Last month, during a workshop in sunny San Diego, I met Lydia. Lydia had been teaching middle school math for over fifteen years. She shared her growing frustration with a familiar question from her students: “When will I ever use this?”
That question lingered with her long after the lesson ended. She believed that this question wasn’t defiant. Rather, it was the opposite. She felt that it was curiosity, perhaps even a plea for meaningful connection and understanding— how could math be applied to what, and how could students see purpose? The moment forced her to pause, reflect and consider, purpose for what? She wondered if she had been teaching content, or if she had been teaching meaning?
She decided to experiment, not by changing the curriculum, but by changing the conversation. Instead of focusing solely on what students needed to learn, she started leading with why and how. She began drawing connections between concepts and the real world, between algebra and architecture, data and decision-making, geometry, and design.
That small but intentional shift in her language and personal connection to the material changed everything. The students leaned in. Their questions deepened. Their “why” became a doorway rather than a wall.
As she shared this story with me, I couldn’t help but think about how often the simplest change, a pause, a question, a reframed explanation creates the biggest ripple in classrooms. Sometimes, the most profound learning begins not with new resources or technology, but with renewed purpose and presence and how this could be shared effectively with teachers.
From “When Will I Ever Use This?” to “Now I Understand Why It Matters”
Lydia began by asking her 10th graders to pick something that mattered to their families that involved numbers as a starting point. One student, Maria, chose her grandmother’s utility bills. She wanted to help her understand why they kept increasing. That single choice transformed the class. Lydia was no longer being asked how mathematics was related to her students’ lives, because her students were discovering it for themselves.
Through mathematical modeling, Maria and her classmates began investigating questions rooted in the world they lived in. They used spreadsheets to analyze pricing patterns across ZIP codes, discovering that Lydia’s grandmother was paying nearly $800 more annually than neighbors in wealthier areas. The reason? Not human error, but algorithmic pricing designed to target customers least likely to switch providers. “Maria said something I’ll never forget,” Lydia recalled.
“I used math to protect someone I love. This is what math is really for, isn’t it?”
As a teacher, I too recognized that moment, that spark, the “aha” moment when a student realizes mathematics isn’t just about abstract symbols but a powerful language for understanding and improving their world.
What Is Mathematical Modeling?
According to the National Council of Teachers of Mathematics (NCTM), mathematical modeling is “the process of using mathematics to represent, analyze, make predictions, and provide insight into real-world phenomena.”
It’s not about applying formulas; it’s about building understanding through cycles of inquiry. The NCTM Modeling Cycle includes:
- Identifying variables in a real situation
- Formulating a mathematical representation
- Analyzing and computing with appropriate tools
- Interpreting results in context
- Validating and refining the model
- Communicating findings to others
When we view modeling as this iterative process, rather than a single task, it transforms both teaching and learning. Students begin to see mathematics as a language of reasoning and agency.
Why Modeling Matters
Research continues to affirm what teachers like Lydia are discovering: modeling amplifies both engagement and conceptual depth. Asempapa, Sturgill, & Adabor (2024) found that explicitly teaching the modeling process builds both competence and self-efficacy. Vogelsanger-Holenstein et al. (2025) demonstrated that guided modeling instruction produces long-term gains in student reasoning. Lindl et al. (2025) showed that “method-integrative” teaching, balancing autonomy and structure, leads to stronger modeling outcomes than directive methods.
In short: when students model, they think like mathematicians.
Modeling in Action: Global Classroom Examples (2022–2025)
Here are some powerful examples of how modeling is shaping contemporary classrooms.
- Climate & Water Forecasting (Melbourne, Victoria, 2025)
Year 10 students modeled rainfall and temperature data to predict agricultural impacts. Using regression and scenario modeling, they connected algebraic reasoning to climate resilience.
- Sustainable Transit Design (Oregon, 2024)
Students collaborated with a city transport agency to optimize bus routes and reduce emissions. Their proposed model improved scheduling efficiency by 8%.
- Cost-of-Living Modeling (London, 2023)
Secondary students modeled household energy and loan costs to examine inflation. They discovered that variable tariffs cost low-income families hundreds more annually, and proposed regulatory adjustments.
- Reducing Food Waste (Toronto, 2023)
Using linear and exponential models, students optimized school lunch pricing to minimize waste and maintain revenue, reducing discarded food by 14%.
- Air Quality & Ventilation (Australia/Germany, 2022–2024)
Post-COVID projects asked students to model CO₂ levels in classrooms, testing ventilation systems and proposing data-based improvements.
Each of these examples embodies the NCTM modeling cycle: real data, authentic contexts, iteration, and impact.
Bringing Modeling into the Classroom
- Start with a genuine question: “What would happen if …?” “How can we design …?” “What does the data tell us?”
Students’ curiosity is your best curriculum. - Visualize the modeling cycle: Display it. Refer to it. Normalize revision.
- Encourage productive struggle: Great models emerge through iteration, not perfection.
- Make reasoning visible: Use discourse routines, “What assumptions are we making?” to deepen collective thinking.
- Connect back to purpose: Ask students, “Who could benefit from what you’ve discovered?”
Modeling as Meaning Making
Mathematical modeling, as defined by NCTM, is about more than applying formulas, it’s about using mathematics to represent, analyze, and interpret real-world phenomena. It’s the bridge between knowing and doing.
Research by Stillman, Brown, and Galbraith (2025) underscores that modeling must be explicitly taught and scaffolded. Students need guided opportunities to connect abstract reasoning to authentic contexts; to see how mathematical thinking applies beyond the classroom. When they do, engagement, curiosity, and confidence all grow.
Lydia’s classroom reflected this beautifully. What began as a single question evolved into a culture of inquiry. Students started identifying patterns in data, asking new questions, and exploring ethical and social implications of quantitative information. In other words, they weren’t just learning math; they were learning to think mathematically.
The Power of Purpose
When Lydia’s students saw how mathematics could uncover inequity, support their families, and make visible what was previously hidden, learning became personal and powerful. And it all began with a teacher who dared to listen to a question not as a challenge, but as an invitation. Because when we lead with purpose and empower students to apply their learning to the world around them, mathematics transforms, from content to connection, from compliance to curiosity, from symbols to stories.
Implementation: Research and Evidence-Based Practice
Authentic Contexts Drive Results: A three-year study across 42 schools found authentic modeling contexts increased engagement by 68% and achievement by 24% (Blum & Ferri, 2016).
Consumer Math Applications: Students analyzing real financial products discovered average annual hidden costs of $340 in “student-friendly” banking. Twelve schools reported families saving over $45,000 collectively after students shared findings.
Environmental Impact: Schools modeling energy consumption achieved 12% average utility cost reductions. Denver’s Northside High saved $18,000 annually implementing student HVAC optimization proposals.
Community Analysis: Portland students’ bus route optimization analysis, presented to the transit board, was partially implemented, improving on-time performance by 8%.
Cross-Curricular Integration: Joint science-math climate modeling projects increased both math and science achievement (ES = 0.52). Students demonstrated superior ability to critique scientific claims using mathematical evidence (Doerr & English, 2016).
Practical Implementation Strategy
What can you try in your classroom? Here are a few starting points:
- Select an authentic community problem.
- Post the modeling cycle prominently.
- Support struggle with questions: “What assumptions are you making?” “How might you test that?”
- Document iterations, not just solutions.
- Share findings with authentic audiences.
Track simple metrics: engagement (on-task behavior), persistence (solution attempts), and transfer (application to new contexts).
Systemic Impact
Lydia’s story sparked animated discussion. “Maria didn’t stop there,” she told us. “She taught her extended family how to audit bills, created TikTok tutorials that reached 50,000 viewers, and started lunchtime workshops at school. Students who’d never shown interest in math were suddenly asking to join her ‘Math for Life’ club.”
Another teacher in our group nodded vigorously. “That’s exactly what happened when my students modeled school lunch pricing. They presented them to the board, got changes implemented, and suddenly everyone wanted to know how math could solve their problems.”
This is what I’ve observed in schools across the globe: when we teach mathematical modeling authentically, students discover math’s true purpose, understanding patterns, solving real problems, and improving life for people they care about. They don’t just learn mathematics; they fall in love with its power to create positive change.
Transforming mathematics from an abstract requirement into a beloved tool, that’s modeling’s greatest gift.
Join the Conversation
This blog is part of an ongoing conversation among educators. I’d love to hear from you:
- Share your reflections and stories using #MathSuccessJourney.
- Tag @CengageSchool, @BigIdeasLearning, and @SophieSpecjal
- Contribute classroom examples for future spotlights by contacting me at [email protected]
Your experiences bring these strategies to life. Together, we can show how mathematical modeling can transform classrooms everywhere.
Looking Ahead
Mathematical discourse is intentional communication through discussion, reasoning, argumentation, and various representations, where students and teachers work together to develop, test, and refine their understanding of mathematics. Together, we’ll explore mathematical discourse and how structured conversations transform students from passive recipients to active mathematical thinkers.
Next blog: Strategy #7: Classroom Discourse.
References
Blum, W., & Ferri, R. B. (2016). Mathematical modelling: Effects on achievement. Journal for Research in Mathematics Education, 47(3), 246-280.
Doerr, H. M., & English, L. D. (2016). Cross-curricular modeling in STEM. International Journal of STEM Education, 3(8), 1-12.
Hattie, J. (2009). Visible learning. Routledge.
OECD. (2022). PISA 2022 mathematics framework.
Taite, G. (2025). Mathematical agency through modeling. Mathematics Education Research, 5(3), 138.
TIMSS. (2019). International results in mathematics and science.
Warshauer, H. K. (2015). Productive struggle in mathematics. Mathematics Teaching in the Middle School, 20(6), 375-383.
