Introductory calculus plays very different roles depending on who is sitting in the room. For some students, it is the start of a mathematics pathway. For many others, it is a required course tied to future work in areas like life sciences, economics, engineering, or the social sciences. A modeling-first approach to calculus that takes students' future aspirations seriously and starts there, rather than assuming every student is aiming for the same outcome.

The Modeling-First Calculus class begins with real questions about the world and uses calculus to understand those questions, rather than treating applications as add-ons at the end of a unit. As Jennifer A. Czocher, Texas State University, explains: “Math modeling means creating a mathematical system to describe, explain, or predict a real-world one.” Doing this requires making choices about what should be included, what should be simplified, and how to justify those choices.

This work is part of a broader Modeling-First Calculus effort that has been developing across multiple institutions. Course materials, problem sets, and emerging AI-supported tools will be shared through COMAP’s website as the project evolves. Faculty are encouraged to attend the Modeling-First Calculus session at JMM (details below) and check back on this page as new resources and updates are added.

How Modeling-First Differs From Traditional Calculus

In many calculus classes, the math is introduced first, and the real-world examples come later. A population model might be added to show how exponential growth works, or an example from epidemiology might be used to practice differential equations. The situation is chosen to fit the mathematics, not the other way around.

A modeling-first approach reverses that order. As Czocher puts it, “This positions mathematics as a tool to question the world, as students are asked to learn mathematics through their inquiry into the phenomenon.” These techniques and representations are developed because they help to answer meaningful questions. The main goal is to help students experience how mathematics generates insight.

Eric J. Deeds, UCLA, sees this shift clearly in his classroom. Rather than starting with derivatives and integrals as formal objects, his courses begin with scientific questions students already care about, such as predicting the spread of disease or understanding how ecosystems respond to change. “Along the way, notions of the derivative and integral, the primary subject of a traditional approach to Calculus, emerge as natural extensions of the questions we are asking in class,” he explains. Students learn advanced ideas because they need them, not because they appear next in a syllabus.

This approach also changes what students focus on. “Our classes have much less of a focus on computation and much more of a focus on conceptual understanding,” Deeds says. Students may know how to compute a derivative, but modeling-first instruction pushes them to explain what that derivative represents and why it matters for the system they are studying.

What Anchors a Modeling-First Course

In the first few weeks of a modeling-first calculus class, many students feel like they are on familiar ground. They know how to calculate change, and they are usually comfortable getting a numerical answer. What often trips students up is not the math itself, but deciding how to describe the change in a particular context. As Czocher points out, even students who feel comfortable doing the calculations can struggle to tell what is the driving force of change in a real-world system.

The situations instructors use to dig into these ideas really depend on the course. In Deeds’ classes, things often start with a simple question: “What happens next?” Most students have an initial reaction right away. But the more they try to explain their answer, the more they realize that instinct alone is not enough. That is usually the point where mathematics starts to matter, not because someone tells them it should, but because they need it to make progress.

At Harvard University, Janet Chen starts with a predator-prey system. Students work with it early, come back to it later, and keep using it as new ideas come up. When they run into other systems, they naturally start asking how those situations compare. Over time, they begin to recognize familiar patterns and interactions, whether they are looking at biological systems, ecological questions, or something else entirely.

Deborah Hughes Hallett, University of Arizona, emphasizes that the mathematics itself does not become less rigorous. It’s a change in the order. “Context, then mathematics,” she explains. “Students are interested in rigorous mathematics when they see it as relevant to something they care about.”

What Changes for Students

Educators often say that when modeling is part of the course from the beginning, students do not treat it like an extra step. As Czocher notes, they are asked to explain why a model makes sense, not just show how they got an answer. That changes what students focus on and how they describe their thinking.

Deeds sees this over and over again. Students who begin the course convinced they are “not a math person” start, weeks later, to describe systems, argue about equilibrium points, and make sense of long-term behavior as they talk. As Deeds explains, "This approach can open students' eyes to the unparalleled power of using mathematics to understand the world around us, and it is incredible to see how this can touch their lives and their future careers."

The Modeling-First Calculus Session at JMM

The COMAP Special Session at the 2026 Joint Mathematics Meetings, hosted by COMAP Executive Director Ben Galluzzo, Brendan Kelly, and Jennifer A. Czocher, brings together faculty who are working on modeling-first calculus in different ways. Some are deep into course redesign. Others are just starting to explore the idea. The session is meant to share what has worked so far and to offer some reassurance to those who are still figuring out where to begin.

“My biggest hope is that attendees will come away with a sense of possibility,” Czocher says. “Yes, it’s possible to teach like this. Yes, students are capable of learning it. Yes, I can think of someone to tell about it.”

If you will be at JMM 2026, make sure you add this session to your itinerary: 

Build Your Own Modeling First Calculus Class
Monday, January 5, 2026, 8:00am-12:00pm
Location: Room 144A, Convention Center