How Big Ideas in Math Build Problem Solvers

By Cheryl Tobey, M.Ed.

When we think about preparing students to be confident problem solvers, we’re really talking about building habits of mind that will serve them well beyond the classroom. Big Ideas, as described in California’s new math framework, give us a structure for this important work. These Big Ideas connect mathematics across grade levels and provide a framework for students to develop reasoning, perseverance, and flexibility in problem solving.

Below, I’ll share some strategies that teachers can use right away to support student learning through the lens of these Big Ideas. These will be applicable for teachers across the country, but particularly important for teachers in California looking to adapt to the state’s new framework.

Big and Small Problems

Not all problems look the same. Some are small, contained tasks that focus on a single idea, while others are larger explorations that encourage students to investigate patterns and make generalizations. Both types are essential for the classroom.

• Small problems build foundational understanding. For example, asking first graders to compare two quantities in a short word problem strengthens their ability to see relationships.
• Big problems invite exploration. Middle schoolers might analyze a set of data and consider how changing one variable affects another. These larger explorations encourage students to connect math to the real world.

When possible, balance your classroom with both small and big problems. Use small tasks to build confidence, then extend learning with open-ended explorations that promote deeper reasoning.

Making Connections Through Representations

One of the most powerful ways to build problem-solving skills is to help students make connections within and across different mathematical representations: physical, visual, symbolic, verbal, and contextual. These five representations comprise a newer framework from NCTM:

For example, when solving a problem about movie ticket costs, don’t start or stop at writing the expression. Ask students to identify what each number represents in the story. Connect the verbal context, the symbolic expression, and a visual model.

You can even consider posting the five representations in your classroom and regularly asking students to show their thinking in more than one way. Encourage them to explain how one representation connects to another.

Building Coherence Across Grades

The Big Ideas also promote coherence, or the idea that students revisit important concepts in increasingly sophisticated ways as they move through the grades. Take the idea of data variation, for example. In grade six, students might compare two data sets for variability. But in grade eight, they expand that understanding by exploring associations between data sets.

This progression helps students see math as a connected story rather than a set of isolated skills.

When introducing a new concept, remind students of earlier work they’ve done. Frame new learning as a continuation of the same Big Idea, building coherence and confidence.

Habits of Confident Problem Solvers

Problem solving is not just about getting the right answer, it’s about developing habits of mind. Here are four habits to prioritize for your students:

1. Make sense of problems: Students learn to slow down and understand the situation before jumping to procedures.

2. Draw on what they know: They connect new challenges to prior knowledge.

3. Try, then revise: They view revision as a natural part of learning.

4. Stay with the struggle: They develop perseverance when problems aren’t immediately clear.

Teachers can use prompts to support these habits. Ask your students: What do you notice? What do you wonder? What part feels tricky? What might you try next? Display these prompts in your classroom to normalize productive struggle.

Back to School with Big Ideas

The beginning of the year is the perfect time to establish problem-solving norms. Introduce routines to create a classroom culture where students feel comfortable exploring ideas and sharing their thinking.

Dedicate time in your first unit to building these habits and norms. Reinforce them throughout the year so students see themselves as mathematicians who can reason, persevere, and explain.

Problem Solving with California Reveal Math

Routines are a good way to consistently make time for problem-solving habits of mind. For example, California Reveal Math’s Be Curious and Notice and Wonder sensemaking routines use multiple representations for problem-solving and mathematical modeling, giving students an opportunity to share their ideas and make connections to math in the world. California Reveal Math incorporates both big and small problems into math instruction, inviting all students to discover the joy of solving problems — all through the lens of Big Ideas. California Reveal Math was built to the California Common Core State Standards for Mathematics and the 2023 California Mathematics Framework. You can learn more about the program here.

Big Ideas Solve Big Problems

Big Ideas aren’t just curriculum anchors. They are opportunities to help students build the habits and confidence of true problem solvers. By weaving together small and big problems, emphasizing connections, promoting discourse, and fostering perseverance, we prepare students to take on math (and more) in their own worlds.

As teachers, we have the privilege of guiding students to see math as a tool for sense-making and exploration. With intentional strategies grounded in the Big Ideas, we can foster classrooms where every student feels capable of tackling challenges and growing as a problem solver.

Cheryl Tobey is a facilitator of strategies that drive informed mathematics instructional decisions and specialist on differentiated professional development to build mathematics knowledge for teaching struggling students. She is an expert on identifying student misconceptions and developing learning targets to help define formative assessment. Cheryl is the author of the Math Probes formative assessment activities and coauthor of 12 books on formative assessment.