Summary:I’m building a calculus course from the ground-up focused onpermanent intuition, not the cram-test-forget cycle we’ve come to expect.
Update: The course is now live at//www.i494.com/calculus
The Problem: We Never Internalized Calculus
首先，今天的微积分教学有什么问题?(Ha!)
Just look at the results. The vast majority of survivors, the STEM folks who used calculus in several classes, haveno lasting intuition. We memorized procedures, applied them to pre-packaged problems (“Say, fellow, what is the derivative of $x^2$?”), and internalized nothing.
想要证据?没有问题。拿一根绳子，把它紧紧地缠在一枚硬币上。再拿一根绳子紧紧地绕着地球。
Ok. Now, lengthen both strings, adding more to the ends, so there’s a 1-inch gap all the way around around the quarter, and a 1-inch gap all the way around the Earth (sort of like having a ring floating around Saturn). Got it?
Quiz time: Which scenario uses more extra string? Does it take more additional string to put a 1-inch gap around the quarter, or to put a 1-inch gap around the Earth?
Think about it. Ponder it over. Ready? It’s the… same.The same!Adding a 1-inch gap around the Earth, and a quarter, uses the same6.28 inches.
And to be blunt: if you “learned” calculus but didn’t have the answer within 3 seconds, you don’t truly know it. At least not deep down.
Now don’t feel bad, I didn’t know it either. Only one engineer in the dozens I’ve asked came up with the answer instantly, without second-guesses (my karate teacher, Mr. Rose).
This question has a few levels of understanding:
Algebra Robot: Calculating change in circumference:
2*pi*(r + 1) - 2*pi*r = 2*pi. They are the same. Calculation complete.Calculus Disciple: Oh! We know
circumference = 2*pi*r. The derivative is2*pi, a constant, which means the current radius has no impact on a changing circumference.Calculus Zen master我看到了事物的本质。我们改变一维的半径观察一维的周长。一进一出，就像把篱笆加长1英尺:初始尺寸并不会改变需要做的工作。这个缺口可以围绕一个圆、正方形、矩形或理查德·尼克松面具，对于类似的形状也是一样的。(而且，我真傻，我忘了周长的方程!)
We can be calculus warrior-monks, cutting through problems with our intuition. Notice how the most advanced approach didn’t need specific equations — it was just thinking about the problem! Equations are nice tools, but are they your only source of understanding?
你看，根据标准化测试和期末考试，我“懂”微积分——但显然只是初级水平。我并没有马上意识到微积分如何帮助解决一个关于使字符串变长的问题。如果你问一个人钱包里有多少现金，里面有6张\ 20美元的钞票，而他们没有想到用乘法，你会说他们内化了算术吗?(“Oh geez, you didn’t tell me this would be amultiplicationquestion! Could you set up the problem for me?”)
I want you to have the intuition-first calculus class I never did.The goal islastingintuition, shared by an excited friend, and built with the test of “If you haven’t internalized the idea, the material must change.”
How Can We Make Learning Intuitive And Interesting?
逐步细化。You may have seen these two methods to download and display an image:
- Baseline Rendering:Download it start-to-finish in full detail
- Progressive Rendering: Download a blurry version, and gradually refine it
Teaching a subject is similar:
- Baseline Teaching:Cover individual concepts in full-depth, one after another
- Progressive Teaching:See the big picture, how the whole fits together, then sharpen the detail
The “start-to-finish” approach seems official. Orderly. Rigorous. And it doesn’t work.
What, exactly, do you know when you’ve seen the first 20% of a portrait in full resolution? A forehead? Do you even know the gender? The age? The teacher has forgotten thatyou’ve never seen the full pictureand likely can’t appreciate that you’re even seeing a forehead!
Progressive rendering (blurry-to-sharp) gives a full overview, arough approximationof what the expert sees, and gets you curious about more. After the overview, we start filling in the details. And because you have an idea of where you’re going, you’re excited to learn. What’s better: “Let’s download the next 10% of the forehead”, or “Let’s sharpen the picture”?
Let’s admit it: we forget the details of most classes. If we’ll have a hazy memory anyway, shouldn’t it be of the entire picture? That has the best shot of enticing us to sharpen the details later on.
How Do We Know If A Lesson Is Any Good?
With the Pizza Box Test. Imagine you pass a dumpster while walking home. You see a message scrawled on a discarded pizza box. Is the note so insightful and compelling that you’d take the pizza box home to finish reading it?
Ignore the sparkle of a lesson being digital, mobile-friendly, gamified, interactive, or a gesture-based hologram.Would you take this lesson home if it were written on a pizza box?
If yes, great! Clean it up and add in the glitz. But if thecore lessonis not compelling without the trimmings, it must be redone.
Everyone’s “pizza box” standard varies; just have one. Here’s a few things I wish were written on the boxes outside my high school:
Psst! Think ofeas a universal component in all growth rates, just like pi is a universal factor in all circles…
Hey buddy! Degrees are from the observer’s perspective.Radians都是搬运工的。这就是弧度更自然的原因。Let me show you…
Yo!Imaginary numbers是另一个维度，乘以I就是在那个维度上旋转90度!两次旋转，你是朝后的，也就是-1。
我们如何知道什么对学生最好?
通过关注未来你会教给现在的你。
Teachers, like all of us, face external incentives which may interfere with their goals (publish or perish, mandated curriculum, need to impress others with jargon, etc.). The test of “What would future-me teach present-me?” helps me focus on the essentials:
Use the shortest lessons possible. There’s no word count to meet. The same insight in fewer words is preferred.
Use the simplest language possible. It’s future-me talking to current-me. There’s nobody to impress here.
Use any analogy that’s memorable. I’m not embarrassed by “childish” analogies. If a metaphor excites me, and helps, I’m going to use it. Nyah.
Be a friend, not lecturer. I want a buddy, a guide who happened to experience the material before I did, not a pompous schoolmarm I can’t question.
Point out the naked emperor. Most calculus classes cover “limits, derivatives, integrals” in that order because… why? Limits are the most nuanced concept, invented in the mid-1800s. Were mathematicians like Newton, Leibniz, Euler, Gauss, Taylor, Fourier and Bernoulli inadequate because they didn’t use them? (Conversely: are you better than them because you do?). Most courses are too timid (or oblivious) to question the strategy of covering the most elusive, low-level topic first.
Learn for the long haul. The elephant in the room is that most math courses are a stepping-stone to some credential. Future-me doesn’t play that game: he only benefits when current-mepermanentlyunderstands something.
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让我们先来学习微积分的直觉。目标是持久提升你的直觉和类比能力。如果这种情况没有发生，课程就不起作用了，它将会被加强，直到它发生。
Sign up for themailing listand I’ll let you know when the course preview is ready, in November.
Happy math.