Differential equations belong in Calculus I

In my calculus textbook you will find differential equations, including modelling of numerous and diverse real-world phenomena, treated much earlier than in conventional books. Putting differential equations front and center from the beginning makes the development of basic differentiation and integration techniques well-motivated and rich in fascinating applications. In this way one also gets right to the core of why calculus is important in the first place. If it wasn’t for differential equations, the calculus would never be anywhere near the staple course it is today; it would be a footnote in the mathematical curriculum studied by a handful enthusiasts rather than millions of people every year. In a nutshell, the calculus matters for one reason and one reason only: differential equations are everywhere. But unfortunately this basic and crucial reason for studying the subject is a secret carefully withheld from students.

Incorporating differential equations from the outset also reflects the way the subject developed historically. Mathematicians did not sit around and catalogue techniques for computing all kinds of artificially complicated limits, derivatives, and integrals, and only then moved on to differential equations. Instead, they tackled technical problems as they arose naturally in the course of investigating genuinely interesting questions, which generally meant differential equations. This did so because this approach made sense and was fun and interesting. One doubts they would have felt the same about the way modern courses butchers the subjects.

Teachers often lament that students do not remember what they learn and, even when they do, do not know how to apply it as soon as the setting deviates from the artificially controlled sandbox examples of end-of-section exercises. No wonder when the calculus is taught as it is. If we teach it instead the way I advocate students will remember it better since the material is embedded in a coherent narrative rich in interconnections between ideas, instead of studied in clinical isolation. Likewise students will learn to use the techniques they study in real problem settings, since they will study them in action in their natural context, showing their actual motivating applications rather than concocted drill problems.

Postponing differential equations for three or four courses, as is traditional, robs the subject of purpose. Students must learn artificial rules in isolation from any meaningful context, thus guaranteeing from the outset that they will become blind formula-crunchers with no sense for what they are doing, or why, or how it all fits together. This is such madness that conspiracy theorists may be excused for suspecting that mathematics educations is purposefully designed to produce mindless, spirit-broken worker drones.

The root of the problem with the traditional approach is the “monkey see, monkey do” teaching paradigm. This paradigm requires that each section of mathematics textbook be defined by one or two replicable problem types that can be repeated ad infinitum with superficial variations. The traditional approach is based on the notion that “teaching” mathematics consists in working out five such examples on the board followed by assigning ten times as many as homework. This very harmful notion is the root of all these evils. It is the sole reason, for example, why students in Calculus I waste their time with minor technicalities like l’Hôpital’s rule and related rates without having ever heard of a differential equation.

The traditionalists shout back at us: But if you don’t know integration you can’t solve any differential equations! This way of thinking makes sense only if you assume, firstly, that it is sensible to teach intricate and artificial integration techniques to innocent children with no credible pretext for doing so except your own authority, and secondly, that teaching differential equations means teaching “tricks” for solving hundreds of replicable drill problems (first separation of variables, then integrating factor, etc.). The real reason why differential equations cannot be taught earlier in traditional classrooms is that doing so would involve genuine, open-minded thinking, rather than mindless plug-and-chug. And, sadly, many people’s idea of mathematics teaching is inconsistent with such a prospect.