Let's start with the language you use all day every day.

You get by, right?

Think back. How did you pick it up?

You learned all about phrasal verbs, for instance? The definition, another name, and a few examples?

No? Nothing?

Then I guess you do without phrasal verbs, and they stump you if you run into them?

Try this quiz. Here's six phrasal verbs. How many can you figure out?

Get by. Think back. Pick up. Do without. Run into. Figure out.

How many did you get?

All six?

No! How is that possible?

Likewise calculus. Calculus is a language. All math is. Maybe you want to learn it the way you learned your first language.

Remember learning your first language? No, you don't. What does that tell you?

Watch for a two-year-old to say "I'm hungry" for the first time. She learns to say "I'm hungry" before she can read or write it, and when predicates are years in the future, along with contractions and pronouns and conjugations of the irregular verb 'to be.' Meanwhile Cheerios are coming right up.

A cheerio, from inside:


If you're learning Spanish now, I hope your Spanish teacher takes this approach. You'll feel the difference between "estoy enfermo" and "soy enfermo" long before you know the grammar rules for Ser and Estar verbs.

Don't take it from me. Here's a Calculus teacher who takes this approach:

Arithmetic is learned intuitively in elementary school and then the logic of it is learned gradually through the work in algebra. Geometry is learned intuitively in junior high school and then the formal deductive approach is presented in senior high school. A difficult subject such as the calculus, therefore, should certainly be introduced by an intuitive approach.

Intuitive means 'by way of examples from physics, often the physics of everyday life.'

This is Morris Kline's Introduction to Calculus: An Intuitive Approach.

In my opinion, a rigorous first course in the calculus is ... too difficult for the students. Beginners are asked to learn a mass of concepts so subtle that they defied the best mathematicians for two hundred years.

Every pedagogue today champions discovery, but few teach it. How does one discover in mathematics? By thinking in physical and geometrical terms....

The intuitive approach may lead to errors ... but "truth emerges more readily from error than from confusion." The student must be allowed to make mistakes, for if he makes no mistakes, he will not progress.

But where do you see everyday examples, everyday physics?

Suppose you hike, or run mountain races, or cycle through hill country.

Someone invites you to cycle trail B. It's only twenty miles, he says, but at a tough seven degree angle.

Four Courses

No way is that a 7 degree angle, you think. A grade of 7 percent is an angle of 4 percent. The grade is the tangent of the angle, given in percent.

And a trail is rated by its steepest grade, not the average. 7 percent is an average of all the ups (positive slopes) and downs (negative slopes). For runners the down slopes are tough in a different way, and use different muscles, and pound the hip and knee and ankle joints. For cyclists they are effortless (wasted?) miles.

Nor is this twenty miles except on a map, as seen from overhead (D here). How far do you actually cycle? A lot more than twenty miles.

Without calculus, you would approximate the distance along B with right triangles. From map distance and altitude you'd figure the hypotenuses and add those.

Four Triangles

More triangles would get you a closer estimate.

More Triangles

Infinitely many triangles would get you even closer, of course, but who could do all that repetitive arithmetic?

What if there were a pattern in all that repetitive arithmetic, and we could jump to the end of it?

Down that road comes calculus.

If the forces that made this terrain can be modeled in a formula, or approximated by some formula, then figure the integral. You don't care about the area under the curve, you're accumulating distance along the curve (arc length). Arc length example here.

Kline again:

... calculus divorced from applications is meaningless.

... the rigorous approach is misleading. Because the introductory calculus course is the student’s first contact with higher mathematics, he obtains the impression that real mathematics is deductive and that good mathematicians think deductively.

You know the feeling. You're given a formula that could easily pass for an elvish inscription found in some tower or cave.

Elvish Magic

You shrug. "OK, I'll take your word for it. One more thing to memorize. Who am I to question the elves?"

I'm testing a python script that reads a problem in LaTeX notation and tells you whether to differentiate using L'Hopital's Rule or integrate by partial fractions. It's just more elvish magic. It's like learning Spanish by studying Spanish grammar, or typing some Spanish into the search box at Try having any kind of conversation that way.

Problems don't come in that form except in a math classroom. Scripts and computers can only do the mechanical parts of math. You're needed where a problem in the world needs an interpretation from physics ("Oh, this is fluid dynamics") and then some approximation in the language of math.

The elves are lost until you come for them.