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Bring Complex Math Down to Earth with Real-World Examples

Using visual modeling and some helpful computer programs and platforms, this professor brings differential equations into clear focus.


Matthew D. Johnston, PhD

Assistant Professor of Mathematics, San Jose State University

PhD in Applied Mathematics, MS and BA in Mathematics

Pull out your paper and pencil: Mathematics is pretty much the same as it has always been for students around the world.

But Matthew Johnston, PhD, knows that grinding out formulas, problems, and equations is only the first step in math’s ultimate and essential contribution. All those numbers, letters, and symbols go into real-life problem-solving that affects our daily existence, even if we are completely unaware of their impact. This is what puts the “applied” in “applied mathematics.”

Not surprisingly, the backbone of applied mathematics is the subject of one of Johnston’s favorite courses: Ordinary Differential Equations.

The differential equation, he explains, is a classic math device we might think of as a set of guardrails on the physical world; these calculations guide the causes and effects generated by energy, motion, gravity, and other dynamic forces.

“Differential equations are everywhere,” says Johnston. “A differential equation, in layperson-speak, has to do with rates of change. Any process that is subject to change over time is a differential equation.”

Differential equations are the mathematical building blocks of man-made structures and devices. Engineers develop and study differential equations related to various projects (bridges, skyscrapers, smartphones, etc.) to understand how various forces work together and to predict how the project might stand up to unexpected stress and strain. Using equations to calculate how a smartphone will react to a battery charger or receive signals from a cell tower, for example, can help designers calculate when a device might overheat, malfunction, or offer buggy service.

While the impact of differential equations may be compelling, Johnston has found that the study of them sometimes is not—particularly if students are presented with the concepts using only pencil and paper.

Challenge: Linking abstract math to real-world projects

Johnston’s course comes at the end of the line for all types of mathematicians, engineers, physics students, and other scholars delving into facts, formulas, math principles, and engineering concepts. Armed with three years of scientific knowledge and numerical training, Johnston’s students must then pass through the gates of the differential-equation universe, a realm where every theory of design or invention is subject to the eternal, pressing question: Will it work in the real world?

Johnston understands it can be challenging for budding engineers to turn a complicated series of math and physics problems—the raw ingredients of every differential equation—into something they can envision in the real world.

“Many students think math is just a tool,” says Johnston. “They put an equation into a computer and get an answer and don’t think about it. But they’re going to run into problems [if they do that], because they need to be able to look at the equations [and what they mean for the project.]”

This, after all, is the point of applied mathematics, says Johnston. “You don’t want to just build something and ‘see how it goes’ when it’s a million-dollar project you’re building, and so many lives are on the line. You want to have a model that predicts [what might cause] something catastrophically bad to happen to something you’re building.” That way, ideally, you can prevent it, or at least reduce the risk.

Innovation: Turn math problems into familiar visuals

Johnston’s differential equations course makes use of a variety of easy-to-use platforms and easy-to-understand visual examples. Though the equations and their applications may be complex, he believes that, by keeping his class organization simple and offering as many visual aids as possible, he can help his engineering students understand the topic on a deeper level. Further, he finds that when they get a better grasp of the subject, they feel more free to use courage and imagination to begin experimenting with the equations.

Here is how he explains it to students: “There’s a light at the end of the tunnel that, hopefully, you’re going to find satisfying. There’s a lot of stuff on the math side that maybe you don’t care much about, but I like you to come away with a list of toolboxes. This is how we do it, this is why we do it, and I try to make it exciting along the way by handling actual examples that will hopefully be relevant.”

Interestingly, his examples are so relatable that one does not need to be an engineering student to understand them!


“If we can handle [differential equations], we can figure out how that thing slides down a ramp and what the acceleration is, what the velocity is, and we can use that in all those classical physics problems. The contextualization is worth it, so stick with us. Physics tells you how [something is] going to change, but you need mathematics to tell you, ‘So what?’”

— Matthew D. Johnston, PhD

Course: MATH 133A Ordinary Differential Equations

Frequency: Two 75-minute class meetings per week for 15 weeks

Class size: 100–250

Course description: First order differential equations, first order linear systems, second order linear equations, applications, Laplace transforms, series solutions.


See resources shared by Matthew D. Johnston, PhD

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Lesson: Let your mind’s eye paint a picture of resonance

One of the most pressing issues addressed by differential equations is the principle of resonance—how much in tune with itself a bridge, building, rocket, or smartphone needs to be to operate at peak efficiency. For his series of lectures on this topic, Johnston presents his students with examples that illustrate how systems that appear to be functioning flawlessly might be thrown out of tune by adding other forces that disturb their “happy resonance.”

“Resonance is just a response to a synergy between a system that is already doing one thing, and then another force comes along and says, ‘I’m going to do the same thing,’” says Johnston. “And you have to be careful when figuring out how these forces interact.”

Here are some of the ways he helps engineering students absorb these complex concepts by using tangible, visual platforms and examples:

Keep all class materials in one place

Johnston stores all of his course assignments, homework, and visual examples on his own website rather than using a larger, more generic software platform. “I always like to [keep everything] on my own site so I have control over the format and how it looks,” he says.

Johnston also makes sure his site includes all grade expectations (rubrics), assignment due dates, and supplemental materials, such as videos and downloadable computer code written in MATLAB or Python.

Provide notes ahead of time

Also residing with the supplemental materials on Johnston’s website are the notes that he has taken on his presentations, which he has accumulated and refined over the years.

“It’s not mandatory that the students read the notes, and most people scribble frantically [during class] because that’s what you do in an undergraduate engineering program,” says Johnston. “But I say, ‘Don’t worry about taking notes if it is a form of stress for you.’”

Using his notes frees up students to keep their eyes on the visual examples he provides on the board.

Solve for sloppy handwriting with computer software

Creating, adjusting, and sharing equations through email can often result in garbled symbols and text. “I get emails all the time from students asking, ‘How do I evaluate this math equation?’” says Johnston. “It looks like gibberish, and half the time I’m asking them to clarify what they mean by it.”

To solve for this, Johnston makes use of the Piazza platform, which allows students to easily format equations and math problems that they can then share with the professor and the rest of the class.

Provide fast (visual) feedback via email

Johnston prefers to handle homework online using the platform WebAssign, which has a quick grading response built into the system. “The students like the immediate feedback,” he says. “They get that little X next to incorrect answers, and they know to go back until they get it right.”

“Sometimes the homework and follow-up tests give them a sense of, ‘Oh, close, but that’s not quite getting the point,’” adds Johnston. “So I’m always helping [to make] things become crystal clear, and they’re able to work through it.”

Put their imaginations to work

When lecturing on resonance in mechanical systems, Johnston uses the example of a child on a swing to illustrate how all forces must be in harmony for a system to perform at maximum efficiency. A child swinging, he explains, roughly duplicates the complex pendulum systems and spring mechanisms that often govern the inner workings of sophisticated machinery.

“If you have a pendulum and displace it to the left or the right, it’s going to wobble left or right,” says Johnston. “This is what happens to electrical circuits, on a much different time scale.”

By observing a child on a swing—and considering how a parent might step in to speed or slow (or stop) the swing—Johnston’s students also get an idea of how forces may come to bear on an efficiently resonant system.

“The output of all this mathematical machinery is the location of the child at any given time,” says Johnston. It answers the question, “Where is my child 10 seconds after I let him go?” and “Where is my child 10 seconds after that?” After students can picture this, Johnston changes the visual to something more engineering-related. For instance, he offers: “We then consider not a child on a swing but the position of a satellite that’s transmitting a signal to your cell phone.”

Dim the lights and fire up the projector

While students can probably imagine that swinging child easily enough, Johnston connects the dots for them. He has created a series of computer renderings of differential equations that unerringly show what is happening—for instance, how a swinging child might be affected by the unexpected push or pull of a parent. He can also render how a slight variation in a differential equation might alter the path of that child—or a rocket, satellite, or even an asteroid hurtling toward Earth.

“[In differential equations lessons, students] always get some equation at the end that’s supposed to represent [something dramatic, like] when the asteroid is going to hit Earth,” says Johnston. “But it’s just an equation. And if you just look at it, even if you know a lot of math, it’s hard to see what’s happening. So it’s nice to be able to go over to a computer and say, ‘Here’s what this actually looks like.’”

“Even in a class of 150 slightly dozing-off, mid-afternoon engineering students, as I’m firing up the computer and bringing the projectors down, you should see their eyes,” says Johnston. “Everybody perks up like they just had a jolt of espresso.”


Johnston wants students to understand that visualizing these equations “in action” will help them with whatever rate-of-change problem they encounter in the future. He tells them, “These models give you a forecast for the future, because whatever system you’re modeling, you know it’s going to change, and physics tells you how it’s going to change. But you need mathematics to tell you, ‘So what?’”

“We have a model that we hope we understand at this point,” says Johnston, “so that when we let our child go, we understand—both from the model and the physical context—that the child is not going to shoot off into outer space or crash to the ground.” And ideally, neither will his students’ future skyscrapers, suspension bridges, or smartphones.

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