The Derivative, Or Why People Hate Math

March 18, 2012 § Leave a comment

Derivative formula, the word boring superimposed

This is a post on math for the non-math-inclined. It’s a study of motion, because numbers are boring.

I heard about something called open-source textbooks, where teachers wanted to provide free high-quality textbooks online to the entire world. This is great — math doesn’t change, for example, and good teaching doesn’t change, so why should there be a booming business for math textbooks? I’m not here to talk about open-source textbooks, though. They’re just the set-up.

I looked for an open-source textbook on calculus and opened it to its explanation of the derivative. The derivative is a good litmus test for a teacher or a textbook, since it’s a simple thing that can be explained in a very complicated way, and has been since it was invented.

Here’s what I found. This is the first picture on the derivative in the textbook I checked. Ask yourself an honest question: does this look simple?

derivative graph

Attribution: Copyright © 2000 by H. Jerome Keisler
(Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License)
http://www.math.wisc.edu/~keisler/calc.html

I’m not ragging on this particular book. As far as I’ve seen, every calculus textbook has a similar explanation of the derivative — something about slope, and tangent, and rise over run, and lots of overlapping lines at different angles.

This is not simple. Why can’t this be simple? Why does a fundamental concept of math, one that applies to every other physical phenomenon on planet Earth, have to be so dry and complicated? Why is math education so plain bad?

So I’m going to show you what the derivative is as simply and truthfully as I can. Because that stuff above, on slopes and tangents, is correct, but not true — in the same way that describing music as sound waves is correct, but not true.

The derivative is about motion. Let’s take ice skating for example. Here’s a sketch (I admit it — it’s mine) of the trails somebody leaves on the ice with their skates. It draws a little picture of their motion. It’s pretty!

ice skating trails

Ice skating

The derivative describes motion at an instant in time. Here I’ve chosen an instant in the skater’s motion. Nothing too fancy, just an ordinary turn:

ice skating trails, red circle around one small curve

An instant of motion

Now we’re going to look close at that instant:

ice skating trail zoomed in

Closer:

ice skating zoomed in further

Closer, darn it!

ice skating zoomed in furthest

What does the skater’s motion look like at that instant in time, when you zoom in close? What shape is this? Take a moment and think about it. Don’t overthink it, but do think.

ice skating zoomed in furthest

What shape is this?

What we’ve got here is a straight line. At this particular instant, if you look really close, the skater was skating in a straight line. And that straight line is the derivative. In other words, the derivative is the straight line of smooth motion when you look really close at a particular instant.

Jagged motion doesn’t have a derivative, because when you zoom in on a jagged instant, it’s not a straight line. Here’s a ball bouncing:

trail of a ball bouncing

Ball bouncing

And when you zoom in on the bounce, it’s not a straight line, it’s V-shaped:

ball bouncing, zoom in on bounce

The instant of the bounce

…So no derivative there. The motion isn’t smooth at that instant.

I don’t understand why calculus is introduced with formulas when it’s the foundation of physical change in the universe. Why isn’t it introduced as an aspect of motion? Newton invented calculus to solve physics problems. Why is it divorced from its natural roots in the classroom? If you hated calculus, or trig, or geometry, or other parts of math because you were forced to understand something physical via numerical formulas, you were right to feel something was off. You were being coerced into unnatural thinking because that’s how the book was written.

You don’t learn to play the piano by memorizing the frequencies of notes. You put your finger on a key, feel its heft, press down and feel it quiet, or loud, learn the difference, hear it quiver and fade. Why don’t we teach kids to get a feel for math?

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