I think that everyone in the world could be a math person if they wanted to.

我觉得如果人们愿意 那每个人都能成为数学专家

The keyword though, I want to say, is if they wanted to.

不过关键在于“他们愿意”

That said, I do think that everyone in America could benefit from having that mathematical background in reasoning

即便如此 我觉得如果人们有了推理的数学背景 那每个美国人都能从中受益

just to help everyone make very good decisions.

这可以帮助他们做出更明智的决策

And here I'm distinguishing already between math as people usually conceive of it, and decision making and analysis, which is actually what I think math is.

在这里 我已经把人们通常认为的数学和我所认为的数学——决策制定和分析区分开了

So, for example, I don't think that being a math person means that you can recite the formulas between the sines, cosines, tangents

比如 我觉得数学专家并不意味着能够背出正弦 余弦 正切之间的公式

and to use logarithms and exponentials interchangeably.

也不是能够交换使用对数和指数

That's not necessarily what I think everyone should try to concentrate to understand.

那并不是我认为大家应该弄懂的东西

The main things to concentrate to understand are the mathematical principles of reasoning.

主要应该弄懂的是推理的数学原理

But let me go back to these sines, cosines and logarithms.

不过我再说一下正弦 余弦和对数

Well actually they do have value.

它们确实也有价值

What they are is that they are ways to show you how these basic building blocks of reasoning can be used to deduce surprising things or difficult things.

它们的作用就是告诉你 如何利用这些推理的基本模块去推论出令人惊讶的东西或很难的东西

In some sense they're like the historical coverages of the triumphs of mathematics,

在某种程度上 它们就像数学成就的历史综合内容

so one cannot just talk abstractly about “yes let's talk about mathematical logic”,

人们不能只是抽象地说“嗯 我们来讨论一下数学逻辑吧”

it's actually quite useful to have case studies or stories, which are these famous theorems.

使用案例学习或故事还是相当有效的 这些案例或故事就是著名的原理

Now, I actually think that these are accessible to everyone.

而我觉得任何人都可以接触这些

I think that actually one reason mathematics is difficult to understand is actually because of that network of prerequisites.

数学之所以难以理解 其中一个原因就是先决条件之间的关联性

You see, math is one of these strange subjects for which the concepts are chained in sequences of dependencies.

数学是一种比较奇怪的科目 它的概念都是以序列相关性连接在一起

When you have long chains there are very few starting points—very few things I need to memorize.

如果序列链很长 那么起始点就会很少——我只需要记很少的东西

I don't need to memorize, for example, all these things in history such as “when was the war of 1812?”

比如 我不需要记住历史上的所有内容 比如“1812年战争是什么时候？”

Well actually I know that one, because that's a math fact—it was 1812—but I can't tell you a lot of other facts, which are just purely memorized.

其实我知道这个 因为那是一个数学事实——是1812年——但我不能告诉你很多单纯依靠记忆的其他事实

In mathematics you have very few that you memorize and the rest you deduce as you go through,

在数学上 你需要记的东西很少 其他内容都是一步步推论出来的

and this chain of deductions is actually what's critical.

这个推论链才是最重要的

Now, let me contrast that with other subjects like say history.

现在 我们把它跟其他学科比如历史比较一下

History doesn't have this long chain, in fact if you fully understand the war of 1812 that's great,

历史没有这样的长链 其实 如果你完全理解1812年战争 那很好

and it is true that that will influence perhaps your understanding later of the women's movement, but it won't to be as absolutely prerequisite.

或许它会影响你后来对妇女运动的理解 但它并不是绝对的先决条件

In the sense that if you think about the concepts I actually think that history has more concepts than mathematics;

如果你想想这些概念 其实我觉得历史的概念比数学还多

it's just that they're spread out broader and they don't depend on each other as strongly.

只不过它们是广泛分布的 并且互相之间没有强烈的依赖关系

So, for example, if you miss a week you will miss the understanding of one unit,

比如 如果你一周没来上课 你会漏掉一个单元的内容

but that won't stop you from understanding all of the rest of the components.

但它不会妨碍你理解其他知识

So that's actually the difference between math and other subjects in my head.

我认为这就是数学和其他科目之间的区别

Math has fewer concepts but they're chained deeper.

数学的概念更少 但它们之间的连接更为紧密

And because of the way that we usually learn when you had deep chains it's very fragile because you lose any one link—

因为我们日常的学习方式 当连接更为紧密时 它就会更脆弱 如果你漏掉其中一个环节

meaning if you miss a few concepts along the chain you can actually be completely lost.

也就是说你错过了这条链上的一些概念 你很可能就完全摸不着北了

If, for example, you're sick for a week, or if your mind is somewhere else for a week, you might make a hole in your prerequisites.

比如 你请了一个星期的病假 或者开了一星期的小差 那你的先决条件就可能出现漏洞

And the way that education often works where it's almost like riding a train from a beginning to an end,

教育的运作方式就像是把一列火车从起点开到终点

well it's such that if you have a hole somewhere in your track the train is not going to pass that hole.

如果轨道上出现了一个漏洞 那么火车就很难越过那个洞

Now, I think that the way to help to address this is to provide a way for everyone to learn at their own pace and in fact to fill in the holes whenever they are sensed.

我觉得解决这一点的方法就是 对每个人因材施教 根据其自己的进度安排学习 一旦意识到哪里有漏洞 及时进行补救

And I actually feel like if everyone was able to pick up every one of those prerequisites as necessary, filling in any gap they have,

我觉得 如果每个人都能补齐这些必要的先决条件 及时填充漏洞

mathematics would change from being the hardest subject to the easiest subject.

那么数学就能从最难学的科目变成最简单的科目

I think everyone is a math person, and all that one has to do is to go through the chain and fill in all the gaps,

我相信每个人都擅长数学 人们需要做的就是在这条链上搜寻并填补漏洞

and you will understand it better than all the other subjects actually.

那么你的数学会比其他任何科目都好