Honest and Realistic Guides for Learning

Summary: Check out theCalculus Learning Guide. I'm sharing an honest, realistic learning plan for what helped me enjoy the subject.

如果你想要一条不期待完美动力的道路，在几分钟内(而不是几周内)分享你的见解，并以终身为目标，这本指南适合你。

First Principles of Education

My learning strategy is to ask honest (sometimes uncomfortable) questions about what's really working.

No games, no kidding ourselves, just:

我正在学习的概念合不合?
如果没有，我能找到更好的解释吗?(让我们一起分享吧。)

Here's my wishlist for a learning guide. Elon Musk talks about thinking fromfirst principles, starting with fundamental truths and working forward from there¹. Who cares what's being done now, what's our goal?

Principle: Avoid teaching hatred of a subject

Priority #1 for any class is:Do not create hate for the subject.

Imagine 99% of people in a skiing class never ski again. They cringe at the thought. We wouldn't console ourselves thinking "Oh, skiing teaches important physical skills that apply to other fields." We'd think "That skiing class is awful and needs to change."

Sure, not everyone willlove滑雪(或烹饪，或数学)，但他们不应该讨厌它。暂时的理解不值得永远的厌恶。

那么，是什么微积分的介绍让我兴奋的想要学习更多呢?

For me, it was seeing how patterns can be cleverly split and re-assembled:

Most courses march you through weeks of theory "appreciate" these diagrams in week 11. Ugh. The big picture helps meappreciate the details, not the reverse.

Principle: Give Realistic Advice

A typical discussion:

"I want to learn Calculus. What should I do?"

"Here's a [full book/course/MOOC]. It's months of effort, I didn't do it myself, but here you go."

In other words, "go the library and read for 100 hours". The real question:

I'm interested in the subject. Is there a plan that worked for you?

Motivation is limited. Traditional classes "work" -- because students are under immense pressure to finish (tuition, peer pressure, fear of not graduating).

Online courses without this pressure havesingle-digit completion rates. We can pretend students "got something" from the experience, just like you "got something" from a movie you walked out of. We can't change the goalposts to "something is better than nothing" halfway through.

Realistic advice on what worked with mylimitedmotivation (even as a math hobbyist!) is:

得到一个啊!moment in minutesthat motivates me to keep going (a cool diagram, example, or simulation).
Take a progressive journeywhere even if I stop after an hour, I have some helpful insights (vs. an hour of stretching in the parking lot).
Maintain a desire to revisit the subjectby having an approachable, gentle introduction. I'll then keep coming back to fill in gaps over time.

Principle: Don't Ignore Difficulties

For fun, find a lesson on imaginary numbers.

Does it acknowledge negative numbers were also distrusted?
Is the name "imaginary" described as an insult, given by people who didn't understand the concept?
Does the teacher mention their own confusion? (Or did imaginary numbers just click?)
Is there a real-world application? (If not, is this because it truly doesn't exist, we haven't tried to look, or it isn't important for learning?)

This type of lesson is a giant pet peeve. The flow is "Here's a confusing concept. I was confused myself, but I won't tell you that. Memorize the definition, apply it in these practice problems, and we'll call it a day."

啊，这把我逼疯了。它强化了人们的刻板印象，即数学课是一个移动符号的游戏。(这个符号乘以另一个符号等于-1。大作。)

It's ok to lack an intuition; I lack it for most things. But hiding our initial confusion implies the subject isn't confusing.

Principle: Expect to get it wrong sometimes

There's a common trope of the smart-aleck student trying to "outsmart" the teacher. Do basketball players try to "outsmart" their coach?

The flawed assumption is teachers must be some omniscient authority giving you access to precious knowledge. The knowledge is out there, it's not like the teacher invented the math herself. Instead, imagine acoachwho is trying to improve your understanding.

Coaches can be wrong, sure. But they've seen many struggle with the same issues you're facing, and are trying to help. It's ok if Lebron James can dunk better than his coach.

The math may be perfect and unchanging, but theway it's taughtis not. Let's make it easy to improve lessonsandnot expect perfection the first time.

Principle: We have Different Goals

Most courses assume you want mastery of the subject. That's fine, but is it necessary?

There are several levels of music understanding:

Intuitive Appreciation:只是在享受音乐。
Natural Description:Humming a tune you heard or made up.
Symbolic Description:读和写乐谱。
Theory:Explaining how harmonies work, why minor scales are somber, etc.
Performance:演奏官方乐器。

In language learning, there is anILR scalefrom no profiency to native fluency. Not everyone studying Calculus needs to become Isaac Newton. Can we have a path that goes as far as we need?

An Honest and Realistic Learning Plan

Combining these insights, I've made aCalculus Learning Guide.

The principles, as I tried to apply them:

It's honest.It's the explanation that actually inspired me, not the theoretical explanation that requires weeks of discipline for some future payoff.
它承认动机有限。How far can you get in 1 minute? 10 minutes? An hour? Pretty far, I think. And getting a win in 10 minutes means you'll come back for more.
It's updatable.With lessons basedprimarily对于文本，我们可以很容易地更新、重新排列、添加、编辑、修复。其他格式基本上就是我们第一次就能成功的一个赌注。
It acknowledges levels of understanding.Most people just want an appreciation for Calculus. Technical performance is a goal we can separate, organize, and build a path to.
I eat the veggies myself.This guide has "gut checks" like "Can I describe an integral in everyday terms?" and "Can I derive the product rule on my own?". This is how I actually refresh my Calculus understanding.

In my ideal world, every Wikipedia topic would have aguide这将使您从1分钟的版本到全面的技术理解。想走多远就走多远，每一步都取得有意义的进步，一路都有乐趣。

Happy math.

马斯克提到的不是“类比推理”，也不是根据另一种情况下发生的事情来假设一个结论是正确的。这不同于“类比理解”，即获得要点，然后进行技术版本的研究。The analogy is a raft to cross the river, to be left behind once you're on land.↩