How to Develop a Mindset for Math

数学使用编造的规则来创建模型和关系。When learning, I ask:

Whatrelationshipdoes this model represent?
What real-world itemsshare this relationship?
Does that relationshipmake sense to me?

这些都是简单的问题，但它们能帮助我理解新话题。If you liked mymath posts, this article covers my approach to this oft-maligned subject. Many people have left insightful comments about their struggles with math and resources that helped them.

Math Education

Textbooksrarely关注理解;它主要是用“即插即用”的公式来解决问题。It saddens me that beautiful ideas get such a rote treatment:

ThePythagorean Theoremis not just about triangles. It is about the relationship between similar shapes, the distance between any set of numbers, and much more.
e isnot just a number. It is about the fundamental relationships between all growth rates.
The2022世界杯预选赛 is not just an inverse function. It is about the amount of time things need to grow.

Elegant, "a ha!" insights should be our focus, but we leave that for students to randomly stumble upon themselves. I hit an "a ha" moment after a hellish cram session in college; since then, I've wanted to find and share those epiphanies to spare others the same pain.

But it works both ways -- I want you to share insights with me, too. There's more understanding, less pain, and everyone wins.

Math Evolves Over Time

I consider math as a way of thinking, and it's important to seehowthat thinking developed rather than only showing the result. Let's try an example.

Imagine you're a caveman doing math. One of the first problems will behow to count things. Several systems have developed over time:

No system is right, and each has advantages:

Unary system:Draw lines in the sand -- as simple as it gets. Great for keeping score in games; you can add to a number without erasing and rewriting.
Roman Numerals:More advanced unary, with shortcuts for large numbers.
Decimals:巨大的意识到数字可以使用带有数位和零的“位置”系统。
Binary:Simplest positional system (two digits, on vs off) so it's great for mechanical devices.
Scientific Notation:Extremely compact, can easily gauge a number's size and precision (1E3 vs 1.000E3).

Think we're done? No way. In 1000 years we'll have a system that makes decimal numbers look as quaint as Roman Numerals ("By George, how did they manage with such clumsy tools?").

Negative Numbers Aren't That Real

Let's think about numbers a bit more. The example above showsour number system is one of many ways to solve the "counting" problem.

罗马人会认为零和分数很奇怪，但这并不意味着“虚无”和“部分对整体”不是有用的概念。但是看看每个系统是如何整合新思想的。

分数(1/3)、小数(.234)和复数(3 + 4i)是表达新关系的方式。它们现在可能没有意义，就像零对罗马人来说没有“意义”。我们需要新的现实世界的关系(比如债务)来让他们产生共鸣。

Even then, negative numbers may not exist in the way we think, as you convince me here:

负数是个好主意，但它本身并不存在。It's a label we apply to a concept.

Me: Sure they do.

You: Ok, show me -3 cows.

Me: Well, um... assume you're a farmer, and you lost 3 cows.

You: Ok, you have zero cows.

Me: No, I mean, you gave 3 cows to a friend.

You: Ok, he has 3 cows and you have zero.

Me: No, I mean, he's going to give them back someday. He owes you.

You: Ah. So the actual number I have (-3 or 0) depends on whether I think he'll pay me back. I didn't realize my opinion changed how counting worked. In my world, I had zero the whole time.

Me: Sigh. It's not like that. When he gives you the cows back, you go from -3 to 3.

他返回3头奶牛，我们从-3跳到3?还有什么我应该知道的新算术吗?What doessqrt(-17)cows look like?

Me: Get out.

Negative numbers canexpress a relationship:

Positive numbers represent a surplus of cows
Zero represents no cows
Negative numbers represent a deficit of cows that are assumed to be paid back

但负数“并不存在”——只有它们所代表的关系(奶牛的盈余/赤字)。我们创建了一个“负数”模型来帮助记账，即使你不能在你的手中持有-3头牛。(我有目的地对“负数”的含义进行了不同的解释:它是一种不同的计数系统，就像罗马数字和小数是不同的计数系统一样。)

By the way, negative numbersweren't acceptedby many people, including Western mathematicians, until the 1700s. The idea of a negative was considered "absurd". Negative numbersdo看起来很奇怪，除非你能看到它们是如何代表复杂的现实关系的，比如债务。

Why All the Philosophy?

我意识到我的心态是学习的关键。**It helped me arrive at deep insights, specifically:

事实知识不是理解。Knowing "hammers drive nails" is not the same as the insight that any hard object (a rock, a wrench) can drive a nail.
Keep an open mind.Develop your intuition by allowing yourself to be a beginner again.

A university professor went to visit a famous Zen master. While the master quietly served tea, the professor talked about Zen. The master poured the visitor's cup to the brim, and then kept pouring. The professor watched the overflowing cup until he could no longer restrain himself. "It's overfull! No more will go in!" the professor blurted. "You are like this cup," the master replied, "How can I show you Zen unless you first empty your cup."

是创造性的。Look for strange relationships. Use diagrams. Use humor. Use analogies. Use mnemonics. Use anything that makes the ideas more vivid. Analogies aren't perfect but help when struggling with the general idea.
Realize you can learn.我们希望孩子们学习代数、三角和微积分，这将震惊古希腊人。我们应该这样做:如果解释正确，我们有能力学到很多东西。不要停止，直到它是合理的，否则数学差距会困扰你。心理韧性是至关重要的——我们常常太容易放弃。

So What's the Point?

I want to share what I've discovered, hoping it helps you learn math:

Math createsmodelsthat have certainrelationships
We try to findreal-world phenomenathat have the same relationship
Our models arealways improving. A new model may come along that better explains that relationship (roman numerals to decimal system).

Sure, some modelsappearto have no use:"What good are imaginary numbers?", many students ask. It's a valid question, with anintuitive answer.

虚数的使用受到我们想象和理解的限制——就像负数“没用”，除非你有债务的想法，虚数可能会令人困惑，因为我们不真正理解它们所代表的关系。

数学提供了模型;理解它们之间的关系，并将它们应用到现实世界的对象中。

Developing intuition makes learning fun -- evenaccountingisn't bad when you understand the problems it solves. I want to covercomplex numbers，微积分和其他难以捉摸的主题的重点是关系，而不是证明和力学。

但这是我的经验——怎样才能学得最好?A few friends have written up their experience:

Math Education

Math Evolves Over Time

Negative Numbers Aren't That Real

Why All the Philosophy?

So What's the Point?

Other Posts In This Series

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