如果我让你去想出一个非常难的问题
If I asked you to think of a really tough problem
你首先想到的会是什么?
what is the first thing that comes to mind? Well,
如果你像我一样 很可能
if you’re anything like me you probably thought
你可能想到一个大而复杂的数学方程 对吧?
of some big complex math equation, right?
对于大多数人而言 似乎数学
For most people, math seems like one
是最难的科目之一
of the most difficult subjects out there.
数学是抽象的 复杂的 不幸的是
It’s abstract, it’s complex, and unfortunately
正因为如此 许多人认为
for those reasons a lot of people adopt the
自己不是学数学的料
belief that they’re just not math people.
这显然是不正确的
Which is patently untrue
因为数学和其它学科一样 是一种能够学会的技能
because math is a skill that can be learned just like any other.
但是因为你点击了本视频
But since you clicked on this video
希望你不是这样的人
hopefully you are not one of those people.
希望你至少在某种程度上相信
Hopefully you have at least some degree of belief
你会学好数学的
that you can become better at math and you
而且你愿意学好
have the motivation to do so.
如果你想学好数学 摆在面前的问题是
And if you do, the obvious question is
怎样学好数学?
how do you get better at math? Well,
幸运的是
fortunately,
这是有一个有非常简单答案的问题
this is one of those questions that has a pretty simple answer.
如果你想要学好数学
If you want to get better
你必须做许多许多的数学题
at math you have to do lots and lots of math.
而且问题越难 越好
And the tougher theproblems are, the better.
因为难的问题会拓展你的思维 让你有新突破
Because tough problems will stretch your understanding and lead you to new breakthroughs. But,
然而在学习数学过程中 做这些难题时
in the course of studying math and working through these
最终你都会遇到难住你的问题
tough problems you are eventually going to come to problems that just
你完全没有思路
stump you, that you get completely stuck on.
这时 最重要的是
And when you get to these points it’s
如何最终解决这些问题
important to know how to eventually solve these problems,
因为这些问题真的能够
because these are the ones that are really
拓展 建立你的(数学)技能体系
going to stretch and build your skill set.
这就是我要在本视频中讲的重点
So that is what I want to focus on in this video.
我会告诉你们使用一些实用的技巧
I want to give you practical techniques for working through,
最终会解决这些最初看起来难以攻破的难题
and eventually solving those problems that seem insurmountable at first.
首先 我想要讲讲匈牙利数学家
To start, I want to focus on a piece of advice the Hungarian mathematician
乔治·波利亚在他的著作《如何解决问题》中分享的一条建议
George Polya shared in his1945 book”How to Solve It.”
他说:“如果你不能解决一个问题 那么你一定能够解决一个比较简单的问题 找到它”
It goes: “If you can’t solve a problem, then there is an easier problem you can solve:find it”
在我看来 这是
This is, in my opinion, the most important
我们要解决数学问题时
technique to understand and put
要懂得的最重要的 实用的技巧”
into practice when you’re trying to solve tough math problems.
因为数学是自己本身的叠加
Because math builds upon itself.
很多复杂的概念是建立在比较简单的概念之上
More complex concepts arebuilt upon simpler concepts.
如果你对基本原则
And if you don’t have a strong grasp
掌握得不牢 那么 复杂的
on the fundamental principles, then a more complex
问题就会难住你
problem is going to likely stump you. So,
所以在你遇到一个
if you come across a problem
解不了的问题时 首先 找到你要解决问题时
that you can’t solve, first, identify the components
必须要用到的基本原则和操作
or the operations that itwants you to carry out.
大多复杂问题都会是多个问题的叠加
A lot of times, complexproblems will have multiple. Now,
现在 这种情况下你能够做的是
what you can do
把这个问题分解成许多小问题
in this case is split the problem into multiple problems
每个小问题只解决其中一方面
that isolate just one of those components or operations.
我想在实际问题中讲讲这一点
I want to show you this concept in action
咱们看一下一个例题
so let’s work through a quick example. Now,
我的确挑了一个
I did have one example picked out
非常简单的问题
that would be pretty easy but it ended up
有点过于简单
being a little bit too easy, so let
我们做一个稍微有点复杂的
‘s do something a little bit more complicated. So,
这是一个求和问题 用到了希腊符号∑
this is a summation problem which uses the Greek symbol, sigma.
这是把一系列
And it essentially says that we’re going to
表达式加在一起
add up a series of expressions that use a
变量是从1到4
variable starting at one and ending at four. But,
你可能注意到了 这个求和问题
if you notice, this summation problem
里面还有一个分数指数
also has a fractional exponent in it. Now,
可能你们其中有些数学奇才
maybe some of you math wizards out there could
闭着眼睛都能做这种题
do this kind of a problem in your sleep but it
但是也很可能
‘s also possibly the case that you don’t have a
你没有完全掌握求和或者分数指数
really firm grasp on either summation or fractional exponents. So,
当你在做一道由这两部分内容组成的问题时
when you’re working a problem that combines the two of them,
你可能会被难住
you might get stuck. So,
想象一下
assuming that’s the case, let’s
能把这个问题分解成两个简单点的问题
break this problem into two simpler problems that each focus
每个问题只针对一方面
on just one of the underlying concepts. First,
首先一起咱们去掉分数指数
let’s create a simpler summation
构成一个简单点的求和问题
problem that just gets rid of that fractional exponent altogether. Now,
我们要做的就是
all we have to do isevaluate that expression
这个表达式加4次
four times and then add up the answers
得到最终答案66
which gets us to a final answer of 66.
现在咱们再进行分数指数运算
And now let’s move on tothe fractional exponent. Now,
我这里会算得非常快
I’m going to go pretty quick here
因为这不是教分数指数的课程
because this is not a lesson on fractional exponents
但是 基本上你们能把4^3/2改写成
but essentially you can rewrite this as four to the power
4^3*1/2
of three times the power of one half.
然后能再写成
And then you can rewrite that again to the square root
√4^3
of four to the power of three.
一旦你算出这些 就得到答案8
And once you evaluate that,you get an answer of eight. Now,
解决这个较简单的单一概念问题的重点是
the whole point of working these simpler
掌握要用的基本
single concept problems is to master the underlying concept
概念或运算
or operation that you’re working on here. So,
如果你解了几道题
if you solve a few and you
仍然觉得心里没底
still don’t feel really confident on that concept
就再做几道题 直到你觉得自己掌握了
keep working it until you do. Remember,
记住:掌握的意思是不能出错
mastery means notbeing able to get it wrong.
不是只做对一次
Not just getting it right once. Anyway,
一旦你掌握了那些单一问题的基本部分
once you’ve mastered those underlying components in
就可以回到
an isolated setting now you can come back to
更复杂的包含那些基本题的问题上了
the more complicated problem that combines them.
这时 你应该闭着眼睛都能
At this point, you should be able to work those
解这些题 也就是说你心理上
isolated concepts in your sleep which means that all of your
有能力面对如何去解决
mental processing power can go towards the new and novel problem
一连串的新问题了
of how they work in tandem. Now,
现在 我要谈另一个简化
there is one additional way of simplifying tough problems that
难题的方法 你可能猜到了
I want to talk about and you might have already guessed it
如果你仔细看这些例题
if you paid really close attention to the examples.
我没有用很复杂的数字
I didn’t use really complex numbers.
没有长数字
I didn’t use long numbers.
没有小数点
I didn’t use decimal points.
没有大的分数
I didn’t use big fractions.
求和的项数也不多
And I stuck to a low limiton my summation problem.
复杂的数字 大数字 有许多小数点的数
Really complex, big numberswith lots of decimal points
会分散你的注意力 不能关注
can distract your attentionaway from the concepts
你应该练习的基本概念和运算
and the operations that you’resupposed to be practicing. So,
所以 如果你遇到的难题有
if you’re stuck on a
这些数字
tough problem that has these kinds of numbers
就去做个类似的 数字不大
go work a similar problem with really small
整个数字容易在脑中
whole numbers that are easy to add or operate
相加或运算 这样 你才能真正
in your head, that way you can really
专注于实际的概念
zero in on the actual concepts.
当然 有时你不确定
Of course, sometimes you have too shaky of an
自己是否清楚解题要用到的
understanding of the concepts and operations themselves for you to
某些概念和运算
actually work with them and solve that problem.
这时 就该去学习了
And in that case, it’s timeto go do some learning.
去看你的课本 翻笔记
Go dig into your book, look through your notes,
或在网上找些例题
or find example problems online that you can follow
带你一步步地看别人
along with step-by-step so you can see how people are
怎么用这些概念解决问题的
getting to the solutions, using these concepts. And,
如果你需要
if you need to,
你也可以找到自己正在解的
you can actually get a step-by-step solution to the exact problem
那道题的解题步骤
you’re working on as well.
你可以有很多工具来做到这点
There are several tools out there that you can use to do this.
本视频中我想重点
The two that I want to focus
推荐我能找到的最好的两个
on in this video which are the best ones I’ve been able to find
WolframAlpha 和 Symbolab
are WolframAlpha and Symbolab.
这两个网站都让你
Both of these websites will allow you to type
输入方程 得到答案
in an equation and get an answer and also Alpha and Symbolabs
Alpha 和Symbolabs你都可以用
that you can follow along with.
两者区别是
The difference between the two is that WolframAlpha,
WolframAlpha更强大 如果你想看分步骤讲解
while being much more power and capable, does require you to be
需要付费
part of their paid plan if you want to get those step-by-step solutions.
而 在Symbolab
By contrast, while I found that typing in
输入方程式费点劲
equations into Symbolab was a little bit slower and
而且不如WolframAlpha直观
less intuitive than it is with WolframAlpha
但是分步骤讲解是免费的
their step-by-step solutions are free.
这里不管你用哪一种工具
Regardless of the tool that you choose to use
重要的是有时
here the underlying point is that sometimes it can
看如何解难住你的难题的
be useful to see a step-by-step solution
步骤是很有用的
for a problem you’re stuck on. But,
但是 这里有两个重要的说明
there are two veryimportant caveats here.
首先最重要的是
First and foremost,before you go running off
在你去搜索问题的答案时 诚实地问自己
to find a solution, ask yourself “ Honestly,
我对这道题
have I pushed my brain to the limit
已经绞尽脑汁了吗?
trying to solve this problem first?”
自己费尽心思地去解题
Expending the mental effort required to solve the
能增强你的能力
problem yourself is going to stretch your capabilities.
和让你更好地解有时看起来
It’s going to make you a better mathematician
解不出的数学题
in a way that just looking through solutions won’t. Now,
如果你的确需要查找答案
if you do need to look up a solution,
可以
that’s fine.
去查吧 按照步骤
Look it up, follow the steps and make sure
一定要明白答案是如何得出的
that you understand how the answer was arrived at. But,
但是 一旦你看完了 考考自己
once you’ve donethat, challenge yourself
让自己不看答案 再重做一遍题
to go back and rework the problem without looking at that reference.
这一点真的很重要
It is really important tostay vigilant about this.
因为如果你想数学变好
Because if you want to get better
重点是要掌握
at math the whole point is to master the concepts
你做的题目的概念
that you’re working with.
查答案会出现的危险是
The danger that comes with looking up solutions is that
数学按步骤看懂
with math it’s really easy to follow along with
真的很容易
a step-by-step solution and comprehend what’s going on.
但是能独自解题与此差别很大
But that is very different than being able to do it on your own.
这就说说最后一个建议
And that brings me tomy final tip for you.
对任何人做数学作业
And this is especially important
都尤其重要
for anybody in a math class working through assigned homework.
在做数学题的时候不要着急
Don’t rush when you workthrough math problems.
我知道大家都想尽快
I know it’s really tempting to try to work
把作业做完
through homework as fast as you can and heck,
我最近就此问题还录了个视频
I even made a video about it pretty recently. But,
但是 对于数学 科学
with math and science
任何真正复杂的学科
and any sort of really complex subject especially
急匆匆只会给你的未来带来坏处
rushing is only going tohurt you down the road.
因为你着急时 就没有掌握基本概念
Because when you rush, youdon’t master the concepts.
你只是强迫自己作答
You just brute force your way to answers or you
你或者去搜索答案 或者其它欺骗的手段
look things up, or you otherwise kind of cheaty-face
去完成作业
your way to a completed homework assignment.
然后 当你参加测试时
And later on, when you’re sitting in a testing room,
或者在现实中 你要运用所学知识时
or you have to apply what you’ve learned
因为你没掌握的知识
in the real world you are going to get a harsh
你会得到残酷的教训
lesson about exactly what it is you don’t know.
所以 咱们扼要重述一下
So let’s recap here.
如果你想学好数学
If you want to get better at math and you
如果你真的想提高数学解题能力
want to improve your ability to solve those really tough
首先 要能辨别出题目中有
problems first, identify the combination of concepts
哪些基本概念和运算 把它们分解出来
or operations being used in a problem and then isolate them.
分别解出较简单的一个问题 掌握每个概念
Work simpler problems that use just one and then master each concept.
你也可以让题目原封不动
You can also simplify the problem by leaving the combination
只是把数字变小点 易于运算
of concepts intact but swapping in smaller, easier to handle numbers.
如果你概念不清
If you need help
翻翻课本 或在网上找找解释
with the concepts themselves go to your book or an explainer article online
找点简单的题目
look up sample problems,
或使用工具 WolframAlpha 或Symbolab找
or use a tool like WolframAlpha or Symbolab to get step-by-step solutions
解题的步骤
to the problem you’re working on.
最后要说的是 不要急匆匆地应付完作业
And finally, don’t rush throughyour homework assignments.
一定要专心掌握这些概念
Make sure that you’re focusing intently on mastering the concepts,
不要只是完成就行
not just finishing.
希望这些技巧能给你信心
Hopefully these tips will give you the confidence to tackle
处理真正难的数学题 拓展你的技能
some really tough math problems and to expand your math skill set.
请注意
And on that note,
我想告诉你们一句名言
I want to leave you with a quote
是伟大的物理学家理查德·费曼说的 他说:
from the great physicist,Richard Feynman, who said,
“我不认为数学只是特定一部分人能理解的 其他人都是普通人
数学是人类的发现 不会复杂到人理解不了的水平”
关键是:最终
The bottom line is this: Ultimately, your ability
你擅长数学和其它技能的能力
to get good at math and anything else for that
始于你有信心学会
matter starts with having the confidence to approach it.
当你解决问题 取得突破时
And as you solve problemsand make mental breakthroughs
自然你的信心就增加了
your confidence is goingto naturally increase.
这成为一个良性循环
It becomes a self-sustaining cycle.
如果你对开始这个循环感兴趣
If you’re interested in starting the cycle of learning now,
可以从这里启程 Brilliant
a great place to begin your journey is at Brilliant.
这是个学习平台 以解决实际问题
A learning platform that uses hands-on problem solving as
为基础帮你学习数学 科学
a basis to help you learn math, science
计算机 真的很有效
and computer science, really effectively.
实际上 我正在上他们的基础计算机课程
I’m actually taking their computer science fundamentals course right now,
我正在参加第一阶段的
and as I was going through the first section on
代数测试 这些问题迫使我学
algorithms the quiz questions they gave me forced me to learn
以前从没学过的新数学知识
new math I had never learned before.
我不得不去查维基百科 找例题
I had to dig into wikisand example problems.
最终 我整理出了一页
And eventually I had to get out a sheet
笔记 基本上得到的每一步算法
of notebook paper and literally draw out algorithms step-by-step
这样我能理解了问题
so I could understand what was going on.
这个过程给了我
This process provided me
比我大学期间任何课上都
with a much more intense and effective learning experience than I got
更具有激情 更有效的学习体验
sitting through mostof my college lectures.
这种困难真的迫使你去查清
And these types of challenges that really force you to dig in,
形成所有课程的
form the foundation
基础知识
for all of their courses.
包括概率 逻辑 微积分 天文
Which include probability, logic, calculus astronomy,
计算机内存 等等
computer memory and more.
除了这些编排好的课程 Brilliant还有每周难题
In addition to the structured courses Brilliant also has weekly challenges that
你可以做做来提高自己的技能 那还有社区
you can use to improve your skills and has a community where
你可以和其他学习者讨论问题
you can talk with other learners and it has a fantastically detailed
还有非常详细的维基百科上面有大量讲解和例题
and helpful wiki with lots of explanations and examples. So,
所以 如果你想试一下Brilliant
if you want to give Brilliant a
点击下面描述框的链接
try click the link the description down below and
免费注册开始学习吧
sign up for free to start learning. And,
作为奖励 如果你是
as a bonus, if you’re among
前200名注册的
the first 200 people to sign up, you’ll
每年有20%的优惠
also get 20 % off your annual premium subscription.
我想要特别感谢Brilliant
I want to give a huge thanks to Brilliant
赞助本视频 资助本频道
for sponsoring this video and helping to support this channel. And,
像以往一样 谢谢大家观看
as always guys, thankyou so much for watching.
如果你觉得本视频对你有帮助 请点赞
If you found this video helpful you can give it a like
支持本频道 也许你可以考虑和
to support this channel and maybe you can consider sharing it with
朋友分享本视频 让他也受益
a friend who could benefit from it as well. Otherwise,
你也可以去那 如果你想第一个知道新视频发布
you can subscribe right there and maybe hit that bell
点击钟形按钮订阅
icon if you want to be the first to know when new
点击这里
videos drop or click right there to get a free copy of
免费得到一册我的关于如何提高成绩的书
my book on how to earn better grades. Alternatively,
或者 听我们有关毁掉求职面试
you can click right there to listen to
的最新podcast
our latest podcast episode on how to crush job interviews
或者你可以立刻得到
or if nothing else, you can click right there
本频道的另一个视频
to get one more video onthis channel right now.
感谢观看 下次节目见
Thanks for watching and I will see you in the next video.

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  • 2021-08-30蚊子嗡嗡叫的真实原因 | DIY神经科学系列翻译:约瑟夫•杰妮校验:克里斯汀•阿帕嗒Translator: Joseph GeniReviewer: Krystian Aparta [蚊子嗡嗡声](Mosquito buzzing) 我们都听过蚊子讨厌的嗡嗡声We’ve all heard the annoying sound of a mosquito, 而且会不惜一切代价将它赶走and we will stop at no..
  • 2021-08-24“最大的成功故事”(马云)我父亲并不富裕 我考了三次大学 都没考上 我向哈佛大学提交了十次的入学申请全都失败了 他们甚至都不想再看到我了 最后一次 我去了一所师范学校 一所三流甚至是四流的大学 我去应聘了30份工作 但都被拒绝了 那段时间很艰难 我很沮丧 因为我在大学里教书 我的工资一个月10美元 因为我找不到一份更好的工作 1994年 说到我准备做一个叫互联网的东西的时候 他们中有23个人反对 他们说 这个想法很愚蠢 ..
  • 2021-08-30质疑的福音我们就在那儿There we were, 灵魂和身体挤在德克萨斯州的一间教堂里souls and bodies packed into a Texas church 那是我们人生的最后一个夜晚on the last night of our lives. 我们就挤在一个像现在这样的房间里Packed into a room just like this, 只不过那里都是吱呀响的长木凳 上面盖着破..
  • 2021-08-27污染是如何改变海洋的化学成分的你可曾想过Do you ever think about 海洋在我们的日常生活中有多重要?how important the oceans are in our daily lives? 海洋覆盖了地球表面的2/3The oceans cover two thirds of our planet. 我们呼吸的一半氧气都是海洋提供的They provide half the oxygen we b..
  • 2021-08-24证明自己小心那些扼杀你的憧憬Beware of vision, 梦想dream, 和激情的人and passion killers. 大多情况下 家人和朋友就是这种人Family and friends, in most cases, 尽管很不想这么说sad to even say this. 大多时候 家人、朋友会是第一批劝你放弃Family and friends, in most cases, w..
  • 2021-08-24现在开始充实的生活天呐 已经不记得你第一次去学校的时间了My goodness don’t remember when you went first to school 你进了幼儿园 在那儿and you went to kindergarten and in kindergarten, the idea was to 一心要努力进入一年级 之后再接再厉 以便push along so to get ..

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