On the Functions of Mathematics Teaching

程序员怎样学数学I've been working for the past 15 months on repairing my rusty math skills, ever since I read a biography of Johnny von Neumann. I've read a huge stack of math books, and I have an even bigger stack of unread math books. And it's starting to come together.自从我读了Johnny von Neumann的传记,我已经为弥补我糟糕的数学技能花了15个月了.读了大量的数学书籍,不过呢,似乎我还有更多没有读.当然我会接着做的.Let me tell you about it.现在我就来告诉你这些.Conventional Wisdom Doesn't Add Up告别传统观念First: programmers don't think they need to know math. I hear that so often; I hardly know anyone who disagrees. Even programmers who were math majors tell me they don't really use math all that much! They say it's better to know about design patterns, object-oriented methodologies, software tools, interface design, stuff like that.首先:程序员不认为他们需要了解数学.我常常听到这样的话;我不知道还有没有不同意的.甚至于以前是主修数学的程序员也告诉我他们真的不是常常使用到数学!他们说更重要的是要去了解设计模式,面向对象原理,软件工具,界面设计,以及一些其他类似的东西.And you know what? They're absolutely right. You can be a good, solid, professional programmer without knowing much math.你了解吗?他们完全正确.你不需要了解很多数学你就能做个很棒,很专业的程序员.But hey, you don't really need to know how to program, either. Let's face it: there are a lot of professional programmers out there who realize they're not very good at it, and they still find ways to contribute.但是呢,同时你也不是真的需要知道如何来编程.让我们面对这样一个事实:有很多专业的程序员发现他们并不是非常擅长数学,但他们仍然能在程序设计上做出贡献.If you're suddenly feeling out of your depth, and everyone appears to be running circles around you, what are your options? Well, you might discover you're good at project management, or people management, or UI design, or technical writing, or system administration, any number of other important things that "programmers" aren't necessarily any good at. You'll start filling those niches (because there's always more work to do), and as soon as you find something you're good at, you'll probably migrate towards doing it full-time.如果你突然觉得自己好烂,周围的人都远远的超过你,你会怎么想呢?好,你可能会发现自己善于项目管理,或人事管理,或界面设计,或技术写作,或系统管理,还有许多其他程序员不必去精通的.你会开始堆积那些想法(因为工作永远干不完),当你发现一些你能掌握的东西时,你很可能会转移去全职的做这个工作.In fact, I don't think you need to know anything, as long as you can stay alive somehow.实际上,我认为有些东西你不需要了解,当前你还能够赖以生存的话.So they're right: you don't need to know math, and you can get by for your entire life just fine without it.所以他们是对的:你不需要了解数学,并且没有数学你也能过的很好.But a few things I've learned recently might surprise you:但是最近我学到一些东西可能会让你也感到惊喜:Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap.在你知道如何编程之后,数学更容易学会.实际上,如果你先学数学,然后半路出家做程序员的话,你会发现编程简直就是小菜一碟.They teach math all wrong in school. Way, WAY wrong. If you teach yourself math the right way, you'll learn faster, remember it longer, and it'll be much more valuable to you as a programmer.学校里教数学的方式都错了.仅仅是教学的方法错了,不是教数学本身错.如果你以正确的方式学习数学的话,你会学的更快,记住这点,对你,作为一个程序员来说很有价值.Knowing even a little of the right kinds of math can enable you do write some pretty interesting programs that would otherwise be too hard. In other words, math is something you can pick up a little at a time, whenever you have free time.哪怕了解一点点相关的数学知识,就能让你写出可爱有趣的程序,否则会有些小难度.换句话讲,数学是可以慢慢学的,只要你有时间.Nobody knows all of math, not even the best mathematicians. The field is constantly expanding, as people invent new formalisms to solve their own problems. And with any given math problem, just like in programming, there's more than one way to do it. You can pick the one you like best.没人能了解所有的数学,就是最棒的数学家也不是.当人们发明新的形式去解决自己的问题时,数学领域就不断的扩展.一些给出的数学问题,也正如编程,不止一种方法可以去解决他.你可以挑个你最喜欢的方式.Math is... ummm, please don't tell anyone I said this; I'll never get invited to another party as long as I live. But math, well... I'd better whisper this, so listen up: (it's actually kinda fun.)数学是......嗯,请别告诉别人我说过这个哈;当然我也不指望谁能邀请我参加这样的派对,在我还活着的时候.但是,数学其实就是......我还是小声的说吧,听好了:(她其实就是一种乐趣啦!)The Math You Learned (And Forgot)你学到的数学(和你忘了的数学)Here's the math I learned in school, as far as I can remember:这儿是我能记得的在学校学到的数学:Grade School: Numbers, Counting, Arithmetic, Pre-Algebra ("story problems")初中:数,数数,算术知识,初级代数("带问题的小故事")High School: Algebra, Geometry, Advanced Algebra, Trigonometry, Pre-Calculus (conics and limits)高中:代数,几何,高等代数,三角学,微积分先修课 (二次曲线论和极限)College: Differential and Integral Calculus, Differential Equations, Linear Algebra, Probability and Statistics, Discrete Math大学:微积分,微分公式,线性代数,概率和统计,离散数学How'd they come up with that particular list for high school, anyway? It's more or less the same courses in most U.S. high schools. I think it's very similar in other countries, too, except that their students have finished the list by the time they're nine years old. (Americans really kick butt at monster-truck competitions, though, so it's not a total loss.)上面那个关于高中数学课程单子上所列的,怎么来着?美国高中几乎都是这样的课程设置.我认为其他国家也会很相似的,除了那些在9岁之前就掌握了这些课程的学生.(美国小孩同时却在热衷于玩魔鬼卡车竞赛,虽然如此,整个来说也算不上什么大损失.)Algebra? Sure. No question. You need that. And a basic understanding of Cartesian geometry, too. Those are useful, and you can learn everything you need to know in a few months, give or take. But the rest of them? I think an introduction to the basics might be useful, but spending a whole semester or year on them seems ridiculous.代数?是的.没问题.你需要代数.和一些理解解析几何的知识.那些很有用,并且在以后几个月里,你能学到一切你想要的,十拿九稳的.剩下的呢?我认为一个基本的介绍可能会有用,但是在这上面花整个学期或一年就显得很荒谬了.I'm guessing the list was designed to prepare students for science and engineering professions. The math courses they teach in and high school don't help ready you for a career in programming, and the simple fact is that the number of programming jobs is rapidly outpacing the demand for all other engineering roles.我现在意识到那个书单列表原是设计来准备给那些以后要当科学家和工程师的学生的.他们在高中里所教的数学课程并不是为你的编程生涯做准备的,简单的事实是,多数的编程工作所需要的数学知识相比其他作为工程师角色的人所需要的数学增长的更快.And even if you're planning on being a scientist or an engineer, I've found it's much easier to learn and appreciate geometry and trig after you understand what exactly math is —where it came from, where it's going, what it's for. No need to dive right into memorizing geometric proofs andtrigonometric identities. But that's exactly what high schools have you do.即使你打算当一名科学家或者一名工程师,在你理解了什么是数学之后-- 数学它如何而来,如何而去,为何而生,我发现这更加容易去学习和欣赏几何学和三角学.不必去专研记住几何上的证明和三角恒等式,虽然那确实是高中学校要求你必须去做的.So the list's no good anymore. Schools are teaching us the wrong math, and they're teaching it the wrong way. It's no wonder programmers think they don't need any math: most of the math we learned isn't helping us.所以这样的书单列表不再有什么用了.学校教给我们的不是最合适的数学,并且方式也不对.不奇怪程序员认为他们不再需要数学:我们学的大部分数学知识对我们的工作没什么大的帮助.The Math They Didn't Teach You他们没有教给你的那部分数学The math computer scientists use regularly, in real life, has very little overlap with the list above. For one thing, most of the math you learn in grade school and high school is continuous: that is, math on the real numbers. For computer scientists, 95% or more of the interesting math is discrete: i.e., math on the integers.在现实中,计算机科学家经常使用的数学,跟上面所列的数学仅有很小的重叠. 举个例子,你在中学里学的大部分数学是连续性的:也就是说,那是作为实数的数学.而对于计算机科学家来说,他们所感兴趣的95%也许更多的是离散性的:比如,关于整数的数学.I'm going to talk in a future blog about some key differences between computer science, software engineering, programming, hacking, and other oft-confused disciplines. I got the basic framework for these (upcoming) insights in no small part from Richard Gabriel's Patterns Of Software, so if you absolutely can't wait, go read that. It's a good book.我打算在以后的博客中再谈一些有关计算机科学,软件工程,编程,搞些有趣的东东,和其他常常令人犯晕的训练.我已经从Richard Gabriel的软件的模式这本书中洞察到一个无关巨细的基本框架.如果你明显的等不下去的话,去读吧.是本不错的书.For now, though, don't let the term "computer scientist" worry you. Itsounds intimidating, but math isn't the exclusive purview of computer scientists; you can learn it all by yourself as a closet hacker, and be just as good (or better) at it than they are. Your background as a programmer will help keep you focused on the practical side of things.到现在为止,不要让"计算机科学家"这个词困扰到你.它听上去很可怕,其实数学不是计算机科学家所独有的领域,你也能作为一个黑客自学它,并且能做的和他们一样棒.你作为一个程序员的背景将会帮助你保持只关注那些有实践性的部分.The math we use for modeling computational problems is, by and large, math on discrete integers. This is a generalization. If you're with me on today's blog, you'll be studying a little more math from now on than you were planning to before today, and you'll discover places where the generalization isn't true. But by then, a short time from now, you'll be confident enough to ignore all this and teach yourself math the way you want to learn it.我们用来建立计算模型的,大体上是离散数学.这是普遍的做法.如果正好今天你在看这篇博客,从现在起你正了解到更多的数学,并且你会认识到那样的普遍做法是不对的.从现在开始,你将有信心认为可以忽略这些,并以你想要的方式自学.For programmers, the most useful branch of discrete math is probability theory. It's the first thing they should teach you after arithmetic, in grade school. What's probability theory, you ask? Why, it's counting. How many ways are there to make a Full House in poker? Or a Royal Flush? Whenever you think of a question that starts with "how many ways..." or "what are the odds...", it's a probability question. And as it happens (what are the odds?), it all just turns out to be "simple" counting. It starts with flipping a coin and goes from there. It's definitely the first thing they should teach you in grade school after you learn Basic Calculator Usage.对程序员来说,最有效的离散数学的分支是概率理论.这是你在学校学完基本算术后的紧接着的课.你会问,什么是概率理论呢?你就数啊,看有多少次出现满堂彩?或者有多次是同花顺. 不管你思考什么问题如果是以"多少种途径..."或"有多大几率的...",那就是离散问题.当他发生时,都转化成"简单"的计数.抛个硬币看看...? 毫无疑问在他们教你基本的计算用法后他们会教你概率理论.I still have my discrete math textbook from college. It's a bit heavyweight for a third-grader (maybe), but it does cover a lot of the math we use in "everyday" computer science and computer engineering.我还保存着大学里的离散数学课本.可能他只占了三分之一的课程,但是它却涵盖了我们几乎每天计算机编程工作大部分所用到的数学.Oddly enough, my professor didn't tell me what it was for. Or I didn't hear. Or something. So I didn't pay very close attention: just enough to pass the course and forget this hateful topic forever, because I didn't think it had anything to do with programming. That happened in quite a few of my comp sci courses in college, maybe as many as 25% of them. Poor me! I had to figure out what was important on my own, later, the hard way.也真是够奇怪的,我的教授从没告诉我数学是用来干吗的.或者我也从来没有听说过.种种原因吧.所以我也从没有给以足够的注意:只是考试及格然后把他们都忘光,因为我不认为她还和编程有啥关系.事情变化是我在大学学完一些计算机科学的课程之后,也许是25%的课程.可怜啊!我必须弄明白什么对于自己来说是最重要的,然后再是向深度发展.I think it would be nice if every math course spent a full week just introducing you to the subject, in the most fun way possible, so you know why the heck you're learning it. Heck, that's probably true for every course.我想,如果每门数学课都花上整整一周的时间,而只是介绍让你如何入门的话,那将非常不错,这是最有意思的一种假设,那么你知道了你正学习的对象是哪种怪物了.怪物,大概对每一门课都合适.Aside from probability and discrete math, there are a few other branches of mathematics that are potentially quite useful to programmers, and they usually don't teach them in school, unless you're a math minor. This list includes:除了概率和离散数学外,还有不少其他的数学分支,可能对程序员相当的有用,学校通常不会教你的,除非你的辅修科目是数学.这些数目列表包括:Statistics, some of which is covered in my discrete math book, but it's really a discipline of its own. A pretty important one, too, but hopefully it needs no introduction.统计学,其中一些包括在我的离散数学课里,她的某些训练只限于她自身.自然也是相当重要的,但想学的话不需要什么特别的入门.Algebra and Linear Algebra (i.e., matrices). They should teach Linear Algebra immediately after algebra. It's pretty easy, and it's amazingly useful in all sorts of domains, including machine learning.代数和线性代数(比如,矩阵).他们会在教完代数后立即教线性代数.这也简单,这但相当多的领域非常有用,包括机器学习.Mathematical Logic. I have a really cool totally unreadable book on the subject by Stephen Kleene, the inventor of the Kleene closure and, as far as I know, Kleenex. Don't read that one. I swear I've tried 20 times, and never made it past chapter 2. If anyone has a recommendation for a better introduction to this field, please post a comment. It's obviously important stuff, though.数理逻辑.我有相当完整的关于这门学科的书没有读,是Stephen Kleene写的,克林闭包的发明者,我所知道的还有就是Kleenex.这个就不要读了.我发誓我已经尝试了不下20次,却从没有读完第二章.如果哪位牛掰有什么更好的入门建议的话可以给我推荐.虽然,这明显是非常重要的一部分.Information Theory and Kolmogorov Complexity. Weird, eh? I bet none of your high schools taught either of those. They're both pretty new. Information theory is (veeery roughly) about data compression, and Kolmogorov Complexity is (also roughly) about algorithmic complexity.I.e., how small you can you make it, how long will it take, how elegant can the program or data structure be, things like that. They're both fun, interesting and useful.信息理论和柯尔莫戈洛夫复杂性理论.真不可思议,不是么?我敢打赌没哪个高中会教你其中任何一门课程.她们都是新兴的学科.信息理论是(相当相当相当相当难懂)关于数据压缩,柯尔莫戈洛夫复杂性理论是(同样非常难懂)关于算法复杂度的.也就是说,你要把它压缩的尽量小,你所要花费的时间也就变的越长,同样的,程序或数据结构要变得多优雅也有同样的代价.他们都很有趣,也很有用.There are others, of course, and some of the fields overlap. But it just goes to show: the math that you'll find useful is pretty different from the math your school thought would be useful.当然,也有其他的一些因素,某些领域是重复的.也拿来说说吧:你所发现有用的那部分数学,不同于那些你在学校里认为有用的数学.What about calculus? Everyone teaches it, so it must be important, right?那微积分呢?每个人都学它,所以它也一定是重要的,不对吗?Well, calculus is actually pretty easy. Before I learned it, it sounded like one of the hardest things in the universe, right up there with quantum mechanics. Quantum mechanics is still beyond me, but calculus is nothing. After I realized programmers can learn math quickly, I picked up my Calculus textbook and got through the entire thing in about a month, reading for an hour an evening.好吧,微积分实际上是相当容易的.在我学习它之前,它听上去好像是世界上最难的一件事,好像和量子力学差不多.量子力学对我来说真的不是那么容易理解,但是微积分却不是.在我意识到程序员能够快速的学习数学时,我拿起一些微积分课本用一个月通读了整本书,一个晚上读一小时.Calculus is all about continuums —rates of change, areas under curves, volumes of solids. Useful stuff, but the exact details involve a lot of memorization and a lot of tedium that you don't normally need as a programmer. It's better to know the overall concepts and techniques, and go look up the details when you need them.微积分都是关于连续统的 -- 变化的比率, 曲线的面积, 立体的体积.是些有用的东西,但是实际细节却包含大量的记忆量并且枯燥,作为一个程序员来说根本不需要这些. 更好的方法是从整体上了解那些概念和技术,在必要的时候再去查询那些细节.Geometry, trigonometry, differentiation, integration, conic sections, differential equations, and their multidimensional and multivariate versions — these all have important applications. It's just that you don't need to know them right this second. So it probably wasn't a great idea to make you spend years and years doing proofs and exercises with them, was it? If you're going to spend that much time studying math, it ought to be on topics that will remain relevant to you for life.几何,三角,微分,积分,圆锥曲线,微分方程,和他们的多维和多元 -- 这些都有重要的应用.不过这时候不需要你去了解它们.这大概不是个好注意让你年复一年的去做证明和它们的练习题,不是吗?如果你打算花大量的时间去学习数学,那也是和你生活相关的部分.The Right Way To Learn Math学习数学的正确方法The right way to learn math is breadth-first, not depth-first. You need to survey the space, learn the names of things, figure out what's what.正确学习数学的方法是广度优先,而非深度优先.你要考察的是整个数学世界,学习每个概念的名字,区分出什么是什么.To put this in perspective, think about long division. Raise your hand if you can do long division on paper, right now. Hands? Anyone? I didn't think so.具体的来看,考虑用长除法?如果你能在纸上做长整除,现在就举起你的手.会有人举手吗?至少我不这么认为.I went back and looked at the long-division algorithm they teach in grade school, and damn if it isn't annoyingly complicated. It's deterministic, sure, but you never have to do it by hand, because it's easier to find a calculator, even if you're stuck on a desert island without electricity. You'll still have a calculator in your watch, or your dental filling, or something,回头看看在学校里学过的长除法,要是不让你觉得烦恼和愤怒才怪.当然,这是显然的,但你不一定要自己亲自去做,因为很容易用计算器来做,即使你不幸在一座没有电力的荒无人烟的小岛上.你起码还有个计算器,在的手表上,补牙的什么东东,或其他什么上面.Why do they even teach it to you? Why do we feel vaguely guilty if we can't remember how to do it? It's not as if we need to know it anymore. And besides, if your life were on the line, you know you could perform long division of any arbitrarily large numbers. Imagine you're imprisoned in some slimy 3rd-world dungeon, and the dictator there won't let you out until you've computed 219308862/103503391. How would you do it? Well, easy. You'd start subtracting the denominator from the numerator, keeping a counter, until you couldn't subtract it anymore, and that'd be the remainder. If pressed, you could figure out a way to continue using repeated subtraction to estimate the remainder as decimal number (in this case, 0.1185678219, or so my Emacs M-x calc tells me. Close enough!)为什么他们还教你这些呢?为什么我们感到含混心虚讷,如果我们不能记住怎样去做?这不是好像我们需要再次知道她.除此以外,如果你命悬一线,你可以运用任意大的数来做长除法.相象你被囚禁在第三世界的地牢里,那儿的独裁者是不会放你出来的,除非你计算出219308862/103503391.你会怎么做呢?好吧,很容易.你开始从分子减去分母,直到不能再减只剩余数为止.若实在有压力,你可以想个办法,继续使用反复减,估算作为十进制的余数(这种情况下,0.1185678219,Emacs M-x calc 告诉我的.够精确了! )You could figure it out because you know that division is just repeated subtraction. The intuitive notion of division is deeply ingrained now.你或许明白,除法就是反复的减.这样从直觉上对除法概念的理解就根深蒂固啦!The right way to learn math is to ignore the actual algorithms and proofs, for the most part, and to start by learning a little bit about all the techniques: their names, what they're useful for, approximately how they're computed, how long they've been around, (sometimes) who inventedthem, what their limitations are, and what they're related to. Think of it as a Liberal Arts degree in mathematics.学习数学的正确方法是忽略实际的算法和证明,对于大部分情况来说, ...:他们的名字,他们的作用,他们计算的大致步骤, (有时是)谁发明了他们,发明了多久了,他们的缺陷是什么,和他们相关的有什么.把数学当文科来学.Why? Because the first step to applying mathematics is problem identification. If you have a problem to solve, and you have no idea where to start, it could take you a long time to figure it out. But if you know it's a differentiation problem, or a convex optimization problem, or a boolean logic problem, then you at least know where to start looking for the solution.为什么呢?因为第一步反应在数学上的是问题的确定.如果你有一个问题去解决,并且假设你没有头绪如何开始, 这将花费你很长的时间来弄明白.但如果你知道这是个变异的问题,或者是一个凸优化问题,或者一个布尔的逻辑问题,然后你起码能知道从哪着手开始寻找解决方案.There are lots and lots of mathematical techniques and entire sub-disciplines out there now. If you don't know what combinatorics is, not even the first clue, then you're not very likely to be able to recognize problems for which the solution is found in combinatorics, are you?现在有许许多多的数学技术和整个的学科分支.如果你不知道组合逻辑是什么,甚至连听都没听说过, 那么你是不可能意识到在组合逻辑中可以找到的解决答案的问题的,难道不是么?But that's actually great news, because it's easier to read about the field and learn the names of everything than it is to learn the actual algorithms and methods for modeling and computing the results. In school they teach you the Chain Rule, and you can memorize the formula and apply it on exams, but how many students really know what it "means"? So they're not going to be able to know to apply the formula when they run across a chain-rule problem in the wild. Ironically, it's easier to know what it is than to memorize and apply the formula. The chain rule is just how to take the derivative of "chained" functions —meaning, function x() calls function g(), and you want the derivative of x(g()). Well, programmers know all about functions; we use them every day, so it's much easier to imagine the problem now than it was back in school.但那实在是个大新闻哪,因为阅读这些领域,学习实际算法,建模和计算结果的方法,记住这些名字都是容易的.在学校里他们教你链式法则,你也能回忆起他们并能运用在考试题上,但有多少学生能真正的了解他们到底意味着什么呢? 所以当他们遇到变种的链式问题时,他们就不懂得如何运用公式了.让人感到讽刺的是,了解这是什么比记住如何运用公式更为容易.链式法则仅仅是如何对链式函数求导的意思,函数 x() 引用函数 g() ,你要求导 x(g()) .好了,程序员知道所有这些函数相关的;我们每天都使用他们,所以现在比过去在学校更加容易能够想象到问题所在.Which is why I think they're teaching math wrong. They're doing it wrong in several ways. They're focusing on specializations that aren't proving empirically to be useful to most high-school graduates, and they're teaching those specializations backwards. You should learn how to count, and how to program, before you learn how to take derivatives and perform integration.这就是为什么我认为他们以错误的方式在教数学. 对大多数高中毕业生来说,他们专门教授的内容,不是可以靠经验来证明数学是如何如何有用的,他们教的那些恰恰是非经验式的内容.在你学习如何求导和做积分之前,你将要学习如何计数,怎样编程.I think the best way to start learning math is to spend 15 to 30 minutesa day surfing in Wikipedia. It's filled with articles about thousands of little branches of mathematics. You start with pretty much any article that seems interesting (e.g. String theory, say, or the Fourier transform, or Tensors, anything that strikes your fancy.) Start reading. If there's something you don't understand, click the link and read about it. Do this recursively until you get bored or tired.我认为学习数学最好的方法是每天花15到30分钟逛维基百科.那上面有数千数学分支的相关文章. 可以从一些你感兴趣的文章着手(比如,弦理论,或者,傅立叶变换,或者张量理论,就是能冲击你相象力的东西) 阅读.如果有什么你不理解的,就去了解那些链接.如此这般直到你累到不行为止.Doing this will give you amazing perspective on mathematics, after a few months. You'll start seeing patterns —for instance, it seems that just about every branch of mathematics that involves a single variable has a more complicated multivariate version, and the multivariate version is almost always represented by matrices of linear equations. At least for applied math. So Linear Algebra will gradually bump its way up your list, until you feel compelled to learn how it actually works, and you'll download a PDF or buy a book, and you'll figure out enough to make you happy for a while.几个月后,这么做会纵向扩展你的数学知识面.你会发现一些模式,好比,数学的每个分支看上去都包括了一个有着复杂的多元的变量,然后线性代数将会慢慢爬满你的书单列表,直到你强迫自己学会他实际上是怎样工作的,你要下载个电子。

数学老师对你的影响英语作文Math teachers play a significant role in shaping students' attitudes and abilities in mathematics. They not only teach mathematical concepts and problem-solving skills but also inspire and motivate students to appreciate the beauty and importance of mathematics.My math teacher has had a profound impact on me. She is not only knowledgeable and passionate about mathematics but also patient and dedicated to helping her students succeed. Her clear explanations and engaging teaching style have made complex mathematical concepts easier to understand and apply. She has encouraged me to think critically, ask questions, and explore different approaches to solving problems.One of the most valuable lessons I have learned from my math teacher is the importance of perseverance and resilience in the face of challenges. She has taught methat making mistakes is a natural part of the learning process and that it is okay to struggle and ask for help. Her support and encouragement have given me the confidenceto tackle difficult problems and overcome obstacles in my mathematical journey.Moreover, my math teacher has instilled in me a love for mathematics and a curiosity to explore its endless possibilities. She has shown me that mathematics is notjust about numbers and equations but also about creativity, logic, and problem-solving. Her passion for the subject has inspired me to pursue further studies in mathematics and consider a career in a math-related field.Overall, my math teacher has been a positive influencein my life, shaping not only my mathematical skills but also my mindset and attitude towards learning. I amgrateful for her guidance, support, and inspiration, and I will always cherish the lessons I have learned from her.数学老师对我有着深远的影响。

OntheFunctionsof...谈情景教学法在中学课堂上的作用On the Functions of the Situational Teaching Methodology in MiddleSchool Classes摘要随着英语教学的开展，教学改革的推进，很多科学而又有效的教学方法展现在课堂教学中。

情景教学法就是一种新颖且极又效果的教学方法。

情景教学法强调语言学习的真实情景。

通过创设语言学习的情景，以情景为中心，培养学生的听说能力。

由于其强调语言和环境的结合，可以增强学生对语言的感性认识。

中学生理解能力不高但擅长对感性认识的记忆,因此情景教学法非常适合教授中学生学习语言时使用。

关键词：小学英语；情景教学法；学生特征；作用AbstractWith English carrying out in middle schools and teaching reforms promoting, lots of scientific and efficient teaching methods appear in classroom teaching. Situational teaching methodology is a novel and effective teaching method.Situational teaching methodology which emphasizes the real scene of language learning cultivate students? listening and speaking abilities by means of creating the scene of language learning and regarding the scene as the center. As the result of its combination of language and environment, situational teaching methodology can enhance students? perceptual understanding of language. For students are not good at understanding but adept in keeping perceptual knowledge in mind, situational teaching methodology is quite suitable to pupils especially the lower grades to study language.Key words: middle English; Situational teaching methodology；Students? characteristics；FunctionContents摘要 (i)Abstract (ii)I.Introduction (1)II.Literature reviewIII.. The Basic Theory of Situational Teaching Methodology⒈The principle of interaction between emotional and cognitive activities⒉The intuiti ve principle of understanding⒋Human's awarenessIV. The Importance of Situational Teaching Methodology1. to develop students? interest2. to cultivate students? practical ability to use English3. to master knowledge and comprehending the difficultiesV. The Foundation and Significance of the Application of Situational Teaching MethodologyFoundation: 1. the students? psychological characteristics2. the structure of textbook3. the standard requirements of new curriculumsSignificance：1. To reduce the difficulty of teaching2. To reduce the degree of fatigue3. To stimulate and improve students? interestVI. carry on Situational Teaching Methodology1. Set up the scene of listening and speaking2. Create visual scenes3. Create a successful scenarioVII.C onclusion………………………………………Bibliography ……………………………………………………………………..Ⅰ. IntroductionSince the purpose of teaching a foreign language is to enable learners to use it, then it should be heard, spoken, read and written according to a realistic situation.Neither translation nor mechanical exercise can help if they are not related to practical life. Drilling of words and sentence structure or make the most out of the substitution table, will inevitably lead to unreality, boredom, and the remoteness of the language. So it lies in the fact that the structure should be presented in a situation where they can be used.As is known to all, the status of English as an international language and a major medium in the Information Age is more and more important in the 21st century. So to start the English education from middle school students has become necessary. Furthermore, it is a national step to make such a start because English education is a systematic project whose different stages are linked with one another just like the rings on a chain. On this chain, middle school is the special ring for it is the starting point and foundation for others. Childhood is the ideal phase to acquire a native or foreign language.So it is vital to start English teaching in middle school. English is a very practical tool for discipline, the use of practical English communication is the main performance. Therefore, English teaching in middle schools must provide students with sufficient English for effective use of the opportunity to practice, So that students can deepen understanding and familiar with the use of a firm grasp, to achieve the purpose of learning. Bloomfield also thinks “the successful foreign language classroom instruction should establish a great many situations in the lesson, let students have opportunities to use the language materials thathave already been learnt.” These have fully reflected that characteristics of children naivety and activeness. The traditional English teaching methods, “Duck-stuffing” and “translat es from English to Chinese”, not only make the atmosphere of the classroom depressive but also have the students accept the knowledge passively, and moreover they don?t fit for the pupil?s psychology and age characteristics at all. In order to change this phenomenon, we can adopt the corresponding interesting and intriguing means based on the different contents of courses to excite students? interest to study English.This thesis is one of the many attempts aimed at the goal to enable to students learn English effective and be willing to learn English.II. Literature ReviewWhat is the situational teaching method? This is an audio-visual method that emerged after World War II in Europe. The so-called scene teaching Method (Situational Method) also called audio-visual Method (the Audio-visual Method), refers to use visual illustration, multimedia, role playing, experiment and many other kinds of means to create the classroom teaching situation, and make the thinking in image and the abstract thinking cognitive and emotional, which combined teaching and learning skills, gave full play to the classroom teaching of middle school, made students enthusiastic, initiative and creative, and changed the old teaching methods that students simply accept a kind of knowledge passively (Richard & Rodgers 10-16). It pays attention to intuitive image, vivid knowledge representation and concrete practice and exercises, to arouse students?positive emotional experience and train students? active learning attitude. To make the students experience learning a foreign language, the seenand heard of the nature should be linked to help students to understand and gain the knowledge or skill, so that the students?psychological function will be developed (Brumfit 186).Situational Teaching Method means a lively or concrete situation or scene the teacher establish with the emotional color by the body of image, in order to draw the students? attitude experience, to help the students unde rstand the textbook, also improve the students? mentality development. The center of Situational Teaching Method is to motivate the students? emotion. Make them learn actively and happily.Its main features:a. Image realityIt inquires the image which is not only the same outside with the real object, but also the same within the spirit. It is not the copy of an object, not as taking a photograph. It is by a simplified figure, suggest technique to get an image which is the same structure with the corresponding object, in order to make the students feel the truth and reality. Then join in to the situation step by step.b.Emotion realityThis means real emotion and genuine purpose. Emotion can join in our learning and studying, it will fully motivate the stud ents? initiative. Emotion join in class is thehighest boundary of teaching and studying. This can come true by using the pattern of Situational Teaching Method easily.c.Broaden situationIt refers to the situation which is wide and broad; this also means enough rooms for the students to exert their imagination, give them a good chance to imagine. Situational Teaching Method does not mean illustrate something and then setsomething up mechanically. It is treated as a whole, and then show it in front of the students, make them feel the “direct impression”; the situation stirred the students? emotion and then become the pusher of their needs.d.Contain theories abundantlyAbundantly contain theories means abstract theory come up with image, this can increase the students? cognition efficiently. Start off from the Centre of the textbook, the teaching form is depending on the teaching content. While teaching, establish situation group by group surrounding the teaching centers.“Making use of the learning environmental factors such as situation, cooperation and conversation and so on to fully give play to students?initiative, positivity and pioneering spirit, and finally to achieve the goal of making students effectively know the significance of what they studied”(Wang Xibao,1996). It will get better results to let students in the specific situations and they would put the knowledge into practice through simulating the situations.To sum up, the situational teaching method bases on the modern psychology, pays much attention to students?feelings, and absorbs the advantages of traditional teaching methods. It stands for the developing trend of modern teaching approach.Ⅲ. The theoretical basis of the Situational teaching methodology1. The principle of the interaction between emotional and cognitive activitiesThe research of emotional psychology shows that the individual emotions have at least three functions: motivating, strengthening and adjusting functions. Motivating function refers to the emotions increase or decrease the efficiency ofcognitive activities, namely the healthy and positive emotions play an active role in cognitive activities while the negative and unhealthy emotions hinder or restrain cognitive activities. The emotions increase the efficiency of cognitive activities, from which we gain new enlightenment on solving the problem that at present the middle students have insufficient motive power. Emotional adjusting function refers that the emotions organize or disintegrate cognitive activities. But too strong or too weak mood and bad mood may cause confusion in thinking and difficulty in memorizing. The situational teaching methodology requires creating the proper situation which makes students feel relaxed, pleasant and calm.Neurophysiological studies have shown that the image of the right brain in charge of thinking, is the emotional brain. Creation of scenarios is the golden key to open the right brain. In specific situations to your liking, that English will not only stimulate the brain, activates the right brain, causing a high degree of excitement. Turn left and right brain so excited, thinking and language not only caused by resonance, and can greatly develop the potential of the brain, the students will not feel tired, but was clear-headed, quick thinking. The studies of cerebral science show that people?s left and right cerebral hemisphere is in charge of logical, rational and analytic thinking, including verbal function. While the right one is responsible for intuition, creativity, and imagination, including emotional activities. The situational teaching methodology always let students feel first and then express in language, or make students feel at the same time innervate the internal language. When feeling, the right cerebral hemisphere which is in charge of the visual ideation excites; when expressing, the left one which hasthe function of abstract thinking excites. In this way, the two hemispheres of the brain excite and restrain alternately or excite andcooperate simultaneously, which can help to tap people cerebral potentialities greatly. And students could study in comfortable and pleasant atmosphere. So the situational teaching method can obviously get better teaching result than traditional teaching. 2. The intuitive principle of cognition According to the methodology, the situational teaching method carries out based on the principle of theory of reflection and the effect of objective reality influencing children?s subjective consciousness. And the images of world just enter the children?s consciousness through visualization. Consciousness is a reflection of objective reality. The context which the situational teaching method creates is conducive to the development of children in the external environment for it is consciously created by man and optimized. This optimized objective context places the children in specific situation under the domination of teachers? words. It not only affects the children?s cognitive psychology but also prompt the children?s emotional activities to inv olve in learning, which leads to the children?s own self-movement.More than 300 years ago, the Czech educator Comenius wrote in the “Didactica magna”: “All the knowledge starts from the sensory organ.” This exposition re flected an important aspect of students understanding the law in the teaching process: visual representation could bring about the reification and visualization of abstract knowledge and it also could help students improve the system of perceptual knowledge. The situational teaching method places the students in the particularcircumstance to show them vivid and specific images (including direct and indirect images) so that the students can achieve the perception of abstract and rational cognition from visual perception on one hand; and on the other hand it arouses students? learning spirits and interests, making learning become students? initiative and conscious activities.4. The person’s cognition is the process of combination of consciousand unconscious mental activities.As we all know, the conscious mental activity is the summation of psychological activities that subject realizes the object, including intentional consciousness, intentional memory, intentional attention, intentional recognition, intentional recollection, intentional imagery, intentional representation, logical and verbal thinking, and intentional experience etc. The final aim of situational teaching is to evoke and utilize the cognitive potential that unconscious mentality provides.The unconscious mental phenomenon has been paid much more attention by scholars since Freud. The so-called unconscious mentality is the sum of all the psychological activities which people are conscious of, the unification of the unconscious cognition and internal experience, and the necessary reactive forms. It consists of unintentional perception, unintentional memorization, unintentional recognition, intentional presentation, and intentional imagery, nonverbal thinking, unintentional attention, unintentional experience etc. This definition emphasizes two functions of unconscious mental activities:(1). an unconscious cognitive function on the object. For example, when we are talking while walking along the road, wewould not have much impression on the roadside scenery and other things. But we will not be stumbled by a pile of stones on the road. The reason is that the stones actually draw our attention and make us generate a physical reaction of “dodging” which is unconscious, unheeded and involuntary.(2). an unconscious internal experienced function on the object. It is said that the "emotional contagion" is the manifestation of unconscious mentality. For instance, we always feel unprovoked happiness or perceptible melancholy, which is often the result of state of mind. The state of mind itself is an emotional state and can affect people?s experience and activities.Ⅳ. The Importance of Situational Teaching MethodologyThe situational teaching methodology refers to that the teachers make use of certain means to create the needed teaching scenes so that the students can understand and apply the knowledge. It can inspire students? cognition from several aspects; enhance their enthusiasm and the awareness of participating. And the class teaching effect will be greatly improved as the situational teaching method applies to the class teaching.1.deve lop and enhance students’ interestThe middle period is the enlightenment phase of students learning English. During this period, the interest is especially important. In our country the pupils lack the real language environment when learning English and therefore they are not very interested in English. It helps to develop and enhance students' interest in learning English to create the language environment of listening and speaking English. It will be very helpful for students to find interest in learning English by stimulating their sensory organs.2. cultivate students’ practical ability to use EnglishThe purpose of we learning knowledge is to make better use of knowledge, and this was particularly prominent in language teaching. The teaching contents of English teaching materials are placed in a certain language environment. We set up a feasible and appropriate language context according to the specific content of teaching materials and our processing and students will naturally play their own part under teacher?s guidance. For instance, in the units of shopping of the textbook of Grade Four, teachers can design the classroom as a shop based on the vocabulary and dialogues in these units. Learning in the designed situation and using in the contexthave students not only understand the meaning of dialogues more clearly but also allow them to experience personally on the scene. In this way their interests are to be aroused and they will then be absorbed in classroom to learn new knowledge.3. master knowledge and comprehending the difficultiesSome key points and difficult points when conveyed directly are quite drab and boring in English teaching materials. Students may not be able to understand them, but it is will be easier for students to learn when those are put in a certain language context. For example, in the first unit of the textbook of Grade Five, there is a dialogue of “Who?s your math teacher? Mr. Zhao. What's he like? He's thin and short. He's very kind”. For students it is awfully boring and difficult to learn it directly. At this time instructor can make use of multimedia to place the teachers? various photos of different subjects on the screen and ask students to describe the photos. The students want to describe the characteristics of their teachers when they see the photos. In this way they will bear the words or sentences in mindunconsciously and can use the flexibly. Because the characteristics of teachers are different, students would make their efforts to learn the new words by heart in order to speak out in front of the classmates.V. The Foundation and Significance of the Application of Situational Teaching MethodologyThe theory of the situational teaching Mr. Li Jilin founded came into being in the practice of middle Chinese teaching, but it is of far-ranging significance and furthermore it has been increasingly and widely applied to different subjects. The author will enumerate the foundation of the application of situational teaching method as following combining with its characteristics.(1). students’ psychological characteristicsThe students are good at imitating and lively and active, have keen sense of hearing and good ideation of specific images, and cannot pay long attention on a thing intentionally. If teacher adopts the traditional and single teaching model in classroom teaching, it cannot bring the teaching into full play in the light of students? psychological characteristics. Attaching importance to interests is another characteristic of the situational teaching method. Teachers should create a variety of scenes by dint of different ways in classroom teaching to make the teaching content visual, specific and vivid and acco rdingly to motivate students? interests. Psychologist Jean Piaget said: “All effective work must take some interests as a prerequisite.” As long as the students become interested in learning, they will go to study on their own initiative and are willing to exchange their feelings and express their own ideas. So the good teaching effect can be achieved if students go through the comprehensive language training in a relaxed and pleasant atmosphere. From perceptualto rational knowledge, this is the universal law of access to knowledge. The integration of linguistic knowledge in a specific scenario, the knowledge and scenarios, abstract closely integrated with the image, set up by the perceptual knowledge for students to understand the bridge leading to the rational, so that students in the learning process to maintain interest.(2). the structure of textbookThe textbooks of elementary school all adopt the system of combining structure and function, taking the structure as the principal thing and supplemented by the scenes, and is permeated the training of communicating function from the beginning to end. The chosen teaching materials also tally with pupils? daily life and sociable learning scenes such as park, school and store etc. The dialogues are always featured with everyday phrases like greeting and thanks and so on. To provide the students the authentic language environment to learn English helps them improve their English level as well as contributes students to cultivate the capacity of using language and ability of commutating.(3). the standard requirements of new curriculumsSince the implementation of the new curriculum standards, middle foreign language teaching more and more attention. Middle English teaching has undergone many changes. However, the majority of foreign language teaching in China still remains in traditional 3P (Presentation-Practice-Production) education level. Many teachers still occurs everyday English, boring classroom atmosphere, students passive acceptance, resulting in tired of psychology. How to make foreign language teaching to adapt to changes in foreign language teaching in the 21st century, how to change the foreign language teaching …time-consuming and inefficient,? T o do this, we must break the traditional mode of teaching, to seek more effective, more suitable for the characteristics of foreign language teaching students.The focus of English curriculum reform is to solve the problem that teacher attaches much importance to explaining and imparting grammar or vocabulary in the course of teaching instead of ignoring the tendency of fostering students? actual ability of using language, insist on that the curricula ought to base on students? interests in learning, life experience and cognitive level, and advocate the learning mode of experiencing, practicing, participating, cooperating and communicating and task-based teaching approach.The core of middle school English teaching is to stimulate the enthusiasm of students to learn English, which requires close contact with the contents of middle school teaching and learning practical life, and continuously enrich the lives of the factors which go into the classroom, and the gradual creation of a reproduction of life, the scene of the expression of life to create a communicative life into the teaching mode. T o enable students to obtain treatment and use information, communicate with others in English, using English to solve practical problems. Attention should be paid off from the daily life of students, training students the ability to actually use the language. From whichever point of view, the standard of new curriculums put forwardthe requirements all-side for English teaching: emphasizing taste, perception, participation, and practice.In fact the use of situational teaching method in middle schools English teaching has great significance:(1). to reduce the difficulty of teachingThe teaching material for students is an unknown area. Situational teaching method reproduces the relevant context of teaching material and makes use of role effect to encourage students to think, try and discover by themselves, which reduces the psychological distance between students and teaching material. Because the situation in the scenes acts on students? sense organs and strengthens their fee lings so that the students feel the teaching material to be authentic even approachable thus giving rise to resonance and immersing themselves in the scene. The situational teaching method can reduce the difficulty of teaching, which ensures that on the one hand students can apperceive and understand teaching content comprehensively, thoroughly and quickly, on the other students could solve the problems with the learned knowledge correctly and rapidly.Especially for the students in lower grades for their superficial and visual thinking, the suitable teaching situation are so necessary that students are used to accept the images first, then establish the relationship between observed objects and words and store them in mind.(2). to reduce the degree of fatigueThe appropriate teaching scenes can reduce difficulty in teaching process, so the students consume less energy when learning, which not only is helpful to students? health but also helps to maintain their attention for a longer time. Learning when playing, playing when learning. Loving to play is the nature of students. Teacher can take advantage of this to make students gradually obtain the new knowledge in all sorts of games instead of learning by rote. To show students a variety of forms of knowledge to be grasped and indoctrinate them thesystematical knowledge as a whole in accordance with their mnemonic features. The students may bear some in mind more or less in any case. On the condition that students are still not very familiar with much knowledge, teacher could strengthen their memory by passing the card, doing the action, finding your friend, drawing and coloring etc. Through the frequent listening and repeating in the games, the impressions are automatically deepened. So the students not only could throw themselves into the games but also reduce their psychological pressure and burden. It is possible for them to enjoy the comfortable and harmonious language environment and meanwhile to make the classroom lively and vivid.(3). to stimulate and i mprove students’ interestTo improve the English teaching quality on a large scale, it must to be good at using a variety of forms, multi-media to create a good language environment and teaching contexts for students? English practice. The purpose of crea ting the teaching situation is to arouse the students? mental activities such as the state being of being surprised and puzzled to promote the students to think actively. To create the teaching situation suits for students? cognitive rule to enable them to participate in the teaching activities by studying students? learning psychology and having a good master of their practical situations.VI. Carry on Situational Teaching MethodTeaching is a scene full use of the image in the teaching, the creation of specific scenes vividly aroused emotions of students in order to guide students to understand and apply the whole language teaching method. The purpose is to stimulate students? interest in learning, develop good study habits. Creation of scenarios for teaching, to better help students understand thelanguage, to improve the actual use of language.Scene of teaching can also significantly reduce teaching time, and achieved good results. Natural language in the real atmosphere, students can actively participate in a bold attempt to greatly enhance the language information input, and produce language internalization; do apply their knowledge, so as to further lay a good foundation for learning English.1. Set up the language scene of listening and speaking(1)．The teacher insists on instructing the class in English.Teachers persist in speaking English in classroom, make the classroom be a small English-speaking world, and increase the chances for students practicing English which may lead to motivate students? interests, promote the development of English language sense and meanwhile review and consolidate the knowledge have learned before. This can also increase the opportunities for students to hear English, so that every student naturally form the auditory memory under the unconscious influence by what they constantly hears or sees and improve the abilities of listening and speaking as well as the self-learning ability.(2). Make use of audio-visual mediaThe tape recorder is the “conventional weapon” and “in dispensable weapon” for English teachers. It effectively provides students with more opportunities for auditory training to foster a sense of language. In the process of English classroom teaching, recording, video tape, projector and other electronic means can be used in the simulated situations. They reproduce the space time of the dialogue visually and vividly to manage to connecting sounds with images, pictures and words. This suits for the characteristics of students: being curious about everything,。

专题06 书面表达（应用文写作）新高考八省最新名校联考试题汇编(解析版)距离高考还有一段时间，不少有经验的老师都会提醒考生，愈是临近高考，能否咬紧牙关、学会自我调节，态度是否主动积极，安排是否科学合理，能不能保持良好的心态、以饱满的情绪迎接挑战，其效果往往大不一样。

以下是本人从事10多年教学经验总结出的以下学习资料，希望可以帮助大家提高答题的正确率，希望对你有所帮助，有志者事竟成！养成良好的答题习惯，是决定高考英语成败的决定性因素之一。

做题前，要认真阅读题目要求、题干和选项，并对答案内容作出合理预测;答题时，切忌跟着感觉走，最好按照题目序号来做，不会的或存在疑问的，要做好标记，要善于发现，找到题目的题眼所在，规范答题，书写工整;答题完毕时，要认真检查，查漏补缺，纠正错误。

总之，在最后的复习阶段，学生们不要加大练习量。

在这个时候，学生要尽快找到适合自己的答题方式，最重要的是以平常心去面对考试。

英语最后的复习要树立信心，考试的时候遇到难题要想“别人也难”，遇到容易的则要想“细心审题”。

越到最后，考生越要回归基础，单词最好再梳理一遍，这样有利于提高阅读理解的效率。

另附高考复习方法和考前30天冲刺复习方法。

1.【广东省深圳市中学2022-2023学年高三试题】假定你是李华，乘坐FL753航班抵达伦教后发现钱包遗失。

请给航空公司写一封邮件说明情况并寻求帮助。

内容包括：1. 行程信息；2. 钱包特征；3. 联系方式。

注意：1. 词数80左右；2. 可以适当增加细节，以使行文连贯。

____________________________________________________________________________________________ ____________________________________________________________________________________________ ____________________________________________________________________________________________ __________________________________【答案】Dear Sir/ Madam,I am writing to report the loss of my wallet. On 1st November, 2018 I flew business class on your flight 753 from China to London.I clearly remember it was with me immediately I boarded the plane. However, on arrival in London Heathrow Airport, I found it gone. I guess it must have slipped out of my pocket during the flight. It is a black leather one with around 200 dollars, my ID card, and 2 credit cards in it. Along with the items mentioned above, the wallet contains a family photo, my precious memory. As you can imagine, these matter a lot to me.Hasanybodyhappenedtofindit?*********************************Thank you for your time and support.Yours sincerely,Li Hua 【分析】本篇考查的写作类型是求助信【详解】第一步：根据提示可知，本文是一封求助信。