This year, communities around the world are celebrating the Year of Math. It’s a chance to acknowledge the role mathematics continues to play in our lives. At COMAP, we’re sharing a series that looks at moments in history when mathematical modeling made a difference. Our goal is to really see how a mathematical modeling mindset has shaped real discussions, problem solving, and outcomes.

Math modeling can sound theoretical until you see what’s at stake. It’s one thing to analyze a classroom problem. It’s another thing to calculate where a spacecraft actually will reenter Earth’s atmosphere.

So picture this: it’s the early 1960s. The United States is trying to send astronauts into orbit with the goal of eventually reaching the Moon. Computers exist, but they aren’t what we think of today. They’re large, limited, and new. Both scientists and engineers are working at the very edge of what’s possible.

But before a spacecraft launches, someone has to answer: Where, exactly, will it land?

That “someone” was often Katherine Johnson.

The Problem Wasn’t Just Getting There

Let's close our eyes and think about spaceflight. Do you automatically think about a rocket launch? We have the same mental image. But we can't forget that once the spacecraft is in the air, there is still a lot that has to happen. Gravity, speed, air resistance, and the Earth’s rotation all start playing a part in where it will go.

A spacecraft's path has to be predicted before its launch. This path accounts for the Earth’s motion and, if all goes well along the way, also for reentry. Too steep, and the spacecraft burns up. Too shallow, and it skips off the atmosphere. This is modeling where there really is no room for error.

From Motion to Mathematics

At NASA, Johnson had to figure out if they launched the spacecraft at a certain speed and angle, where would it end up? She helped build the math model that answered those questions.

But it wasn't that she just “did the math.” It’s how she approached it. Johnson had to:

• Figure out what actually mattered
• Decide what could be simplified
• Set up the math so it reflected what was really happening
• Double-check the results

After all, a model is only as good as the choices behind it.

Modeling Under Constraints

Spaceflight modeling, like most modeling, comes with constraints and complexity. You cannot include every detail of reality, or the system would become unsolvable. So, you have to simplify without ignoring the constraints.

You need to decide which forces can be approximated and which cannot, and which effects are negligible and which are not. You work within time, computational, and physical constraints. And you work with a degree of uncertainty you hope to minimize.

It has to be simple enough to work through, but not stripped down so much that it stops reflecting reality.

That was the expectation every time she sat down to work. The numbers had to reflect what was really happening, but they also had to run on the computers they had.

Interpreting the Work 

When NASA began relying more on computers, Johnson was still part of the process. She went over the results, and if a number didn’t make sense, she didn’t ignore it.

A math model can’t interpret itself. Someone has to look at the result and ask, “Does this actually make sense?”

More than a Moon Landing

Students working on math modeling problems today, including contests like HiMCM®/MidMCM and MCM®/ICM®, run into many of the same kinds of issues.

They have to:

• Work through messy problems
• Identify what matters
• Decide what belongs in the model
• Explain their choices
• Make their thinking clear
• Validate their results 

The context may be sustainability, economics, public health, or engineering design. The context is different. The thinking isn’t.

When students build models, they’re practicing that same kind of thinking. We remember the Moon landing. But it worked because people made sure the math worked first.