, and as a result, I understood absolutely nothing about calculus. When I became a teacher, I made it my mission to ensure that I would teach kids to have a full understanding of math concepts. 

7 Steps for a CGI-Based Approach to Math

If you want to give CGI-inspired math problem-solving a shot, consider trying these steps toward building a more student-centered, if noisy, learning environment.

  1. Begin each new concept with a real-world problem your students care about. Design it so that you can learn what students know and understand.
  2. Brainstorm possible misconceptions and difficulties that students might face as they work to solve the problem.
  3. Prepare questions that you can use to help students work through challenges.  
  4. Observe students and ask questions (as they solve the problem) that will help you understand their mathematical thinking.
  5. Take notes on student understanding and misconceptions. You can use these to plan your next steps. 
  6. Ask students to share their strategies with the class. Start with the most basic strategies, and then introduce more complex ones. 
  7. End by facilitating a class discussion about the various strategies that students used and what they learned.

I hope this approach sparks joy, excitement, and progress in your math class the way it has in mine. My fifth graders are heading off to middle school next year, and I’m intent on helping each and every one of them embark on that chapter of their academic life with the confidence and skills to lead their learning and excel in math.

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  • Formative Assessment
  • Math
  • 3-5 Upper Elementary

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