为什么数学老师这么不擅长解释(一)
正文翻译
Why are math teachers so bad at explaining things?
为什么数学老师这么不擅长解释?
Why are math teachers so bad at explaining things?
为什么数学老师这么不擅长解释?
评论翻译
Nico Lindsay
Some of these other answers are laughably egotistical and negative. None of them consider the uniqueness of mathematics as a discipline. The primary answer to this question is just that – that mathematics is genuinely a hard subject to teach.
Think about any realization you’ve had while studying math. Some problems seem daunting, but once you’ve had a look at the answer often times it seems like it should’ve been obvious. Usually because there exists a nicely elegant and perhaps rather creative solution being presented.
Think about every time you’ve seen that “beautiful solution” and convinced yourself the problem is easier than it actually is. Well duh, everything seems easier in hindsight. A similar thing happens when teaching math. The more mathematics you understand, the harder it is to gauge how difficult the material is. 3Blue1Brown makes a good point of this in his one video
by the end he’s talking about the author of math textbooks who must come up with practice problems. He noted that:
“Across a wide variety of contributors there is one constant: Nobody is able to tell how difficult their exercises are. Knowing when math is hard is waaayyyy harder than the math itself.”
And I think this is something to keep in mind both as an instructor and as a student.
其他一些答案都是可笑的自我主义和消极的。他们都没有考虑到数学作为一门学科的独特性。这个问题的主要答案就是——数学确实是一门很难教的学科。
想想你在学习数学时的任何体会。有些问题看起来让人望而生畏，但一旦你看到了答案，往往看起来应该是显而易见的。通常是因为存在一个非常优雅的，也许是相当有创意的解决方案。
每当你看到“美丽的解决方案”并确信问题比实际更容易时，想想吧。事后看来，一切似乎都更容易。在教数学时也会发生类似的事情。你理解的数学越多，就越难衡量材料的难度。科普博主3blue1brown在他的一个视频中很好地说明了这一点。
最后，他谈到了数学教科书的作者，他必须提出实践问题。他指出：
“在各种各样的贡献者中，有一点是不变的:没有人能说出他们的练习有多难。知道数学什么时候难比数学本身更难。”
我认为这是作为一名教师和学生都应该牢记的。
Some of these other answers are laughably egotistical and negative. None of them consider the uniqueness of mathematics as a discipline. The primary answer to this question is just that – that mathematics is genuinely a hard subject to teach.
Think about any realization you’ve had while studying math. Some problems seem daunting, but once you’ve had a look at the answer often times it seems like it should’ve been obvious. Usually because there exists a nicely elegant and perhaps rather creative solution being presented.
Think about every time you’ve seen that “beautiful solution” and convinced yourself the problem is easier than it actually is. Well duh, everything seems easier in hindsight. A similar thing happens when teaching math. The more mathematics you understand, the harder it is to gauge how difficult the material is. 3Blue1Brown makes a good point of this in his one video
by the end he’s talking about the author of math textbooks who must come up with practice problems. He noted that:
“Across a wide variety of contributors there is one constant: Nobody is able to tell how difficult their exercises are. Knowing when math is hard is waaayyyy harder than the math itself.”
And I think this is something to keep in mind both as an instructor and as a student.
其他一些答案都是可笑的自我主义和消极的。他们都没有考虑到数学作为一门学科的独特性。这个问题的主要答案就是——数学确实是一门很难教的学科。
想想你在学习数学时的任何体会。有些问题看起来让人望而生畏，但一旦你看到了答案，往往看起来应该是显而易见的。通常是因为存在一个非常优雅的，也许是相当有创意的解决方案。
每当你看到“美丽的解决方案”并确信问题比实际更容易时，想想吧。事后看来，一切似乎都更容易。在教数学时也会发生类似的事情。你理解的数学越多，就越难衡量材料的难度。科普博主3blue1brown在他的一个视频中很好地说明了这一点。
最后，他谈到了数学教科书的作者，他必须提出实践问题。他指出：
“在各种各样的贡献者中，有一点是不变的:没有人能说出他们的练习有多难。知道数学什么时候难比数学本身更难。”
我认为这是作为一名教师和学生都应该牢记的。
Joseph M.
Quick answer: culture and administrative practices.
America has both awe and disdain for mathematics. On one hand, people are impressed by long, complicated-looking equations on blackboards; on the other, they say “What the hell am I gonna need this for in real life?” This influences the outlook of many math teachers: one subset will look to impress; another will look for the easiest way out—both at the expense of efficacy.
District guidelines strangle K-12 teachers in public schools; so, even the ones who are good are compromised by bureaucracy. Whatever passes the most students is the enforced teaching methodology. In most cases, that’s memorization.
In college, professors are just as susceptible to the memorization disease as their students. To make it worse, they’re often responsible for reaffirming that ideology and oiling the gears for another cycle when their students become instructors.
Memorization leads to poor explanation.
快速回答：文化和管理实践。
美国对数学既敬畏又鄙视。一方面，人们对黑板上长而复杂的方程印象深刻；另一方面，他们说“我在现实生活中要这个干什么？”这影响了许多数学教师的看法：其中一部分人希望给人留下深刻印象；另一部分则是寻找最简单的方法——以牺牲功效为代价。
地区指导方针扼杀公立学校的K-12教育体系的教师；因此，即使是优秀的人也会受到官僚主义的影响。最让学生失望的是强制的教学方法。在大多数情况下，这是死记硬背。
在大学里，教授和学生一样容易患上记忆疾病。更糟糕的是，他们经常负责重申这一意识形态，并为学生成为教师的另一个周期加油。
死记硬背导致解释力差。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
Quick answer: culture and administrative practices.
America has both awe and disdain for mathematics. On one hand, people are impressed by long, complicated-looking equations on blackboards; on the other, they say “What the hell am I gonna need this for in real life?” This influences the outlook of many math teachers: one subset will look to impress; another will look for the easiest way out—both at the expense of efficacy.
District guidelines strangle K-12 teachers in public schools; so, even the ones who are good are compromised by bureaucracy. Whatever passes the most students is the enforced teaching methodology. In most cases, that’s memorization.
In college, professors are just as susceptible to the memorization disease as their students. To make it worse, they’re often responsible for reaffirming that ideology and oiling the gears for another cycle when their students become instructors.
Memorization leads to poor explanation.
快速回答：文化和管理实践。
美国对数学既敬畏又鄙视。一方面，人们对黑板上长而复杂的方程印象深刻；另一方面，他们说“我在现实生活中要这个干什么？”这影响了许多数学教师的看法：其中一部分人希望给人留下深刻印象；另一部分则是寻找最简单的方法——以牺牲功效为代价。
地区指导方针扼杀公立学校的K-12教育体系的教师；因此，即使是优秀的人也会受到官僚主义的影响。最让学生失望的是强制的教学方法。在大多数情况下，这是死记硬背。
在大学里，教授和学生一样容易患上记忆疾病。更糟糕的是，他们经常负责重申这一意识形态，并为学生成为教师的另一个周期加油。
死记硬背导致解释力差。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
Sycophantic memorizers can complete doctoral degrees, and they are precisely the types preferred for the task: they’re easily molded to fit the department’s every whim. Their personalities don’t magically turn when they’re the professors and running the operation.
Math professors work at research universities for the prestige, for churning out papers and securing funding. It’s a show to impress, not to teach. As such, they’ll cut every pedagogical corner they can, conveniently blaming their students when it all goes down.
“The course is hard”—All the more reason to prepare to address the points of difficulty as smoothly as possible.
“Students don’t read the textbook”—A lot of good reading the textbook will do them if the professor is incapable of answering questions coherently.
Efficient, clear communication skills are apparently neglected in math departments. For several months, I worked for a proofreading company that services even top-ranking colleges in the United States, and most of the mathematics submissions I encountered were atrocious. I actually lost that job because I could not bring the last paper up to standard without rewriting it, and my suggestions for improvement “weren’t indulgent enough” for the professor.
When I was a math instructor in college, I was sorely disappointed at how lazy and apathetic many of the other instructors and professors were. Yes, there are students who don’t want to lift a finger, but too often this image is wrongly assumed, especially by professors. They’re minimizing the flaws of their in-group and maximizing the flaws of an out-group.
阿谀奉承的记忆者可以完成博士学位，他们正是这项任务的首选类型：他们很容易被塑造，以适应组织中的每一个突发奇想。当他们成为教授并经营公司时，他们的性格不会神奇地转变。
数学教授在研究型大学工作是为了声望，为了大量撰写论文和获得资金。这是一场给人留下深刻印象的表演，而不是教学。因此，他们会尽可能地削减教学方面的每一环节，当一切都失败时，他们会很方便地指责学生。
这门课很难——所以更有理由准备好尽可能顺利地解决难点
学生不读课本”——如果教授不能条理清晰地回答问题，那么大量的教材阅读对他们来说是有帮助的
数学系显然忽视了高效、清晰的沟通技巧。几个月来，我在一家校对公司工作，这家公司甚至为美国一流的大学提供服务，我遇到的大多数数学作业都很糟糕。实际上，我丢掉了那份工作，因为我不能在不重写的情况下使上一篇论文达到标准，而且我的改进建议对教授来说“不够宽容”
当我在大学当数学老师时，我对其他老师和教授的懒惰和冷漠感到非常失望。是的，有些学生不想动一根手指，但这种形象往往是错误的，尤其是教授。他们最小化了自己群体的缺陷，最大化了其他群体的缺陷。
Math professors work at research universities for the prestige, for churning out papers and securing funding. It’s a show to impress, not to teach. As such, they’ll cut every pedagogical corner they can, conveniently blaming their students when it all goes down.
“The course is hard”—All the more reason to prepare to address the points of difficulty as smoothly as possible.
“Students don’t read the textbook”—A lot of good reading the textbook will do them if the professor is incapable of answering questions coherently.
Efficient, clear communication skills are apparently neglected in math departments. For several months, I worked for a proofreading company that services even top-ranking colleges in the United States, and most of the mathematics submissions I encountered were atrocious. I actually lost that job because I could not bring the last paper up to standard without rewriting it, and my suggestions for improvement “weren’t indulgent enough” for the professor.
When I was a math instructor in college, I was sorely disappointed at how lazy and apathetic many of the other instructors and professors were. Yes, there are students who don’t want to lift a finger, but too often this image is wrongly assumed, especially by professors. They’re minimizing the flaws of their in-group and maximizing the flaws of an out-group.
阿谀奉承的记忆者可以完成博士学位，他们正是这项任务的首选类型：他们很容易被塑造，以适应组织中的每一个突发奇想。当他们成为教授并经营公司时，他们的性格不会神奇地转变。
数学教授在研究型大学工作是为了声望，为了大量撰写论文和获得资金。这是一场给人留下深刻印象的表演，而不是教学。因此，他们会尽可能地削减教学方面的每一环节，当一切都失败时，他们会很方便地指责学生。
这门课很难——所以更有理由准备好尽可能顺利地解决难点
学生不读课本”——如果教授不能条理清晰地回答问题，那么大量的教材阅读对他们来说是有帮助的
数学系显然忽视了高效、清晰的沟通技巧。几个月来，我在一家校对公司工作，这家公司甚至为美国一流的大学提供服务，我遇到的大多数数学作业都很糟糕。实际上，我丢掉了那份工作，因为我不能在不重写的情况下使上一篇论文达到标准，而且我的改进建议对教授来说“不够宽容”
当我在大学当数学老师时，我对其他老师和教授的懒惰和冷漠感到非常失望。是的，有些学生不想动一根手指，但这种形象往往是错误的，尤其是教授。他们最小化了自己群体的缺陷，最大化了其他群体的缺陷。
Aaron Brown
Without agreeing with your premise, I will throw out one common issue. Most pre-college math is taught by people who are not particularly interested in or good at math. There are many exceptions, of course, but math is a requirement in every year of pre-college schooling, and there aren’t enough math-lovers who want the jobs to teach them all.
Imagine a music teacher with no appreciation and talent for music. He could tell students how to hold their instruments and how to produce different notes. He could lecture on musical styles, history and theory. All these things can be learned and taught by rote. But he couldn’t explain the nature or value of music, nor would he be likely to inspire students nor extract enjoyable performances from them.
While there are good and bad teachers in every subject, it’s easier to find English teachers who love literature, history teachers who are fascinated by the past and gym teachers addicted to sports—than math teachers who appreciate math.
在不同意你的前提下，我将抛出一个共同的问题。大多数大学预科数学是由对数学不特别感兴趣或不擅长数学的人教授的。当然，也有很多例外，但数学是大学预科每一年的必修课程，而且没有足够的数学爱好者来教所有人。
想象一下，一个音乐老师对音乐没有鉴赏力和天赋。他可以告诉学生如何握住乐器，如何发出不同的音符。他可以讲授音乐风格、历史和理论。所有这些东西都可以通过死记硬背来学习和传授。但他无法解释音乐的本质或价值，也无法激励学生，他们也无法进行令人愉快的表演。
虽然每个学科都有好老师和糟糕的老师，但比起欣赏数学的数学老师，更容易找到热爱文学的英语老师、迷恋过去的历史老师和对体育上瘾的体育老师。
Without agreeing with your premise, I will throw out one common issue. Most pre-college math is taught by people who are not particularly interested in or good at math. There are many exceptions, of course, but math is a requirement in every year of pre-college schooling, and there aren’t enough math-lovers who want the jobs to teach them all.
Imagine a music teacher with no appreciation and talent for music. He could tell students how to hold their instruments and how to produce different notes. He could lecture on musical styles, history and theory. All these things can be learned and taught by rote. But he couldn’t explain the nature or value of music, nor would he be likely to inspire students nor extract enjoyable performances from them.
While there are good and bad teachers in every subject, it’s easier to find English teachers who love literature, history teachers who are fascinated by the past and gym teachers addicted to sports—than math teachers who appreciate math.
在不同意你的前提下，我将抛出一个共同的问题。大多数大学预科数学是由对数学不特别感兴趣或不擅长数学的人教授的。当然，也有很多例外，但数学是大学预科每一年的必修课程，而且没有足够的数学爱好者来教所有人。
想象一下，一个音乐老师对音乐没有鉴赏力和天赋。他可以告诉学生如何握住乐器，如何发出不同的音符。他可以讲授音乐风格、历史和理论。所有这些东西都可以通过死记硬背来学习和传授。但他无法解释音乐的本质或价值，也无法激励学生，他们也无法进行令人愉快的表演。
虽然每个学科都有好老师和糟糕的老师，但比起欣赏数学的数学老师，更容易找到热爱文学的英语老师、迷恋过去的历史老师和对体育上瘾的体育老师。
JetBrains
When you start working on a new project, you will often need to set up a whole new infrastructure. This time-consuming process can require setting up multiple tools that all require integration and maintenance. Using different tools often creates silos, leading to miscommunication, less efficient collaboration, and a loss of information.
With JetBrains Space, a single, integrated out-of-the-box solution, you can do this much more quickly, keeping you fast and reactive to the needs of your business.
JetBrains Space is a unified platform that covers the entire software development pipeline, including hosting Git repositories, automating CI/CD, publishing packages, orchestrating cloud dev environments, and managing issues, documents, and chats – all in one place.
With its first-class integration with IntelliJ-based IDEs, you can perform advanced Space code reviews, clone your Git repositories hosted in Space, and track your automation job’s progress, all from your IDE.
当你开始着手一个新项目时，通常需要建立一个全新的基础架构。这个耗时的过程可能需要设置多个工具，这些工具都需要集成和维护。使用不同的工具通常会造成贮仓，导致沟通不畅、协作效率低下和信息丢失。
借助JetBrains Space（一体化的完整协作平台），一个单一的、集成的开箱即用解决方案，你可以更快地完成这项工作，使你能够快速响应业务需求。
JetBrains Space是一个涵盖整个软件开发流程的统一平台，包括托管Git存储库、自动化CI/CD、发布软件包、协调云开发环境以及管理问题、文档和聊天——所有这些都在一个地方。
通过与基于 多个项目的集成开发环境的一流集成，可以执行高级空间代码审查，克隆托管在空间中的Git存储库，并跟踪自动化作业的进度，所有这些都可以通过集成开发环境完成。
When you start working on a new project, you will often need to set up a whole new infrastructure. This time-consuming process can require setting up multiple tools that all require integration and maintenance. Using different tools often creates silos, leading to miscommunication, less efficient collaboration, and a loss of information.
With JetBrains Space, a single, integrated out-of-the-box solution, you can do this much more quickly, keeping you fast and reactive to the needs of your business.
JetBrains Space is a unified platform that covers the entire software development pipeline, including hosting Git repositories, automating CI/CD, publishing packages, orchestrating cloud dev environments, and managing issues, documents, and chats – all in one place.
With its first-class integration with IntelliJ-based IDEs, you can perform advanced Space code reviews, clone your Git repositories hosted in Space, and track your automation job’s progress, all from your IDE.
当你开始着手一个新项目时，通常需要建立一个全新的基础架构。这个耗时的过程可能需要设置多个工具，这些工具都需要集成和维护。使用不同的工具通常会造成贮仓，导致沟通不畅、协作效率低下和信息丢失。
借助JetBrains Space（一体化的完整协作平台），一个单一的、集成的开箱即用解决方案，你可以更快地完成这项工作，使你能够快速响应业务需求。
JetBrains Space是一个涵盖整个软件开发流程的统一平台，包括托管Git存储库、自动化CI/CD、发布软件包、协调云开发环境以及管理问题、文档和聊天——所有这些都在一个地方。
通过与基于 多个项目的集成开发环境的一流集成，可以执行高级空间代码审查，克隆托管在空间中的Git存储库，并跟踪自动化作业的进度，所有这些都可以通过集成开发环境完成。
IWISHTOLVE
I read a lot, and something stuck with me. . . If you can explain it to an 8 year old, you understand it PERFECTLY. None of these teachers can explain it to an 8 year old , they can barely explain it to a college student and they all have access to answer keys.
Why hasn’t anyone come to the conclusion that these teachers just don’t understand calculus very well, and come up with a bevvy of excuses for the professors who have both tenure and play judge jury and executioner with our futures?
我读了很多书，有些东西让我难以释怀，如果你能向一个8岁的孩子解释，就说明你已经完全理解。这些老师没有一个能向一个8岁的孩子解释，他们几乎不能就他们自己找到答案的关键向一个大学生解释。
为什么没有人得出这样的结论，这些老师只是不太懂微积分，为那些既拥有终身教职，又对我们的未来扮演法官、陪审团和刽子手角色的教授们想出一大堆借口。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
I read a lot, and something stuck with me. . . If you can explain it to an 8 year old, you understand it PERFECTLY. None of these teachers can explain it to an 8 year old , they can barely explain it to a college student and they all have access to answer keys.
Why hasn’t anyone come to the conclusion that these teachers just don’t understand calculus very well, and come up with a bevvy of excuses for the professors who have both tenure and play judge jury and executioner with our futures?
我读了很多书，有些东西让我难以释怀，如果你能向一个8岁的孩子解释，就说明你已经完全理解。这些老师没有一个能向一个8岁的孩子解释，他们几乎不能就他们自己找到答案的关键向一个大学生解释。
为什么没有人得出这样的结论，这些老师只是不太懂微积分，为那些既拥有终身教职，又对我们的未来扮演法官、陪审团和刽子手角色的教授们想出一大堆借口。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
Rory Coker
Math and physics teachers always get a lot of criticism and hate from students. Here is what’s at issue. Most students never learn how to study with the aim of understanding, at best they just memorize a bunch of garbage for the upcoming test. Unless you genuinely study and keep up in a math or physics class, reading the relevant chapter BEFORE the lecture, you will get very little out of the lecture… it will go right over your head. This is not a defect of the lecture, in general, it is a defect found in much of the intended audience.
数学和物理老师总是受到学生的批评和憎恨，这就是问题所在。大多数学生从来没有为了理解而学习，充其量他们只是为了即将到来的考试记住一堆垃圾。除非你真的在数学或物理课上学习并跟上进度，在课前阅读相关章节，否则你从课堂上得到的东西很少，你完全无法理解。这不是教学法的缺陷，一般来说，这是在许多目标受众中发现的缺陷。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
Math and physics teachers always get a lot of criticism and hate from students. Here is what’s at issue. Most students never learn how to study with the aim of understanding, at best they just memorize a bunch of garbage for the upcoming test. Unless you genuinely study and keep up in a math or physics class, reading the relevant chapter BEFORE the lecture, you will get very little out of the lecture… it will go right over your head. This is not a defect of the lecture, in general, it is a defect found in much of the intended audience.
数学和物理老师总是受到学生的批评和憎恨，这就是问题所在。大多数学生从来没有为了理解而学习，充其量他们只是为了即将到来的考试记住一堆垃圾。除非你真的在数学或物理课上学习并跟上进度，在课前阅读相关章节，否则你从课堂上得到的东西很少，你完全无法理解。这不是教学法的缺陷，一般来说，这是在许多目标受众中发现的缺陷。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
William Rose
Some think just because they find math fascinating and/or fun everyone else does. Or worse, they don’t find it fascinating or fun. For many people it is scary, hard, or boring. It has to be made interesting and fun at least through introductory calculus. (Advanced math in college is different though it still can be made more interesting than most professors made it). A physics teacher I had in HS would, for example, draw a cliff and stick figures with names of students. Then he’d ask, if Alice pushed Bob with a force of X off a 50′ high cliff, where will Bob go splat?
Different people learn through different senses and methods. A friend’s kid was having trouble learning the multiplication tables. He told me they had tried drills, flash cards, etc. Nothing worked. I told a great math teacher I knew about it. From my descxtion she immediately knew what to do. She said, have the kid write out his own flash cards. A few days after I told the friend he said it was a miracle. His child had learned them almost immediately. It was the combination of physical (writing the flash cards) and visual (seeing the cards he had written out himself) that made the connection. You can’t teach every person the same way. You need to present the material in several different ways. Then, if several don’t get it yet, work with them individually to find out what works for them. (See “Engage the students” below).
Make it relevant. Give real world examples of applications for the subject matter. Algebra, trig, geometry, calculus all have lots of fascinating applications in the real world. Yet they are all often taught simply as dry equations rather than applied to our lives and to nature. There are thousands of amazing applications all around us. Show them how math is everywhere and they won’t ask “Why do I need to know this?”.
有些人认为这仅仅是因为他们觉得数学很有趣和/或有吸引力。更糟糕的是，他们不觉得它有趣或有吸引力。对许多人来说，这是可怕的、艰难的或无聊的。至少通过微积分入门，它必须变得有趣或者有吸引力。（大学里的高等数学是不同的，尽管它仍然可以比大多数教授展现的更有趣）。例如，我在高中的一位物理老师会画一个悬崖，并在上面画上学生的名字。然后他会问，如果爱丽丝用X的力把鲍勃推下50英尺高的悬崖，鲍勃会摔到哪里?
不同的人通过不同的感觉和方法学习。一个朋友的孩子对学习乘法感到有困难。他告诉我，他们试过操练、快闪卡等，但都没用。我将这件事告诉一位伟大的数学老师。根据我的描述，她立刻知道该怎么办。她说，让孩子自己写记忆卡。我告诉朋友此方法的几天后，他说这是一个奇迹。他的孩子几乎马上就学会了。正是物理（写闪存卡）和视觉（看到他自己写的记忆卡）的结合，形成了这种联系。你不能以同样的方式教所有人。你需要以几种不同的方式呈现信息。然后，如果有个别人还没有效果，则与他们单独合作，找出适合他们的方法。（见下文“吸引学生参与”）。
使其相关。给出该主题的实际应用示例。代数、三角、几何、微积分在现实世界中都有许多吸引人的应用。然而，它们通常都被简单地当作枯燥的方程式来教授，而不是应用于我们的生活和自然。我们周围有成千上万令人惊叹的应用程序。向他们展示数学是如何无处不在的，如此他们不会问“为什么我需要知道这个？”。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
Some think just because they find math fascinating and/or fun everyone else does. Or worse, they don’t find it fascinating or fun. For many people it is scary, hard, or boring. It has to be made interesting and fun at least through introductory calculus. (Advanced math in college is different though it still can be made more interesting than most professors made it). A physics teacher I had in HS would, for example, draw a cliff and stick figures with names of students. Then he’d ask, if Alice pushed Bob with a force of X off a 50′ high cliff, where will Bob go splat?
Different people learn through different senses and methods. A friend’s kid was having trouble learning the multiplication tables. He told me they had tried drills, flash cards, etc. Nothing worked. I told a great math teacher I knew about it. From my descxtion she immediately knew what to do. She said, have the kid write out his own flash cards. A few days after I told the friend he said it was a miracle. His child had learned them almost immediately. It was the combination of physical (writing the flash cards) and visual (seeing the cards he had written out himself) that made the connection. You can’t teach every person the same way. You need to present the material in several different ways. Then, if several don’t get it yet, work with them individually to find out what works for them. (See “Engage the students” below).
Make it relevant. Give real world examples of applications for the subject matter. Algebra, trig, geometry, calculus all have lots of fascinating applications in the real world. Yet they are all often taught simply as dry equations rather than applied to our lives and to nature. There are thousands of amazing applications all around us. Show them how math is everywhere and they won’t ask “Why do I need to know this?”.
有些人认为这仅仅是因为他们觉得数学很有趣和/或有吸引力。更糟糕的是，他们不觉得它有趣或有吸引力。对许多人来说，这是可怕的、艰难的或无聊的。至少通过微积分入门，它必须变得有趣或者有吸引力。（大学里的高等数学是不同的，尽管它仍然可以比大多数教授展现的更有趣）。例如，我在高中的一位物理老师会画一个悬崖，并在上面画上学生的名字。然后他会问，如果爱丽丝用X的力把鲍勃推下50英尺高的悬崖，鲍勃会摔到哪里?
不同的人通过不同的感觉和方法学习。一个朋友的孩子对学习乘法感到有困难。他告诉我，他们试过操练、快闪卡等，但都没用。我将这件事告诉一位伟大的数学老师。根据我的描述，她立刻知道该怎么办。她说，让孩子自己写记忆卡。我告诉朋友此方法的几天后，他说这是一个奇迹。他的孩子几乎马上就学会了。正是物理（写闪存卡）和视觉（看到他自己写的记忆卡）的结合，形成了这种联系。你不能以同样的方式教所有人。你需要以几种不同的方式呈现信息。然后，如果有个别人还没有效果，则与他们单独合作，找出适合他们的方法。（见下文“吸引学生参与”）。
使其相关。给出该主题的实际应用示例。代数、三角、几何、微积分在现实世界中都有许多吸引人的应用。然而，它们通常都被简单地当作枯燥的方程式来教授，而不是应用于我们的生活和自然。我们周围有成千上万令人惊叹的应用程序。向他们展示数学是如何无处不在的，如此他们不会问“为什么我需要知道这个？”。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
Engage the students. The same teacher mentioned above divides the class into small work groups so they can go over homework and review exams to see where they got it right or wrong, and so they can teach and learn from each other rather than just listen to her talk. She also has students come to the board, do problems, and explain how they solved them. Each student has their own way of thinking and solving a problem and chances are there are others in the class that think the same way.
A lot of lower level math teachers really don’t know their math. Some don’t even seem to enjoy it. If you can’t see the beauty in math, you shouldn’t be teaching it. Its one thing if you are teaching arithmetic: +. -, *, /. But once you hit algebra, trig and geometry you should be totally proficient. Sure, there will be those students who will stump you with a question. But you should be able to come in the next day with an answer. Yet there are a lot of teachers who really don’t know the subject matter very well. You should know these subjects inside out, be able to derive any formula and explain why it is so. I also believe you should know, and know well, at least one level above whatever it is you are teaching. And again, you must know the real world applications and be able to explain them clearly.
The worst teachers I had were the ones who just regurgitated the same problems as those in the text books. That does nothing to help students see alternative ways to think about and solve a problem, see different forms of problems, and it certainly does nothing to bring math to life. Most text books are dry and formulaic. The teachers who taught that way were the same.
The shame of it is I had very few math or science teachers who were great. In fact, most were not even good.
One last item that really peeved me in college. A teacher or professor has to be able to speak the language clearly and intelligibly. I had foreign-born professors in college who were nearly unintelligible. If you have to spend all of your energy just trying to understand the words, you miss the meaning. I would immediately transfer to another class.
让学生参与进来。上面提到的同一位老师将全班分成小组，这样他们就可以检查家庭作业和复习考试，看看他们在哪里做对了或错了，这样他们可以互相教和学习，而不仅仅是听她的演讲。她还让学生走上讲台，做问题，并就他们如何解决问题进行解释。每个学生都有自己的思考和解决问题的方式，班上其他人很有可能也有同样的想法。
很多低水平的数学老师真的不了解他们的数学。有些人甚至似乎不喜欢它。如果你看不到数学的美，你就不应该教它。如果你在教算术，这是一件事：+、-、*、/。但一旦你接触代数、三角和几何，你应该完全精通。当然，有些学生会用一个问题难住你。但你应该能在第二天带着答案来告诉学生。然而，有很多老师对这门学科并不十分了解。你应该彻底了解这些主题，能够推导出任何公式，并解释为什么是这样才行。我也相信，你应该知道，而且很清楚，至少比你所教的东西高一个层次。同样，必须了解真实世界的实例，并能够清楚地解释它们。
我遇到的最糟糕的老师是那些只会重复课本上的问题的老师。这无助于帮助学生看到思考和解决问题的替代方法，看到不同形式的问题，当然也无助于将数学带入生活。大多数教科书枯燥无味，公式化。以这种方式授课的老师都是一样的。
遗憾的是，我的数学和科学老师很少是优秀的。事实上，大多数都不好。
在大学里，最后一件让我非常恼火的事：教师或教授必须能够清楚、易懂地讲这种语言。我在大学里有一些外国出生的教授，他们几乎让人难以理解。如果你把所有的精力都花在理解单词上，你就错过了意思。遇到此情况我会立即转到另一个班。
A lot of lower level math teachers really don’t know their math. Some don’t even seem to enjoy it. If you can’t see the beauty in math, you shouldn’t be teaching it. Its one thing if you are teaching arithmetic: +. -, *, /. But once you hit algebra, trig and geometry you should be totally proficient. Sure, there will be those students who will stump you with a question. But you should be able to come in the next day with an answer. Yet there are a lot of teachers who really don’t know the subject matter very well. You should know these subjects inside out, be able to derive any formula and explain why it is so. I also believe you should know, and know well, at least one level above whatever it is you are teaching. And again, you must know the real world applications and be able to explain them clearly.
The worst teachers I had were the ones who just regurgitated the same problems as those in the text books. That does nothing to help students see alternative ways to think about and solve a problem, see different forms of problems, and it certainly does nothing to bring math to life. Most text books are dry and formulaic. The teachers who taught that way were the same.
The shame of it is I had very few math or science teachers who were great. In fact, most were not even good.
One last item that really peeved me in college. A teacher or professor has to be able to speak the language clearly and intelligibly. I had foreign-born professors in college who were nearly unintelligible. If you have to spend all of your energy just trying to understand the words, you miss the meaning. I would immediately transfer to another class.
让学生参与进来。上面提到的同一位老师将全班分成小组，这样他们就可以检查家庭作业和复习考试，看看他们在哪里做对了或错了，这样他们可以互相教和学习，而不仅仅是听她的演讲。她还让学生走上讲台，做问题，并就他们如何解决问题进行解释。每个学生都有自己的思考和解决问题的方式，班上其他人很有可能也有同样的想法。
很多低水平的数学老师真的不了解他们的数学。有些人甚至似乎不喜欢它。如果你看不到数学的美，你就不应该教它。如果你在教算术，这是一件事：+、-、*、/。但一旦你接触代数、三角和几何，你应该完全精通。当然，有些学生会用一个问题难住你。但你应该能在第二天带着答案来告诉学生。然而，有很多老师对这门学科并不十分了解。你应该彻底了解这些主题，能够推导出任何公式，并解释为什么是这样才行。我也相信，你应该知道，而且很清楚，至少比你所教的东西高一个层次。同样，必须了解真实世界的实例，并能够清楚地解释它们。
我遇到的最糟糕的老师是那些只会重复课本上的问题的老师。这无助于帮助学生看到思考和解决问题的替代方法，看到不同形式的问题，当然也无助于将数学带入生活。大多数教科书枯燥无味，公式化。以这种方式授课的老师都是一样的。
遗憾的是，我的数学和科学老师很少是优秀的。事实上，大多数都不好。
在大学里，最后一件让我非常恼火的事：教师或教授必须能够清楚、易懂地讲这种语言。我在大学里有一些外国出生的教授，他们几乎让人难以理解。如果你把所有的精力都花在理解单词上，你就错过了意思。遇到此情况我会立即转到另一个班。
Alexander Farrugia
This isn’t true for all maths teachers, of course. But let me give a few possible reasons:
Some teachers may think that if something is obvious to them, then it’s obvious to everyone. From my experience teaching mathematics, this is something I used to believe is true that I later learned is very far from the truth. Perhaps your teacher thinks that sin2x and 2sinx are obviously different, so it’s useless explaining the obvious. But you cannot see why you can’t write sin2x instead of sinx+sinx .
Some teachers believe that everyone learns through step-by-step guides. People who are good at mathematics tend to be sequential, structured thinkers. But not everyone has a brain that functions sequentially and orderly. Not everybody requires a set of instructions to build a piece of furniture, for example. If you are in the ‘not requiring instructions to do things’ category, you may find your teacher’s step-by-step instructions to solve a particular equation confusing or even maddening.
Some aspects of mathematics are difficult, or even impossible, to teach. I wonder how Art teachers teach art. How do artists know where to place the brush, which colours to use, whether to use brush or pencil? I don’t, and I’ll never know. The same thing is true for mathematics. If you’re stuck solving a trigonometric equation, then perhaps you have chosen the wrong brush. That’s something your teacher cannot always teach you.
Some teachers can be just plain bad. Perhaps teaching mathematics was never what they intended to do, or they are unmotivated for some reason or another. In other words, they hate their job. No wonder they are terrible at explaining things — they’d rather be in a different job.
In short, the problem may be the teacher, it may be you, it may be the nature of the subject, or it may be a combination of all three.
当然，并非所有的数学老师都是这样，但让我给出几个可能的原因：
一些老师可能认为，如果某件事对他们来说是显而易见的，那么对每个人来说都是显而易见的。从我教数学的经验来看，我过去认为这是真的，但后来我发现这与事实相去甚远。也许你的老师认为sin2x和2sinx是明显不同的，所以解释显而易见的是没有用的。但是你看不出为什么不能把sin2x写成sinx+sinx。
一些教师认为，每个人都通过循序渐进的指导来学习的。擅长数学的人往往是循序渐进、有条理的思考者。但并不是每个人的大脑都是有序运转的。例如，并不是每个人都需要一套指示来制作一件家具。如果你属于“不需要指导去做事情”类别，你可能会发现老师的一步一步地指导你去解决一个特定的方程会让你困惑甚至发狂。
数学的某些方面很难教，甚至不可能教。我想知道艺术教师是如何教授艺术的。艺术家如何知道画笔放在哪里，使用什么颜色，是用画笔还是铅笔？我不知道，我永远不会知道。数学也是如此。如果你在解一个三角方程时卡住了，那么也许你选错了画笔。这是你的老师不能总是教你的。
有些老师可能很糟糕。也许教数学从来不是他们想要做的，或者他们出于某种原因而教授而不是出于动机。换句话说，他们讨厌自己的工作。难怪他们在解释数学方面很糟糕——他们宁愿换一份工作。
简言之，问题可能是老师，可能是出在你这，可能是学科的性质，也可能是这三者的结合。
This isn’t true for all maths teachers, of course. But let me give a few possible reasons:
Some teachers may think that if something is obvious to them, then it’s obvious to everyone. From my experience teaching mathematics, this is something I used to believe is true that I later learned is very far from the truth. Perhaps your teacher thinks that sin2x and 2sinx are obviously different, so it’s useless explaining the obvious. But you cannot see why you can’t write sin2x instead of sinx+sinx .
Some teachers believe that everyone learns through step-by-step guides. People who are good at mathematics tend to be sequential, structured thinkers. But not everyone has a brain that functions sequentially and orderly. Not everybody requires a set of instructions to build a piece of furniture, for example. If you are in the ‘not requiring instructions to do things’ category, you may find your teacher’s step-by-step instructions to solve a particular equation confusing or even maddening.
Some aspects of mathematics are difficult, or even impossible, to teach. I wonder how Art teachers teach art. How do artists know where to place the brush, which colours to use, whether to use brush or pencil? I don’t, and I’ll never know. The same thing is true for mathematics. If you’re stuck solving a trigonometric equation, then perhaps you have chosen the wrong brush. That’s something your teacher cannot always teach you.
Some teachers can be just plain bad. Perhaps teaching mathematics was never what they intended to do, or they are unmotivated for some reason or another. In other words, they hate their job. No wonder they are terrible at explaining things — they’d rather be in a different job.
In short, the problem may be the teacher, it may be you, it may be the nature of the subject, or it may be a combination of all three.
当然，并非所有的数学老师都是这样，但让我给出几个可能的原因：
一些老师可能认为，如果某件事对他们来说是显而易见的，那么对每个人来说都是显而易见的。从我教数学的经验来看，我过去认为这是真的，但后来我发现这与事实相去甚远。也许你的老师认为sin2x和2sinx是明显不同的，所以解释显而易见的是没有用的。但是你看不出为什么不能把sin2x写成sinx+sinx。
一些教师认为，每个人都通过循序渐进的指导来学习的。擅长数学的人往往是循序渐进、有条理的思考者。但并不是每个人的大脑都是有序运转的。例如，并不是每个人都需要一套指示来制作一件家具。如果你属于“不需要指导去做事情”类别，你可能会发现老师的一步一步地指导你去解决一个特定的方程会让你困惑甚至发狂。
数学的某些方面很难教，甚至不可能教。我想知道艺术教师是如何教授艺术的。艺术家如何知道画笔放在哪里，使用什么颜色，是用画笔还是铅笔？我不知道，我永远不会知道。数学也是如此。如果你在解一个三角方程时卡住了，那么也许你选错了画笔。这是你的老师不能总是教你的。
有些老师可能很糟糕。也许教数学从来不是他们想要做的，或者他们出于某种原因而教授而不是出于动机。换句话说，他们讨厌自己的工作。难怪他们在解释数学方面很糟糕——他们宁愿换一份工作。
简言之，问题可能是老师，可能是出在你这，可能是学科的性质，也可能是这三者的结合。
Gerry Rzeppa
Probably because they wrongly assume that:
(a) your brain works the same way theirs does,
(b) you have the same prerequisite knowledge in your head that they have, and
(c) you care about math as much as they do.
For example, this math joke…
…is funny only if:
(a) you’ve got the kind of brain that likes puzzles and likes to check the math whenever you see numbers, and
(b) you know that a nickel is worth five cents and that 43 is not evenly divisible by five; and
(c) you cared enough to recall the necessary information and do the required processing to get a mere chuckle.
Note also that most teachers these days don’t know the difference between teaching and testing. This is how we explain the difference to our Plain English teachers:
可能是因为他们错误地认为：
（a） 你的大脑的思维模式和他们的一样；
（b） 你头脑中有与他们相同的先决知识，并且
（c） 你和他们一样关心数学。
例如，这个数学笑话：
只有在以下情况下才有趣：
（a） 你有一种喜欢拼图的大脑，每当你看到数字时，就喜欢检查数学；
（b） 你知道一个五分硬币值五美分，43不能被五整除；和
（c） 你很在意回忆必要的信息，并进行必要的处理，以获得一个简单的笑声。
还要注意的是，现在的大多数老师都不知道教学和考试之间的区别。以下是我们如何向我们的老师用通俗易懂的方式解释其中的差异:
Probably because they wrongly assume that:
(a) your brain works the same way theirs does,
(b) you have the same prerequisite knowledge in your head that they have, and
(c) you care about math as much as they do.
For example, this math joke…
…is funny only if:
(a) you’ve got the kind of brain that likes puzzles and likes to check the math whenever you see numbers, and
(b) you know that a nickel is worth five cents and that 43 is not evenly divisible by five; and
(c) you cared enough to recall the necessary information and do the required processing to get a mere chuckle.
Note also that most teachers these days don’t know the difference between teaching and testing. This is how we explain the difference to our Plain English teachers:
可能是因为他们错误地认为：
（a） 你的大脑的思维模式和他们的一样；
（b） 你头脑中有与他们相同的先决知识，并且
（c） 你和他们一样关心数学。
例如，这个数学笑话：
只有在以下情况下才有趣：
（a） 你有一种喜欢拼图的大脑，每当你看到数字时，就喜欢检查数学；
（b） 你知道一个五分硬币值五美分，43不能被五整除；和
（c） 你很在意回忆必要的信息，并进行必要的处理，以获得一个简单的笑声。
还要注意的是，现在的大多数老师都不知道教学和考试之间的区别。以下是我们如何向我们的老师用通俗易懂的方式解释其中的差异:
TEACHING IS TELLING
When you tell a student exactly what to do, and he does it, you're teaching — transferring tried-and-true neural patterns from your working brain into your student's fledgling brain. So you tell the student over and over exactly what you want him to know and do; and before long you find him saying, without prompting, "Alright already! I've got it." This is teaching.
Asking a student to solve problems that he hasn't seen before is not teaching; it's testing. Asking him to hypothesize about future results is not teaching, it's testing. In fact, the whole so-called "Socratic Method" is not teaching; it's testing. And when you use such techniques, you're essentially asking the student to "reinvent the wheel" — to intuitively come up with something that you already know. Why not simply let him in on the secret?
So when you're teaching Plain English programming, don't ask your students to solve problems, or guess what will happen if this or that code is run — tell them. Show them. Have them type it in (so it will enter their brains through their hands as well as their eyes and ears), and have them run the programs they've entered (for the fun of it); but don't ask them to "come up with" what's already in your head: hand it to them on a silver platter. Teaching is telling.
If a student wants to experiment on his own, that's fine. But don't hand out an assignment unless the answer is provided and the student is told to study and copy the answer (not try to reinvent it on his own). A student who is typing in a correct solution is being taught; Teaching is telling.
教学是讲述：
当你告诉一个学生该做什么，而他做了，你就是在教学-将你的工作大脑中经过验证的真实神经模式转移到你对此无经验的人学生的大脑中。所以你一遍又一遍地告诉学生你想让他知道什么，做什么；不久，你会发现他在没有提示的情况下说：“好吧，我已经知道了。”这就是教学。
要求学生解决他以前没有见过的问题不是教学，这是测试。让他对未来的结果做出假设不是教学，而是测试。事实上，整个所谓的“苏格拉底方法”不是教学，这是测试。当你使用这些技术时，你基本上是在要求学生“重新发明轮子”-直觉地想出一些你已经知道的东西。为什么不让他知道这个秘密呢？
因此，当你在用通俗易懂的方式教授时，不要要求学生解决问题，也不要猜测如果这段或那段代码运行会发生什么，告诉他们就好，给他们看。让他们吸收（这样它会通过他们的手、眼睛和耳朵进入他们的大脑），让他们运行他们输入的程序（为了好玩）；但不要要求他们“想出”你头脑中已经存在的东西，相当于就是把它放在银盘上交给他们—教学就是讲述。
如果一个学生想自己做实验，那没关系。但是，除非提供了答案，并且要求学生学习并复制答案（而不是试图自己重新创造），否则不要分发作业。正在吸收正确答案的学生正在接受教育—教学就是讲述。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
When you tell a student exactly what to do, and he does it, you're teaching — transferring tried-and-true neural patterns from your working brain into your student's fledgling brain. So you tell the student over and over exactly what you want him to know and do; and before long you find him saying, without prompting, "Alright already! I've got it." This is teaching.
Asking a student to solve problems that he hasn't seen before is not teaching; it's testing. Asking him to hypothesize about future results is not teaching, it's testing. In fact, the whole so-called "Socratic Method" is not teaching; it's testing. And when you use such techniques, you're essentially asking the student to "reinvent the wheel" — to intuitively come up with something that you already know. Why not simply let him in on the secret?
So when you're teaching Plain English programming, don't ask your students to solve problems, or guess what will happen if this or that code is run — tell them. Show them. Have them type it in (so it will enter their brains through their hands as well as their eyes and ears), and have them run the programs they've entered (for the fun of it); but don't ask them to "come up with" what's already in your head: hand it to them on a silver platter. Teaching is telling.
If a student wants to experiment on his own, that's fine. But don't hand out an assignment unless the answer is provided and the student is told to study and copy the answer (not try to reinvent it on his own). A student who is typing in a correct solution is being taught; Teaching is telling.
教学是讲述：
当你告诉一个学生该做什么，而他做了，你就是在教学-将你的工作大脑中经过验证的真实神经模式转移到你对此无经验的人学生的大脑中。所以你一遍又一遍地告诉学生你想让他知道什么，做什么；不久，你会发现他在没有提示的情况下说：“好吧，我已经知道了。”这就是教学。
要求学生解决他以前没有见过的问题不是教学，这是测试。让他对未来的结果做出假设不是教学，而是测试。事实上，整个所谓的“苏格拉底方法”不是教学，这是测试。当你使用这些技术时，你基本上是在要求学生“重新发明轮子”-直觉地想出一些你已经知道的东西。为什么不让他知道这个秘密呢？
因此，当你在用通俗易懂的方式教授时，不要要求学生解决问题，也不要猜测如果这段或那段代码运行会发生什么，告诉他们就好，给他们看。让他们吸收（这样它会通过他们的手、眼睛和耳朵进入他们的大脑），让他们运行他们输入的程序（为了好玩）；但不要要求他们“想出”你头脑中已经存在的东西，相当于就是把它放在银盘上交给他们—教学就是讲述。
如果一个学生想自己做实验，那没关系。但是，除非提供了答案，并且要求学生学习并复制答案（而不是试图自己重新创造），否则不要分发作业。正在吸收正确答案的学生正在接受教育—教学就是讲述。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
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