Why teach calculus in the age of AI

If you’re reading this, you’ve probably taken a calculus course before. How was it? Chances are you found it frustrating or, at least, complicated.

You may remember one of the most feared parts of calculus courses: the computation of derivatives and integrals. For most students, this is the first time they face a mathematical problem whose solution is not a number, but a much more complicated object: a function. Computing derivatives and integrals involves memorizing tables, applying complicated composition rules and avoiding sneaky arithmetic errors. In short, it is hard.

Enters technology

For several decades now, it is possible to solve these problems using symbolic computation software packages, available for computers and even for some pocket calculators. Since the explosion of Artificial Intelligence (AI) engines, access to such tools is easier than ever. So, if you want to know the derivative of some function, you can just ask your favourite AI engine via your smartphone, and you’ll (most likely) get the correct answer.

All of this raises a valid question: why bother learning all those tedious methods?

Calculus is not only about calculations

One of the reasons is that differentiation and integration involve much more than the algorithms. Like all mathematical operations, they have a meaning and applications. As a matter of fact, a huge number of applications: basically all physics and engineering relies on them.

While it’s true that one doesn’t need to know how to compute derivatives or integrals in order to understand their meaning, it is quite hard to do it without wrestling with some exercises. They are a great way to internalize and create a solid understanding.

For instance, by learning some of the algorithmic rules, we’ll also learn that:

• Integration by substitution is just the chain rule for derivatives, read backwards.
• Integration by parts is the product rule for derivatives, just written differently.

Even the quintessential tedious task, that of reading (and eventually) memorizing the tables, can teach valuable lessons, such as:

• A table of integrals is basically a table of derivatives read backwards.
• Derivatives destroy information, hence the arbitrary constant in the table of integrals.

Other concepts that are difficult to grasp without “getting your hands dirty” are:

• Integrals and derivatives can be estimated.
• Mathematics can manipulate objects more complicated than numbers.

Perhaps more importantly, if you look deeper, you’ll see that studying calculus teaches you more than just mathematics. For instance, solving calculus problems shows you that:

• Some questions have a clear, uncontroversial and verifiable answer.
• You can find out the solution to complicated-looking problems by yourself, using nothing more than pen and paper.

Conclusion

AI is here to stay, and it is a wonderful tool for many applications. But it comes with some dangers too. One of the main criticisms that AI engines receive from the scientific and teaching communities is that they work as black boxes: they spit out results and encourage users to accept them without further scrutiny.

Being able to reach conclusions with pen and paper (quite the opposite of a black box) is certainly not less important now than before. It is a crucial skill for responsibly using AI as a learning companion.

Education is not only about getting the right answers. Education is also about learning how to ask the right questions, to reason clearly and to trust your own judgment.

Technology can solve problems for you. Calculus teaches that, at least some of them, you can solve yourself. That kind of confidence, curiosity, and mental discipline is something no black box can give you.

You earn it.