A Gentle Introduction To Learning Calculus

I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education.

微积分以一种优雅的、令人挠头的方式联系主题。我最接近的类比是达尔文的进化论:一旦理解了，你就开始从生存的角度来看待自然。你知道为什么药物会导致耐药细菌(适者生存)。你知道为什么糖和脂肪尝起来是甜的(鼓励人们在稀缺时期食用高热量食物)。这些都是相互关联的。

Calculus is similarly enlightening. Don’t these formulas seem related in some way?

They are. But most of us learn these formulas independently. Calculus lets us start with $\text{circumference} = 2 \pi r$ and figure out the others — the Greeks would have appreciated this.

Unfortunately, calculus can epitomize what’s wrong with math education. Most lessons feature contrived examples, arcane proofs, and memorization that body slam our intuition & enthusiasm.

It really shouldn’t be this way.

Math, art, and ideas

I’ve learned something from school:Math isn’t the hard part of math; motivation is.Specifically, staying encouraged despite

Teachers focused more on publishing/perishing than teaching
Self-fulfilling prophecies that math is difficult, boring, unpopular or “not your subject”
Textbooks and curriculums more concerned withprofitsand test results than insight

‘A Mathematician’s Lament’ [pdf]is an excellent essay on this issue thatresonated with many people:

“…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”

Imagine teaching art like this:孩子们，在幼儿园不能画手指画。相反，让我们来学习绘画化学、光的物理和眼睛的解剖学。12年后，如果孩子们(现在是青少年)已经不讨厌艺术了，他们可能会开始自己着色。毕竟，他们有“严格的、可测试的”基础来开始欣赏艺术。Right?

Poetry is similar. Imagine studying this quote (formula):

This above all: to thine own self be true, And it must follow, as the night the day, Thou canst not then be false to any man. — William Shakespeare, Hamlet

It’s an elegant way of saying “be yourself” (and if that means writing irreverently about math, so be it). But if this were math class, we’d be counting the syllables, analyzing the iambic pentameter, and mapping out the subject, verb and object.

数学和诗歌是指向月亮的手指。不要把手指和月亮混淆。Formulas are ameans to an end，一种表达数学真理的方式。

We’ve forgotten that math is about ideas, not robotically manipulating the formulas that express them.

Ok bub, what’s your great idea?

Feisty, are we? Well, here’s what I won’t do: recreate the existing textbooks. If you need answersright awayfor that big test, there’s plenty ofwebsites,class videosand20-minute sprintsto help you out.

Instead, let’s share the core insights of calculus. Equations aren’t enough — I want the “aha!” moments that make everything click.

形式数学语言只是交流的一种方式。图表、动画和简单的对话通常比一页的证明更能提供深刻的见解。

但是微积分太难了!

I think anyone can appreciate the core ideas of calculus. We don’t need to be writers to enjoy Shakespeare.

It’s within your reach if you know algebra and have a general interest in math. Not long ago, reading and writing were the work of trained scribes. Yet today that can be handled by a 10-year old. Why?

因为我们期待它。期望在可能性中扮演着重要的角色。Soexpectthat calculus is just another subject. Some people get into the nitty-gritty (the writers/mathematicians). But the rest of us can still admire what’s happening, and expand our brain along the way.

It’s about how far you want to go. I’d love for everyone to understand the core concepts of calculus and say “whoa”.

So what’s calculus about?

Somedefine calculusas “the branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables”. It’s correct, but not helpful for beginners.

Here’s my take: Calculus does to algebra what algebra did to arithmetic.

Arithmeticis about manipulating numbers (addition, multiplication, etc.).
Algebra finds patterns between numbers: $a^2 + b^2 = c^2$是一个著名的关系式，用来描述直角三角形的两条边。Algebra finds entire sets of numbers — if you know a and b, you can find c.
Calculus finds patterns between equations: you can see how one equation ($\text{circumference} = 2 \pi r$) relates to a similar one ($\text{area} = \pi r^2$).

Using calculus, we can ask all sorts of questions:

How does an equation grow and shrink? Accumulate over time?
When does it reach its highest/lowest point?
我们如何使用不断变化的变量?(Heat, motion, populations, …).
And much, much more!

Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them.就像进化论一样，微积分扩展了你对自然如何运作的理解。

An Example, Please

Let’s walk the walk. Suppose we know the equation for circumference ($2 \pi r$) and want to find area. What to do?

要知道，一个填充的圆盘就像一套俄罗斯玩偶。

Here are two ways to draw a disc:

Make a circle and fill it in
Draw a bunch of rings with a thick marker

在每种情况下，“空间”(面积)的数量应该是相同的，对吗?一个环需要多少空间?

Well, the very largest ring has radius “r” and a circumference $2 \pi r$. As the rings get smaller their circumference shrinks, but it keeps the pattern of $2 \pi \cdot \text{current radius}$. The final ring is more like a pinpoint, with no circumference at all.

现在事情开始变得奇怪了。Let’s unroll those rings and line them up.What happens?

我们得到一堆线，形成一个锯齿三角形。但是如果我们使用更薄的圆环，三角形就会变得不那么锯齿状(在以后的文章中会有更多的介绍)。
One side has the smallest ring (0) and the other side has the largest ring ($2 \pi r$)
我们有半径为0到r的环。对于每个可能的半径(0到r)，我们只需将展开的环放在那个位置。
“环三角形”的总面积= $\frac{1}{2} \text{base} \cdot \text{height} = \frac{1}{2} (r) (2 \pi r) = \pi r^2$，这就是面积的公式!

Yowza! The combined area of the rings = the area of the triangle = area of circle!

(Image from Wikipedia)

This was a quick example, but did you catch the key idea? We took a disc, split it up, and put the segments together in a different way. Calculus showed us that a disc and ring are intimately related: a disc is really just a bunch of rings.

This is a recurring theme in calculus:Big things are made from little things.And sometimes the little things are easier to work with.

A note on examples

Many calculus examples are based on physics. That’s great, but it can be hard to relate: honestly, how often do you knowthe equation for velocity一个对象呢?如果是的话，一周不到一次。

我更喜欢从物理的、视觉的例子开始，因为这是我们的大脑工作的方式。我们做的那个戒指/圆圈?你可以用几个管道清洁器构建它，将它们分开，并将它们拉直成一个粗糙的三角形，看看数学是否真的有效。That’s just not happening with your velocity equation.

A note on rigor (for the math geeks)

我能感觉到那些数学迷们在敲键盘。关于“严谨”我只说几句。

你知道我们学微积分的方式不是牛顿和莱布尼茨发现它的方式吗?They used intuitive ideas of “fluxions” and “infinitesimals” which were replaced with limits because“Sure, it works in practice. But does it work in theory?”.

We’ve created complex mechanical constructs to “rigorously” prove calculus, but have lost our intuition in the process.

我们从脑化学的角度来看待糖的甜味，而不是把它看作是大自然在说“这有很多能量。”吃它。”

I don’t want to (and can’t) teach an analysis course or train researchers. Would it be so bad if everyone understood calculus to the “non-rigorous” level that Newton did? That it changed how they saw the world, as it did for him?

A premature focus on rigor dissuades students and makes math hard to learn. Case in point: e is technically defined by a limit, but theintuition of growth它是如何被发现的。The natural log can be seen as an integral, or the2022世界杯预选赛 . Which explanations help beginners more?

Let’s fingerpaint a bit, and get into the chemistry along the way. Happy math.

(PS: A kind reader has created ananimated powerpoint slideshowthat helps present this idea more visually (best viewed in PowerPoint, due to the animations). Thanks!)

Note: I’ve made an entire intuition-first calculus series in the style of this article:

//www.i494.com/calculus/lesson-1