♪ theme music ♪
【教育无边界字幕组 Makroner QhelDIV | 砖头Man | Milite42】 
Good day uni students.
I'm Norman Wildberger.
日安 各位大学生们 我是诺曼·威尔德伯格 (Norman Wildberger)
So, we've reached the end
of this little course.
这个小小的课程也接近了尾声
This is our last lecture today,
今天是最后一课
and in this lecture I want you
to tell you about
在这一课中 我想要讲讲
how to write mathematics at the college
or university level.
怎样进行大学水准的数学书写
So, we're talking about writing
mathematics in the context of a test,
就是在测验 考试或者作业时
or an exam, or perhaps an assignment,
or problem set.
如何使用准确的数学语言
My experience is that most students
coming to university
根据我的经验 很多大学生没有经过足够练习
don't have enough practice
writing out good solutions.
写不出好的解题答案
So, I want to give you just a few pointers
or suggestions that might help you
所以我想就数学语言的表达给诸位几点建议
write mathematics more clearly
and get better marks,
希望能够帮助你们用数学语言更清晰的表达
and better results as well.
取得更好的成绩和结果
So, I think we can summarize
what's necessary
我认为我们可以把重点总结成三个c
with the three c’s.
我认为我们可以把重点总结成三个c
We aim to be correct, clear and concise.
即正确 (correct) 清晰 (clear) 和简洁 (concise) 
When we're solving a mathematical problem,
writing down a solution,
当我们在解决数学题写下答案的时候
we certainly first of all want
the solution to be correct.
我们首先当然希望解法正确
That almost goes without saying.
这自不必说
But we also want to be clear.
但是我们也会想要表达清晰
We want to lay things out
in such a way that somebody else
想要将语言组织起来
can come along and without our help
just read it and understand
其他人可以无需我们的帮助 
what we've written.
就能阅读并理解我们所写的答案
So, it should be logical,
it should be well organized,
所答案需要具有逻辑性 经过精心组织
it should be easy to read.
方便他人阅读理解
And we want it to be concise.
而且答案应该简洁明了
Concise means rather compact;
简洁意味着紧凑
not long and drawn out,
going on and on and on.
不是冗长的 重复不断的
We want to say what has to be said,
but no more.
我们应该只说需要说的 其余不必多言
So these are the things
that we're aiming for
这些就是我们用数学语言表达时的目标
when we're writing mathematics.
这些就是我们用数学语言表达时的目标
And the advantage of being able
to do this is that, first of all,
这样做有诸多好处
it's easier for someone to mark
your assignment, or problem set,
首先这方便了他人对你作业和问题集的批改
or test or exam solution.
And if it's easier for the person to mark
或者考试的阅卷 相信我
you're more likely to get a better mark,
believe me.
方便了阅卷 就意味着你能得更高的分 
So, it will improve your marks.
总之 这会提高你的成绩
But more than this, this kind of skill,
being able to express yourself
但更重要的是 能够用技术性语言更好地表达自己
in a technical subject well
is highly valued
是一项非常有价值的技能
when you go out and get a job.
尤其是当你走向社会找工作的时候
This is a skill that employers value.
雇主们很看重这项技能
And there is also advantage for you
that later on, maybe a few years later,
这对你今后生活也有好处 可能是几年之后
or maybe just only a few months later,
when you want to understand
或是几个月之后 当你想了解自己之前写下的东西时
what you have written, if your solution
is correct, clear and concise,
如果你的解答是正确清晰又简洁的话
you will be able to look at it and say,
"A-ha, that's what I learned there.
你就能看着它说 “啊哈 那就是我之前弄明白的”
That's what I was doing there.
I understand what I did."
“那就是我以前做的 我看懂自己以前的工作了“
So, it's a benefit to you when you're
going over your accomplishments later on.
所以当将来的你浏览之前完成的工作时 会对你很有好处
Let's talk about the overall structure
of a solution to a mathematical problem.
让我们来谈谈数学题解答的整体结构
In general there are three parts.
There is an introduction,
完整的解通常有三个部分 有引子
there's the main body
and there's the conclusion.
主体 还有结论
Most of the time the introduction
and the conclusion will be very short.
大部分时候 引子和结论会很短
They don't need to be very long,
just one line will often do.
它们也不需要很长 通常一行就足够了
In the introduction we want
to define the terms that we're gonna use.
在引子中 我们将定义要使用的术语
Sometimes, but not always, we might
want to review some relevant fact
有时 并不是总是 可能会需要提到一些相关的事实
or definition that we're going to use.
或者将要用到的定义
And we might also just give
an overall direction to the solution,
如果答案很复杂的话
if the solution is actually
quite complicated.
我们也需要对解答给定一个大致方向
We might instruct the reader and say,
"Well, we're gonna do this first,
我们可能需要引导读者 告诉他们 “我们先会这样”
and then we're going to do this,
and then we're going to do this
“然后我们会这样做  之后我们会这样”
to give an overview
of the direction of the solution.”
这样就给出了解答的大致方向
The main body is the majority
of the solution.
主体是解答的主要部分
You're going to derive a solution.
You're going to lay it out
你需要推导出一个解法  
in a clear and logical way.
然后将其清楚地 有逻辑地写下
You're going to use, hopefully,
complete English sentences,
希望你们能用完整的英文句子表达
with careful diagrams
and of course correct formulas.
辅之以细心准备的图表 还有正确的公式
So, there are all aspects
of the actual laying out
当然 也要注意答案主体的呈现结构布局
of the body of the solution.
当然 也要注意答案主体的呈现结构布局
And finally at the end, you should give
a concise summary
最后在结尾 你需要给出一个简洁的总结 
of what you've accomplished.
You should say,
描述一下你所完成的事项
"Here's the answer to the question.
Here we've solved the following problem."
比如说“这是问题的答案” “这里我们解答了以下问题”
And hopefully you are actually
addressing the original question
你们应该提到原始的问题
or problem that was asked.
也就是一开始被问的问题
If you were asked how many rabbits
are there gonna be at time t=7,
如果你的问题是在 t=7 的时候有多少兔子
well, hopefully your last sentence will say,
那么你答案的最后一句应该是
"Therefore, the total number or rabbits
at time t=7 is..."
“因此 在 t=7 时兔子的数量为...” 然后是你的结果
whatever it is. So, you want to address
the actual original question
总而言之 你要在最后一句回答最初的问题
in that last line.
总而言之 你要在最后一句回答最初的问题
So, this is the rough structure
of the solution.
这就是解答的大致结构
Now, let's have a little bit closer look
at some of the features.
现在 让我们来仔细了解一些具体内容
So, let's have a look at this idea
of defining the terms.
我们看看如何定义术语
This is one of the first things
that you should do
这是在你解决一个问题时 首先要做的事情之一
when you're solving a problem.
这是在你解决一个问题时 首先要做的事情之一
Most mathematical problems
involve some variables,
多数的数学问题涉及到变量或未知量
or some unknowns.
多数的数学问题涉及到变量或未知量
Not always, but most of them do,
并非一定有 但在很多情况下如此
and it's important to define
the meanings of those variables
为这些解题中使用的变量下定义很重要
that you're going use in the problem.
为这些解题中使用的变量下定义很重要
So, variables like x or y or n
or a or capital A, etcetera,
所以 像 x y 或者 n 或 a 或者大写 A 等变量
they should be defined
before they are used.
在使用它们之前应该要先进行定义
So, when you're reading someone's solution
and you see an n somewhere,
当阅读某人的解题答案 看到某处出现了一个n时
you shouldn't have to ask yourself,
"What is that n? What does that mean?"
不应该需要自己去猜 “那个n是什么？” “它代表了什么？”
"Where is that n coming from?"
“那个n是从哪里来的？”
There should be a little brief statement
somewhere before that tells you
在n出现之前 应该有一个简短的陈述来说明 
what the n actually means.
n所代表的含义
So, for example, you might start by saying,
"Let n be the number of rabbits
比如说 你可以这样开始  
at time t=0.
“令n等于兔子在t=0时的数量”
That might be a quantity
that you're interested in,
这可能是你在解题中需要的值
that's involved in the problem.
这可能是你在解题中需要的值
So, you're being very precise
about what n means.
这样一来关于n代表的含义就十分明确了
Another example might be: consider
an arbitrary vector v=(x, y).
另一个可能的例子是 “对于任意向量v=(x,y)”
Good. So, you're introducing a vector,
you're making it clear that the vector's name
那么 你引入了一个向量 清楚地标明了向量的名称v
is gonna be v, and that its components
are gonna be x and y.
它的构成元素为x和y
And that's an arbitrary one.
而且这个向量是任意的
Another example. If the initial and final
speeds were S sub I and S sub F
另一个例子 “如果初始速度和最终速度分别为s下标I和s下标F 那么…”
 respectively then you might say
something else.
然后你可以写一些其他的东西
It's just the first part of this line
is introducing these two quantities.
这一行的开始部分介绍了这两个量
And it's telling us that these
two quantities are the initial
这一部分告诉我们这两个量分别代表了
and final speeds respectively.
初始速度和最终速度
This is an important word
that's used a lot in mathematics.
分别 (respectively) 是在数学中很常用的词
And it's used in this kind of situation
where we have a number of different
当有很多对象或变量
objects or variables, and a number
of different descriptions of them.
而它们都分别对应不同的描述的情况下 我们会使用这个词语
And we want to make sure that the reader
understands that the first description
我们想要确保读者能够明白
goes with the first variable.
第一个描述对应第一个变量
And that the second description
goes with the second variable.
第二个描述对应第二个变量
If we didn't put this "respectively"
in here, then the reader
如果我们不使用 “分别” 这个词 
might be confused and say,
"Okay, the initial and final speeds
读者可能会困惑地问 “这两个变量是最初速度和最终速度“
are these two, but which one
is the initial one,
“但是到底哪个代表最初 哪个代表最终呢？”
and which one is the final one?
I don't know."
“我不知道”
When you say "respectively",
it's telling the reader that the first one here
当你说 “分别地” 时 它会让读者明白
corresponds to the first one here,
the second here corresponds
第一个描述对应这里的第一个变量 (sI)
to the second thing here.
第二个对应这里的第二个 (sF) 
Now, notice that in this case
you probably could get away
注意在这里你可能不需要使用 “分别” 
without using "respectively",
because it's pretty obvious
读者也能明白你的意思 因为这很明显 
that S sub I  and S sub F refer to
the initial and final speeds respectively.
si和sf分别代表了最初 (initial) 和最终 (final) 速度 
Because these indices that we're using...
this is called an index,
是因为我们所使用的下标 (index)
this little thing underneath
the main variable.
就是标在主要变量下面的小东西
This index, and this index here,
are obviously related to initial and final:
这里这两个下标很明显指代了“最初”和“最终” 
"I" for initial and "F" for final.
“I”代表最初 ”F”代表最终
And the plural of index is indices.
index的复数形式是 “indices”
So, here, the indices here help us
remember which variable is which.
所以 下标帮助我们记住哪个变量是哪个
This is a very good idea
to try to do in a problem,
在解决数学问题的时候 这是个很好的方法
if there are several variables around
you want to keep them separate.
当问题中出现了多个你想区分开的变量的时候
Make an index that has some logical
meaning so it's easy to remember.
使用一个有逻辑含义的下标会比较方便记忆 
So, you can remember that
S sub I is the initial speed,
这样你就能记住sI代表初始速度
and S sub F is the final speed.
sF代表最终速度
So, a very good thing to do at the beginning,
make sure that all the variables
总之在解题过程的开头来定义变量很有好处
or unknowns are pinned down.
确保所有的变量或未知量都被明确定义
You're telling the reader
what those symbols stand for.
你要告诉你的读者们这些符号代表了什么
So, now when we get to the main body
of your solution,
下面就到答案的主体部分了
it's very important to have a clear
and logical layout.
有一个清晰有逻辑的布局是很重要  
Layout means how we present
the solution on the paper.
布局指的是将你的解答展现在纸上的方式
So, one very important thing to suggest
to you is to write neatly.
一条很重要的建议是 你的书写要工整 
Make sure that your writing is legible,
clearly legible, that a person doesn't have
保证别人能非常清楚地阅读你的卷面
to squint and guess what
your letters are meaning.
阅卷的人不需要眯着眼来猜你写下的字母
Cross out wrong work neatly.
如果你写错了 整洁地划掉
Okay, here I was starting to write
"meatly". Obviously it's wrong.
这里我一开始写成 “meatly” 很明显是错的
So, I'm going to cross it out.
But instead of scratching it wildly,
所以我要把它划掉 但不要胡乱涂抹 
I'm just going to write
a neat line through it.
只用简单地画一条线穿过它就行
That's a nice way of crossing
something out.
这是划掉错误的好方法
We use a box for bigger errors.
So, if I made a whole big paragraph,
对于大段的错误 画个框包住它 
or bigger sentence, and it's all wrong,
instead of just scrawling it out,
如果写错了一大段或者很长的句子 比起将其涂去
I prefer to put a box around it
and neatly cross it out.
我更推荐用方框框住后 干净地划掉
So, the reader knows, okay, it's wrong,
but the wrongness is contained.
这样读者们就会知道 这是错的 但是错误的地方被框起来了
It's just there, that wrong thinking
is not affecting the things around it.
错误的想法只在那一部分 并没有影响其他部分
So, be careful how you cross work out.
You don't have to worry about
所以当你划去错误的时候要认真小心
doing everything perfect. It's all right
to make mistakes in mathematics.
你不需要将一切做到完美 在数学中允许出错
But if you make a mistake,
cross it out and carry on.
但是当你犯错的时候 划掉它然后继续写
Please, don't skimp on paper.
不要吝啬纸张
Paper is cheap. There's lots of it around.
纸并不贵 也很充裕
Use enough paper so that
you are not cramped.
使用足够多的纸 来避免你的字都挤在一起
So that you have room
to express yourself.
留下足够的空间来表达你的思路
Have a look at these two solutions
from a distance.
这两份答案从远处看 无须细想你也能注意到
Even without looking at any of the details,
you can see that this person
这两份答案从远处看 无须细想你也能注意到
doesn't really understand
what they're talking about.
这个人并不明白自己在讲什么 
This person is probably confused.
可能自己都很困惑
This solution is very likely wrong.
这个答案很有可能是错的
You can see that without even
looking at the details.
不用细看也能得出这样的结论
The solution is going
in different directions,
这个答案写得横七竖八
there's lots of crossing out,
there's arrows.
有很多地方被划掉了 也有箭头
There's not a sense of
logical clear layout.
这个答案的布局逻辑性并不强
Here the person has scrawled
a lot of stuff in a little corner.
还把很多字都挤在这个角落
It's unattractive. It may be right,
but the marker is going to be prejudiced
解答可能是对的 但令人非常厌烦 
against that even before
he starts, or she starts reading it.
评分人在下结论之前就会保持这样的偏见
This solution, on the other hand,
looks logical.
而这边的答案看起来逻辑性很强
There's a clear layout. There's one,
there's two, there's three.
布局清晰 这里分成了一二三点
The person has compartmentalized things.
He has written relatively neatly---
这个人将答案内容分布得很合理 书写也比较工整
maybe there's something crossed out here
and there, but it's contained.
有些地方被划掉了 但是划掉的地方被围起来了
We give a look at the solution like that
and say, "Yes, it looks reasonable."
我们第一眼看上去就会想 “嗯 它看起来很合理”
You would be surprised.
So, I've been marking university level
你可能会觉得不可思议  
mathematics for decades now.
我已经批改了几十年的大学数学题
I'm pretty good at guessing,
even before I actually look at the details
很擅长通过看一个人的书写来预测大概会得多少分
of a solution, just by how it looks,
what kind of mark is this person
甚至是在我还没仔细看其答案之前  
is likely to get.
甚至是在我还没仔细看其答案之前 
I'm not always right,
but I'm often pretty close.
我并不总是对的 但猜测和事实经常非常接近
If you see a solution which is neatly,
clearly laid out in an organized way,
如果你看到一个干净整洁 布局合理的答案
chances are the person's thinking
is also clear and organized.
那么有很大可能这个人的思路也同样的清晰合理
If you see a solution that's messy,
that's disorganized,
如果你到的是一个杂乱无章的答案
then very likely that person's thinking
is also messy and disorganized.
那么很可能那个人的思路也同样毫无章法
It's the way it is. So, try to present
yourself in this way.
这就是事实 所以 请尽可能像右边这位这样答题
Structure your argument.
对你的论据进行合理布局
Think about, especially if it's a more
complicated problem, think about
尤其是面对十分复杂的问题的时候
breaking it up into parts
to make it clearer:
想想怎样将其分成几个部分 让它更加清晰
there's this, and there's this,
and then we're going to do this.
首先是这个部分 其次是这一部分 接下来我们会做这个
Organization is important
in lots of things,
组织 在很多方面都是一项十分重要的技能 
but it's also important
when you're writing mathematics.
在进行数学书写时 它也同样发挥着重要的作用
When you're actually writing
it's important to use
当你在实际写作时 
complete English sentences.
使用完整的英文语句很重要
So, sentences start with a capital letter.
句子由一个大写字母开始
A capital "I", capital "N".
(这里是)一个大写的 “I” 大写的 “N”
They end with a period.
(句子)以一个句号结束
There's a period, there's a period.
这是个句号 这也是个句号
Even if the mathematics has
a lot of formulas in it,
尽管数学语言中包含了很多公式
has a lot of variables in it,
still you want to organize things
也有许多变量 但你仍然需要组织好自己的语言
in terms of English sentences
so that the reader can read it
用完整的英文句子表达 这样读者才能更方便地
in the sense of reading a book.
像读书一样阅读你的答案
So, make formulas parts of sentences.
所以 让公式成为你句子中的一部分
You can use connectives to do that.
Connectives like so, thus, therefore, but,
可以用词语连接来做到这一点 比如 所以 因此 故 但是
now we see that, observe...
可知 注意到 等等
Those kind of little words can tie
formulas together and into
这些简短的词语可以将公式连接并且融入到整个句子当中去
the main body of the sentence.
这些简短的词语可以将公式连接并且融入到整个句子当中去
Let me illustrate with some examples.
So, here is just some statements
让我举几个例子来解释一下
that might occur somewhere in your proof,
你的证明当中也许会用到
somewhere in your solution.
也可能是你的解答里
If p1, p2, up to pk, are all
the prime numbers less than N,
若 p1 p2 一直到 pk 都是小于N的素数
then define K=p1+p2 all the way to pk+1.
则定义 K = p1+p2 +...+ pk + 1
So, this is just a single statement
taken out of context.
这是脱离原文单独拿出来的一个陈述
But we're just looking
at that statement itself.
我们先考虑这句话本身
We see that it reads
like an English sentence.
它读起来就像是自然的英语句子
It has a start, it has a finish.
有开头 有结尾
There are some variables
and some formulas in the sentence,
虽然句中有些公式和变量
but still there's this linear aspect
of the sentence.
但句子的结构仍然是线性的
We start here, and we end there.
从这里开始 在那里结束
Notice that there's the "if/then" construction.
Those two words like to go together.
注意这个”若...则” (if...then) 的结构 这两个词经常一起出现
Notice a few things about the mathematics here,
that when we're creating a sequence
注意这里的数学表达形式 当我们在写一个数列的时候
we have commas, and between these commas
there are three dots.
我们用逗号 以及两个逗号之间的三个点
So it goes comma, then three dots,
then comma, and then the last fellow.
先是逗号 再是三个点 接下来还是逗号 然后是最后一项
Here's another example
of part of a solution.
再看一个解的部分
Now, we see that that, there's a formula.
a plus b squared minus a minus b squared
“可知” 接下来是公式 (a+b)^2 - (a-b)^2 = a^2+2ab+b^2-(a^2-2ab+b^2)=4ab
equals a square plus two ab, 
plus b squared minus a squared minus two ab
“可知” 接下来是公式 (a+b)^2 - (a-b)^2 = a^2+2ab+b^2-(a^2-2ab+b^2)=4ab
plus b squared equals 4 ab,
which is greater than zero,
“如果a和b都大于0”
if a and b are greater than zero.
“整个式子也是大于0的”
So, this is all one English sentence.
Even though it contains this rather
即使写在中间的公式十分显眼
prominent formula written
in the middle of it.
所有的东西都还是在一个英语句子当中
The formula is still part of the story,
and the sentence is telling us
公式仍然是句子的一部分
something about the formula
and how it connects with the argument.
同时句子也在解释公式 以及它和证明的关系
So, try to do that. Try to put
your formulas inside sentences.
所以尽量把公式放在句子里
And make sure that it reads
like a sentence.
而且让它读起来是个句子
Use careful diagrams.
细心使用图表
So, a picture is worth
a thousand words they say.
人常说一图胜千言
And in mathematics, which is often
a very visual subject,
对于数学这样注重视觉的学科
a picture or diagram can help us
understand things very much.
图象能帮助我们对概念进行更好地理解
So, don't be afraid of using diagrams
and figures in your solution
为了解释清楚你的思路 别害怕在解答中使用图表
to clarify what you're trying to explain.
为了解释清楚你的思路 别害怕在解答中使用图表
There are some little hints
that help doing this.
这里有些小窍门
So, first of all, try to be careful
with the diagrams.
首先要注意画图时要细心
Rulers can help with straight lines.
可以用直尺画直线
If you have to make axes like this…
如果你要画这样的座标轴
Some people can draw a straight line
free hand very well.
某些人徒手画的直线就很直了
If you're one of them, great.
如果你能像这些人徒手画直线那就手画
If you can't, then just have a ruler
and use a ruler.
否则就最好借助尺子画
It makes it look a lot better.
这样会好看多了
If you have axes, like the x y axes,
which we often do,
如果要画这样的x y轴
label the axes. This is the x axis,
this is the y axis.
要标出轴名 这是x轴 这是y轴
These arrows here are telling us
the direction of positive or increase.
这些箭头指出了它们的正方向
So, this arrow is telling us
that the x variable is getting bigger
也就是说x在这个方向上增加
if you go in that direction.
And here the y variable
y在这个方向上增加
is getting bigger
if we go in that direction.
y在这个方向上增加
And put in some units. At least
put in a "1" somewhere,
标出单位 至少标出1在哪里
so that you can see that there's
a sense of scale on your axis.
这样就可以对座标轴的比例有所把握
Color can help a lot.
颜色能帮上大忙
If you're working under test conditions,
don't bother with color,
如果是在考试那就不要用各种颜色了
you don't have enough time.
But if you're preparing a problem set
你没那个时间 但如果你是在写作业题的话
solution or an assignment
then having a diagram with some color
用一些彩色的图表会很有益处
can help a lot.
用一些彩色的图表会很有益处
There are a few basic shapes
that are useful to be able to draw.
最好会画一些常用的基本图形
So it helps to be able to draw a sphere,
or a cylinder, or a parallelepiped.
球体 圆柱体或是平行六面体
And I remind you that when you're
drawing a sphere like this, the equator
注意画球的时候 大圆应该画成一个椭圆
should actually look like an ellipse.
注意画球的时候 大圆应该画成一个椭圆
Often we make it a little bit pointy,
but it should be actually be round there.
我们有时会把这两端画得尖尖的 但它应该是圆滑的
It's not so easy to do that.
比较不好画
And just to let you know that
if you're looking at a sphere
还应当注意 当看到一个球的大圆这样的时候
and the equator is like this,
it means that it's tilted down,
意味这球是向下倾斜的
the sphere is tilted in a way.
So the north pole will actually be up here
它向下旋转了一些 所以北极点应该在这里
and not exactly up at the top.
不是在最上端
Here's a picture of a cylinder.
这是一个圆柱体
Note the dotted lines that suggest
the back of the cylinder,
后面这条虚线表明这是圆柱体的背面
even though we can't see it.
虽然我们是看不到的
Here is a parallelepiped.
这个是平行六面体
The basic idea is, you start with
three lines, and then you draw
基本画法是先画三条线 
other lines which are parallel
and equal to those ones.
然后再画出与这三条平行等长的其他线
And the thing sort of builds itself.
然后它自然地就成形了
So, once you have this line,
this line is just a copy of this one,
如果你已经画好了这条线 平移过来就变成了这条
this line is just a copy of that one,
moved over in the right spot.
再平移一下就是这条了 
So practice making some parallelepiped,
that's a good thing to practice.
练习画平行六面体会很有用
So, you don't have to be very
artistically inclined.
当然你没有必要画得很艺术
I'm hopeless at drawing things.
我不是画画的料
But I still produce half-decent diagrams,
because I take a little bit of care.
但我还是可以画出差不多像样的图的
Not too much, but a little bit of care.
And I'm precise about where the…
只是用了点心罢了 认真一点就够了
positions of things are.
而且我对位置的把握比较准
Not a lot of skill is involved,
but good diagrams can really enhance
画个像样的图并不需要很多技巧
your solutions a lot.
但它却能使你的解答更出彩
And so now coming to the end
of your solution, the conclusion:
现在到了你解答过程的末尾——结论部分
well, the conclusion should make it clear
what the answer is.
结论应该明确地说明最终得到的答案是什么
You've just solved the problem, hopefully.
既然你已经解决了问题
And so you should say clearly
what the answer to the problem is.
你就该说清楚问题的答案是什么
And of course you have to make sure
that you really are answering
当然了你要回答原问题
the original question and not some
question that you've made up.
而不是回答你胡乱编的问题
So, for example a solution might end:
举个例子吧 现在是解答的结尾
So, x equals three over two plus
five over seven, which is
故x=3/2 + 7/11 也就是 (21+10)/14 即 31/14
twenty-one plus ten over fourteen,
see that calculation.
故x=3/2 + 7/11 也就是 (21+10)/14 即 31/14
And then simplify to
thirty-one over fourteen.
故x=3/2 + 7/11 也就是 (21+10)/14 即 31/14
And I've circled the final answer
so that someone looking at this,
我把最终答案圈了出来 
if they're only interested in
whether I've gotten the right answer,
所以要是有人只关注是不是得到了正确答案
they can quickly see what the answer is.
就可以马上看到这个答案了
And if that is the right answer,
well they're probably not going to worry
如果答案是正确的 就不用操心推导过程了
too much about the details.
如果答案是正确的 就不用操心推导过程了
Another example. You might end:
"And so the number of rabbits
举另一个例子 可能你会这么结尾：
at time t=7 is 1438.
当t=7时 兔子的个数为1438只
So, you're answering the question.
Presumably the question was,
你在回答问题 问题也许是
you know, what's the number
of rabbits at time t=7 given such and such.
在时刻t=7时兔子的数量是多少？
And here finally you're
answering the question.
你在这里对该问题作出解答
So, you're repeating the question back
a little bit, just making it clear
为了让答案明了 应该对题目稍作重复后作答
what exactly you've found.
为了让答案明了 应该对题目稍作重复后作答
You've found that the number of rabbits
at time t=7 is this number here.
你算出在t=7时兔子的数量就是这个值
Now, sometimes we work very hard
at a solution and we can't get it.
有时候我们非常努力去解题 但是就是没办法算出答案
That happens to everybody.
每个人都会经历这种情况
So, you can still get something
out of that kind of situation.
但你仍然可以从求解过程当中得到一些信息
A partial solution is also worth
something.
部分解也是有价值的
So, you have to make it easy
for the marker to give you
写出部分解才能让改卷人给你部分的分数
some part marks.
写出部分解才能让改卷人给你部分的分数
And if you haven't been able to solve
a question, be honest about it.
如果你实在解不出题
And say what you have done.
Try to present what you have done.
就写出来你已经完成的部分 把它们展示出来
There's no better way of losing marks
and losing the goodwill of the marker
如果你并没有做完却假装完成了 就会失去阅卷人的信任 
if you're pretending that you've solved
the problem and really you haven't.
那么对方当然不会手下留情
And you're going to hope that the marker
doesn't know or can't see that.
不要侥幸希望阅卷人没有注意到
Usually we can tell, and we are merciless
to people who try to fool us.
对于这种想愚弄我们的行为 我们一般都会发现并绝不留情
So, for example you could say
something at the end.
其实你可以这么写 举个例子
So, I can't solve this for x,
but at least x is positive,
我无法解出x 但是我断定x大于等于0
and I suspect that x is between
two and three.
而且我认为x应该介于2和3
So, first of all you are admitting that
whatever formulas you had,
首先 你承认了你用了各种公式 但没办法解出答案
okay you can't solve it.
首先 你承认了你用了各种公式 但没办法解出答案
But you have some information about x,
some partial information,
但是你得到了关于x的部分信息
and you may have some suspicion,
or you may have some intuition
这说明你对解有些想法和猜测
about this solution.
这说明你对解有些想法和猜测
Maybe you haven't proven this,
you haven't demonstrated it,
也许你证明不了这个想法
but you have some feeling for it,
which is also worth something.
但说出你的感觉也是有部分价值的
So, this is not about problem solving
but let me just say a few very quick words
虽然这门课不是讲如何解题的
about problem solving.
但是我还是简短地说几句解题的事情
So, here is some general advice for
attacking the writing of a solution.
这里列出来了几条攻克问题的建议
Make sure that you understand what
the words in a problem mean first.
首先要明白题干描述都是什么意思
You'd be surprised how many students
come to my office sometimes
你可能很难想像有多少学生做过这样的事
for some help, and we're looking
at a particular problem,
他们去我的办公室问我某道题目怎么解
they don't know how to solve
a particular problem.
他们卡在这些问题上了
They say, "How am I going to do this?"
And I ask them, "Well, what do these
他们问：“我该怎么办呢？” 
various words mean?" And sometimes
it becomes clear that they don't understand
我问他们：“这些词都是什么意思？” 
what the words mean.
有的时候能看出来他们不懂那些词语
If you don't understand what the words mean
of course you're going to have a hard time
不知道题干的意思 当然会卡住
tackling that problem.
不知道题干的意思 当然会卡住
It's a necessary prerequisite.
That's part of the reason why,
读懂题干是解题的一个基本前提
you know, it's important to know
the terminology.
这也说明了为什么了解术语非常重要
Especially in mathematics,
there's no getting around that.
这点是毋庸置疑的 尤其是在数学当中
Don't be afraid of looking at
special cases.
别怕考虑特殊情况
Special cases means when you restrict
yourself to perhaps a simpler situation,
特殊情况指的是
or instead of looking at things
in generality for all x,
除了最普遍会考虑到的情况之外的可能
you'll say, "Well, what happens
if x is 5?"
你还要考虑 比方说 要是x=5时会怎样？
"What happens in that special case?"
That can be instructive.
在这个特殊情况时会怎样？ 那可能是很用的
Drawing a diagram can often
shed some light on things.
画出图来往往会有帮助
Give you a pictorial understanding
or feeling for what's going on.
给你对问题本质的直观理解和印象
And I guess it's important to realize
that mathematical problem solving
再有很重要的是 要明白解决数学问题是一种技能
is a skill. It's a skill that you don't
just have at birth.
它不是一个人与生俱来的
It's something that is developed
with practice, with repetition,
这个技巧需要时间去反复练习
with seeing and watching
what other people do.
需要看他人是怎么做的 才能得到提升
So, it's a skill that can be learned.
所以它还是可以被学会的
And finally it's important to have
kind of an honest approach to things.
最后要确保做事情别骗人
Just to make sure that you're...
you know, you're not trying to present
不要不懂装懂地作答
your solution as more
than what it really is.
要实事求是
So, some basic ideas about
structuring and writing mathematics.
好 这就是我要讲的 如何组织和书写数学的内容
So that brings our little course to an end.
And I'd like to thank you all for watching.
这也意味着我们这门小课就到此结束了 感谢大家的观看
♪ theme music ♪
【教育无边界字幕组 Makroner QhelDIV | 砖头Man | Milite42】
