I think that everyone in the world could be a math person if they wanted to.
我认为只要大家想成为数学家 任何人都能成功
The keyword though, I want to say, is if they wanted to.
然而我想强调的是如果他们愿意
That said, I do think that everyone in America could benefit from having that mathematical
简言之 我真的认为每个美国人都能因有一定的数学背景而受益匪浅
background in reasoning just to help everyone make very good decisions.
理由仅仅是它可以帮助人们做出更好地选择
And here I'm distinguishing already between math as people usually conceive of it,
在此我已经区分了通常情况下人们理解的数学的意义

and decision making and analysis, which is actually what I think math is.
和我认为的数学，即数学是用来做决策和分析的
So, for example, I don't think that being a math person means that
例如我认为数学达人并不意味着
you can recite the formulas between the sines, cosines, tangents and to use logarithms and exponentials interchangeably.
你能够背诵正弦余弦正切之间的切换公式和对数和指数的互换公式
That's not necessarily what I think everyone should try to concentrate to understand.
我认为没必要每个人都要理解那些公式
The main things to concentrate to understand are the mathematical principles of reasoning.
我们需要集中精力去理解的是数学的推理原则
But let me go back to these sines, cosines and logarithms.
但让我们回到正弦余弦和对数
Well actually they do have value.
它们确实有它们存在的价值
What they are is that they are ways to show you how these basic building blocks of reasoning
它们很重要的原因在于它们给我们展示了基本的推理模块
can be used to deduce surprising things or difficult things.
是如何推理出令人惊讶的或复杂事物的方法
In some sense they're like the historical coverages of the triumphs of mathematics,
从某些方面它们很像是数学成功的历史保障
so one cannot just talk abstractly about “yes let's talk about mathematical logic”,
所以人们不能仅仅是抽象地说 好让我们来谈论一下数学逻辑吧
it's actually quite useful to have case studies or stories, which are these famous theorems.
这些着名的定理真的是非常有用的研究案例或故事
Now, I actually think that these are accessible to everyone.
现在 我觉得这对大家来说是可以理解的
I think that actually one reason mathematics is difficult to understand is actually because of that network of prerequisites.
我认为数学很难理解的真正原因在于它网状的预备知识
You see, math is one of these strange subjects for which the concepts are chained in sequences of dependencies.
数学是按概念相关性顺序被连接起来的奇怪学科之一
When you have long chains, there are very few starting points—very few things I need to memorize.
当你有很长的网链时出发点就会很少 需要记住的东西也会非常少
I don't need to memorize, for example, all these things in history such as “when was the war of 1812?”
我不需要记忆 例如 所有这些东西从历史的角度来说就好比1812年的战争是什么时候发生的
Well actually I know that one, because that's a math fact—it was 1812—but I can't tell
我当然知道 因为这是一个数学事实 就是1812年啊 但我不能告诉你
you a lot of other facts, which are just purely memorized.
一些别的什么事实 这只是单纯的被记住了
In mathematics you have very few that you memorize
在数学中你需要记忆的东西很少
and the rest you deduce as you go through, and this chain of deductions is actually what's critical.
其余的都是你通过推理得出的 这个学习链的简化实际上是至关重要的
Now, let me contrast that with other subjects like say history.
现在 让我来将数学和其它学科对比一下 譬如历史
History doesn't have this long chain, in fact if you fully understand the war of 1812
历史没有很长的网链关联 事实上如果你完全了解1812年的战争
that's great, and it is true that that will influence perhaps your understanding later of the women's movement,
当然很好也许你后来才了解到它确实影响了女权运动
but it won't to be as absolutely prerequisite.
但这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;
在这个意义上如果你考虑概念 事实上我觉得历史的概念多于数学
it's just that they're spread out broader and they don't depend on each other as strongly.
只是历史的概念流传更广 而且它们不像数学概念一样概念之间的相互依赖性很强
So, for example, if you miss a week you will miss the understanding of one unit,
例如你错过一节历史课 那么你只是理解不了这个单元
but that won't stop you from understanding all of the rest of the components.
这并不会影响你理解历史的其它模块
So that's actually the difference between math and other subjects in my head.
所以那就是我所理解的数学与其他学科之间的不同之处
Math has fewer concepts but they're chained deeper.
数学较其他学科概念较少 但是概念之间的联系更加密切
And because of the way that we usually learn when you had deep chains it's very fragile
由于我们通常的学习方法的问题 当你对数学的学习进入到深层次的关联理解时 这个环节是很脆弱的
because you lose any one link—meaning if you miss a few concepts along the chain you can actually be completely lost.
因为任何一个环节的丢失 也就是说如果你漏掉了链中很小的一个概念你就完全跟不上进度了
If, for example, you're sick for a week, or if your mind is somewhere else for a week,
例如如果你生病了一个星期或者你一个星期上课都心不在焉
you might make a hole in your prerequisites.
你的预备知识就可能有一个大漏洞了
And the way that education often works where it's almost like riding a train from a beginning to an end,
教育的方式就好比在驾驶一辆火车
well it's such that if you have a hole somewhere in your track the train is not going to pass that hole.
就好比如果你的轨道中有了一个窟窿火车就无法通过这个窟窿了
Now, I think that the way to help to address this is
现在 我认为处理这个问题的方法就是
to provide a way for everyone to learn at their own pace and in fact to fill in the holes whenever they are sensed.
使每个人以他们自己的学习进度学习 一旦他们发现哪儿有漏洞就要及时填补上
And I actually feel like if everyone was able to pick up every one of those prerequisites as necessary,
我想如果每个人都能捡起每一个必需的预备知识
filling in any gap they have, mathematics would change from being the hardest subject to the easiest subject.
填补他们有的每一个漏洞 数学将会从最难的学科变成最简单的学科
I think everyone is a math person, and all that one has to do is to go through the chain
我认为每个人的数学都可以很棒 我们所要做的就是将链子串起来
and fill in all the gaps, and you will understand it better than all the other subjects actually.
填补所有漏洞 这样你就会发现实际上它比任何学科都好学