Dr. Karen Fuson has spent her career researching how children think. Her work reveals a simple truth: when students begin with their own ideas, they build understanding that lasts.
In classrooms shaped by Fuson’s approach, math instruction doesn’t begin with a formula. It begins with a question. “Math Expressions starts each new topic by eliciting student thinking. It’s very important to know where kids are for teachers to teach appropriately,” she says.
Each Big Idea in every Math Expressions unit begins with a question that students pursue. Throughout learning, students also have the chance to raise their own questions.
This starting point is more than just a strategy. It’s a philosophy. Students share their methods, and teachers respond with guidance that builds on what they already know. “But then Math Expressions does not let the kids just do those methods.” Fuson explains what happens after students share their initial approaches: “We move on to ‘phase two’ in which teachers and other students present more advanced ‘better methods,’” Fuson says.
This learning environment where students feel safe to explore and grow is a fundamental part of Math Expressions. Fuson developed this idea for the Nurturing Math Talk Community from the Children’s Math Worlds Research Project, which was instrumental in developing the Math Expressions curriculum.
How can we help students build conceptual understanding and discover the “why” behind math?
Develop Understanding and Inquiry
Building understanding is a core component of the Nurturing Math Talk Community. Students use drawings and visuals to show their thinking share ideas with peers, and ask each other questions.
The classroom becomes a space for exploration. “During all this time, this is occurring in a Nurturing Math Talk Community in which students make math drawings. They point to their work. They explain their work. They discuss. They help each other. They say, ‘Any questions?’” says Fuson.
The ability to explore is essential to inquiry-based learning, a student-centered approach that involves formulating questions and constructing resolutions to those questions through experience, discussion, and reflection.
Inquiry-based learning invites students to:
- Pose questions that they are interested in, with guidance from the teacher
- Investigate those questions individually or in groups, typically during in-class or out-of-class time
- Present their findings using a variety of methods, such as a website, podcast, movie, or whole-class presentation
- Reflect on their learning to build metacognition and become better investigators in the future
Similar to a Nurturing Math Talk Community, an inquiry environment is filled with students who feel confident asking questions and investigating topics.
Build Fluency and Practice
Fuson’s model doesn’t stop at developing understanding and inquiry. Once understanding is built, students move into fluency and practice.
Fluency is about so much more than memorizing facts. The National Council of Teachers of Mathematics describes procedural fluency as being able to use strategies efficiently, flexibly, and accurately—and knowing when one approach works better than another. It’s not just recalling steps; it’s being able to apply them in different situations.
That’s why strong fluency starts with strong understanding. Students can’t be truly fluent in something they don’t deeply grasp. Practice helps build fluency and understanding, and Math Expressions provides the tools and activities to guide students through this process, making the learning stick.
Fuson’s Model in Action
Let’s explore how building understanding, inquiry-based learning, fluency, and practice work together in a math classroom. Consider a third-grade classroom encountering the following multi-digit subtraction problem: 52−27. Rather than demonstrating the standard procedure, the teacher invites students to solve the problem using any method that makes sense to them.
Building Understanding
Students begin by sharing their approaches to the problem. One student draws 52 tally marks and crosses out 27 of them. Another breaks down the problem into smaller numbers, such as taking away 20 first to get 32, then 7. A third student counts from 27 to 52, tracking the distance between the numbers. The teacher listens carefully, asking students to explain their thinking and draw representations of their work. The teacher explains how there can be multiple valid pathways to a solution.
Inquiry-Based Learning
The classroom then moves into a guided discovery phase. The teacher introduces more ways to solve the problem. Students compare these methods with their own, discussing which approaches feel the most efficient and why. Students test different methods on new problems and discuss their effectiveness based on the problem. For example, one student realizes that counting up works particularly well when the numbers are close together. This exploration phase isn’t just about finding an answer but about understanding the structure of subtraction itself.
Fluency
After building conceptual understanding, students practice their different methods until they can execute them smoothly and accurately. Students are building automaticity with strategies they genuinely understand. The goal isn’t speed, but confident execution.
Practice
Students practice subtraction in a variety of ways—through word problems, puzzles, and multi-step challenges—so they can apply their skills in different situations. This practice reinforces the procedure and the reasoning. Students don’t just learn how to subtract, they understand why the methods work and when to use them.
Students are capable of so much more than memorization. Math Expressions taps into that potential, giving them space to explore, ask questions, test ideas, and discover strategies that truly make sense. Every “aha” moment comes from understanding, not repetition, and practice strengthens skill and confidence. By combining curiosity, fluency, and guided exploration, students don’t just solve problems, they think, reason, and grow like mathematicians.
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Learn how Fuson’s Math Expressions turns student thinking into a launchpad for inquiry-based learning. Download the program overview to see how guided exploration, visual models, and collaborative problem solving build deep student understanding.
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