费曼技巧】——1年自学完MIT 33门课，10天内掌握线性代数
斯考特·杨在12个月内自学完成了4年麻省理工学院计算机科学的33门课程，并通过了MIT的实际测试。 平均算来，杨修完每门课程大概只需要一个半星期。诀窍在于，他有一套加速学习的策略，而且这套策略，并不只是天才们的专利。最快的学习方法就是——【费曼技巧】 斯考特·杨（Scott Young）博客
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——————————————————————————————————如何在十天内掌握线性代数
最近，我的朋友斯考特·杨（Scott Young）
成就了一个惊人的壮举：他在一年之内，完成了传说中的MIT计算机科学课程表的全部33门课，从线性代数到计算理论。最重要的是，他是自学的，观看在线教程讲座，并用实际的考试作自我评估。（到斯考特的FAQ页面，看看他如何完成这个挑战）
按照他的进度，读完一门课程大概只需要1.5个星期。我坚信，能快速掌握复杂信息，对成就卓越事业至关重要。因此，我很自然地问起斯考特，让他给我们分享他的学习奥秘。所幸他答应了。接下来是一份斯考特的详细解说稿，深入剖析他的学习技巧（包括具体例子），展示他如何拿下这MIT挑战。以下时间交给斯考特……
看我怎么驾驭MIT计算机科学的课程
我老想着学快一点，再快一点，并为此兴奋不已。掌握那些重要的学问吧，专业知识与娴熟技艺将是你的职业资本，帮你赚取金钱与享受生活。如果过得好是你的目标，学问能引你到向往之地。
尽管学得更快有很多好处，但大多数人并不愿意学习“如何学习”。大概是因为我们不肯相信有这种好事，在我们看来，学习的速度只取决于好基因与天赋。确实总有些人身怀天赋本钱，但研究表明你的学习方法也很重要。更深层次的知识加工，与时而反复的温故知新，在某些情况下会加倍你的学习效率
。是的，“刻意练习”方面的研究表明，没有正确的方法，学习将永远停滞。
今天，我想分享一下学习策略，看看我如何在12个月内完成
4年MIT计算机科学的课程。这套策略历经33门课的锤炼，试图弄清楚学得更快的窍门，哪些方法有用，哪些没用。
为什么临时抱佛脚没用？
很多学生可能嘲笑我，妄想只花1年的时间学会4年的课程。毕竟，我总可以临时抱佛脚，什么都不懂还能顺利通过考试，不是吗？ 很可惜，这个策略在MIT行不通。首先，MIT的考试苛求解决问题的技巧，还经常出些没见过的题型。其次，MIT的课程讲究循序渐进，就算你能死记硬背侥幸通过一次考试，同系列课程的第七课可能就跟不上了。除了死记硬背，我不得不另辟蹊径，加速理解过程。
你能加速理解吗？
“啊哈！”当我们终于想通了，都曾经这样恍然大悟地欢呼过。问题是，大多数人都没有系统地思考。经典的学生求学之路，就是听讲座，读书；如果还不懂，只好枯燥地做大量习题（题海）或重看笔记。没有系统的方法，想更快地理解似乎是天方夜谭。毕竟，顿悟的心理机制，还全然不知。
更糟的是，理解本身，很难称得上是一种开关。它像洋葱的层层表皮，从最肤浅的领会到深层次的理解，逐层巩固对科学革命的认知。给这样的洋葱剥皮，则是常人知之甚少、易被忽略的理解过程。
加速学习的第一步，就是揭秘这个过程。如何洞悉问题，加深你的理解，取决于两个因素：
- 建立知识联系；
- 自我调试排错。
知识联系很重要，因为它们是了解一个想法的接入点。我曾纠结于傅里叶变换，直至我意识到它将压强转化为音高、或将辐射转化为颜色。这些见解，常在你懂的和你不懂的之间建立联系。调试排错也同样重要，因为你常常犯错，这些错误究根到底，还是知识残缺，胸无成竹。贫瘠的理解，恰似一个错漏百出的软件程序。如果你能高效地自我调试，必将大大提速学习进程。建立准确的知识联系与调试排错，就足够形成了深刻的问题见解。而机械化技能与死记硬背，通常也只在你对问题的本质有了肯定的直觉以后，才有所裨益。
钻研（The Drilldown Method）：你学得更快
经年累月，我完善了一个方法，可以加速逐层增进理解的过程。这个方法至今已被我用于各科目的课题，包括数学、生物学、物理学、经济学与工程学。只需些许修改，它对掌握实用技能也效果很好，比如编程、设计或语言。这个方法的基本结构是：知识面、练习、自省。我将解释每个阶段，让你了解如何尽可能有效率地执行它们，同时给出详细的例子，展示我是怎么应用在实际课程的。
第一阶段：知识面覆盖
你不可能组织一场进攻，如果你连一张地形图都没有。因此，深入研习的第一步，就是对你需要学习的内容有个大致印象。若在课堂上，这意味着你要看讲义或读课本；若是自学，你可能要多读几本同主题的书，相互考证。
学生们常犯的一个错误，就是认为这个阶段是最重要的。从很多方面来讲，这个阶段却是效率最低的，因为你每单位时间的投入只换来了最少量的知识回报。我常常加速完成这个阶段，很有好处，这样，我就可以投入更多时间到后面两个阶段。
如果你在看课程讲座的视频，最好是调到1.5x或2x倍速快进。这很容易做到，只要你下载好视频，然后使用播放器（如VLC
）的“调速”功能。我用这法子两天内看完了一学期的课程视频。如果你在读一本书，我建议你不要花时间去高亮文本。这样只会让你的知识理解停留在低层次，而从长远来看，也使学习效率低下。更好的方法是，阅读时只偶尔做做笔记，或在读过每个主要章节后写一段落的总结。
这里
有个例子，是我上机器视觉这门课时的笔记。
第二阶段：练习
做练习题，能极大地促进你的知识理解。但是，如果你不小心，可能会落入两个效率陷阱：
- 没有获得即时的反馈：研究表明，如果你想更好地学习，你需要即时的反馈。因此，做题时最好是答案在手，天下我有，每做完一题就对答案，自我审查。没有反馈或反馈迟来的练习，只会严重牵制学习效率；
- 题海战术：正如有人以为学习是始于教室终于教室，一些学生也认为大多数的知识理解产自练习题。是的，你总能通过题海战术最终搭起知识框架，但过程缓慢、效率低下。
练习题，应该能凸显你需要建立更好直觉的知识领域。一些技巧，比如我将会谈到的费曼技巧（the Feynman technique），对此则相当有效。对于非技术类学科，它更多的是要求你掌握概念而不是解决问题，所以，你常常只需要完成最少量的习题。对这些科目，你最好花更多的时间在第三阶段，形成学科的洞察力。
第三阶段：自省
知识面覆盖，与做练习题，是为了让你知道你还有什么不懂。这并不像听上去那么容易，毕竟知之为知之，不知为不知，难矣。你以为你都懂了，其实不是，所以老犯错；或者，你对某综合性学科心里没底，但又看不确切还有哪里不懂。
接下来的技巧，我称之为“费曼技巧”，将帮助你查漏补缺，在求知路上走得更远。当你能准确识别出你不懂的知识点时，这个技巧助你填补知识的缺口，尤其是那些最难以填补的巨大缺口。这个技巧还能两用。即使你真的理解了某个想法，它也能让你关联更多的想法，于是，你可以继续钻研，深化理解。
费曼技巧（The Feynman Technique）
这个技巧的灵感，源于诺贝尔物理奖获得者，理查德·费曼（Richard Feynman）。在他的自传
里，他提到曾纠结于某篇艰深的研究论文。他的办法是，仔细审阅这篇论文的辅助材料（supporting material），直到他掌握了相关的知识基础、足以理解其中的艰深想法为止。
费曼技巧，亦同此理。对付一个知识枝节繁杂如发丝、富有内涵的想法，应该分而化之，切成小知识块，再逐个对付，你最终能填补所有的知识缺口，否则，这些缺口将阻挠你理解这个想法。对此，请看这个简短的教程视频
。
费曼技巧很简单：
- 拿张白纸；
- 在白纸顶部写上你想理解的某想法或某过程；
- 用你自己的话解释它，就像你在教给别人这个想法。
最要紧的是，对一个想法分而化之，虽然可能重复解释某些已经弄懂的知识点。但你最终会到达一个临界点，无法再解释清楚。那里正是你需要填补的知识缺口。为了填补这个缺口，你可以查课本、问老师、或到互联网搜寻答案。通常来说，一旦你精准地定义了你的不解或误解，找到确切的答案则相对而言更轻松。
我已经使用过这个费曼技巧有数百次，确信它能应付各种各样的学习情境。然而，由于学习情境各有特点，它需要灵活变通，似乎显得难以入门，所以，我将尝试举些不同的例子。
对付你完全摸不着头脑的概念
对此，我仍坚持使用费曼技巧，但翻开课本，找到解释这个概念的章节。我先浏览一遍作者的解释，然后仔细地摹仿它，并也试着用自己的思维详述和阐明它。如此一来，当你不能用自己的话写下任何解释时，“引导式”费曼技巧很有用处。这里
有个例子，展示我如何理解摄影测量学。
对付各种过程
你也能通过费曼技巧去了解一个你需要用到的过程。审视所有的步骤，不光解释每一步在干什么，还要清楚它是怎么执行的。我常这样理解数学的证明过程、化学的方程式、与生物学的糖酵解过程。这里
有个例子，展示我如何想到怎么实现网格加速。
对付各种公式
公式，应该被理解，而不只是死记硬背。因此，当你看到一个公式，却无法理解它的运作机理时，试着用费曼技巧分而化之。这里
有个例子，展示我如何理解傅里叶分析方程。
对付需要记忆的内容
费曼技巧，也可以帮你自查是否掌握非技术类学科那些博大精深的知识概念。对于某个主题，如果你能顺利应用费曼技巧，而无需参考原始材料（讲义、课本等），就证明你已经理解和记住它。这里
有个例子，展示我如何回忆起经济学中的掠夺性定价概念。
形成更深刻的直觉（Deeper Intuition）
结合做习题，费曼技巧能帮你剥开知识理解的浅层表皮。但它也能帮你钻研下去，走得更远，不只是浅层的理解，而是形成深刻的知识直觉。直观地理解一个想法，并非易事。它看似有些许神秘，但这不是它的本相。一个想法的多数直觉，可作以下归类：
类比、可视化、简化
类比：你理解一个想法，是通过确认它与某个更易理解的想法之间的重要相似点；可视化：抽象概念也常成为有用的直觉，只要我们能在脑海为它们构筑画面，即使这个画面只是一个更大更多样化想法的不完全表达；简化：一位著名的科学家曾说过，如果你不能给你的祖母解释一样东西，说明你还没有完全理解它。简化是一门艺术，它加强了基础概念与复杂想法之间的思维联系。
你可以用费曼技巧去激发这些直觉。对于某个想法，一旦你有了大致的理解，下一步就是深入分析，看能不能用以上三种直觉来阐释它。期间，就算是借用已有的意象喻义，也是情有可原的。例如，把复数放到二维空间里理解，很难称得上是新颖的，但它能让你很好地可视化这个概念，让概念在脑海中构图成型。DNA复制，被想象成拉开一条单向拉链，这也不是一个完美的类比，但只要你心里清楚其中的异同，它会变得有用。
学得更快的策略
在这篇文章里，我描述了学习的三个阶段：知识面、练习、与自省。但这可能让你误解，错以为它们总在不同的时期被各自执行，从不重叠或反复。实际上，随着不断地深入理解知识，你可能会周而复始地经历这些阶段。你刚开始读一个章节，只能有个大概的肤浅印象，但做过练习题和建立了直觉以后，你再回过来重新阅读，又会有更深刻的理解，即温故而知新。
钻研吧，即便你不是学生
这个过程不只是适用于学生，也同样有助于学习复杂技能或积累某话题的专业知识。学习像编程或设计的技能，大多数人遵循前两个阶段。他们阅读一本相关的基础书籍，然后在一个项目里历练。然而，你能运用费曼技巧更进一步，更好地锁定与清晰表述你的深刻见解。积累某话题的专业知识，亦同此理；唯一的差别是，你在建立知识面以前，需要搜集一些学习材料，包括相关的研究文章、书籍等。无论如何，只要你弄清楚了想掌握的知识领域，你就钻研下去，深入学习它。
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费曼技巧——高速学习的方法
这个技巧是我几个月前在译言上看到的，作者非常NB，用了一年完成了MIT课程，且还是自学完成的。他总结了自己的学习方法，这里我也复制简化成文。[进入作者的博客]
[三个阶段]
一 知识面覆盖
- 了解学习的大致内容：看目录
- 看讲义
- 读课本
- 阅读同主题书
Notice：加速完成这个阶段，把书过一遍。不必画重点，也就偶尔做做笔记，或在读过每个主要章节后写一段落总结。利用指读来加速阅读速度。不会同时阅读两本书。
二 练习
做练习能加速知识的理解。
Notice：两个效率陷阱
- 没有及时获得反馈。(做完题马上对答案。)
- 题海战术
三 自省
上面两个阶段是为了让你明白你还有什么不懂的。
然后有目的的查缺补漏。
[费曼技巧]
分化和梳理
一旦你精准地定义了你的不解或误解，找到确切的答案则相对而言更轻松。
“费曼技巧”的要点正在于，不需要对教科书照本宣科，而是按自己的需要重构一个理论系统。
- 拿张白纸；
- 在白纸顶部写上你想理解的某想法或某过程；
- 用你自己的话解释它，就像你在教给别人这个想法。
下面是作者根据自己的经验，针对不同学习内容的具体应用方法：
概念
翻开课本，他首先会找到解释这个概念的章节，浏览一遍作者的解释，然后仔细地摹仿它，并试着用自己的思维详述和阐明。
过程
审视所有过程和步骤，不光解释每步在干什么，还要清楚如何执行。
公式
公式应该被理解，而不是死记硬背。若看到公式却无法理解其运动机理或其过程。请用费曼技巧分而化之。
需记忆的内容
针对某主题，如果你能顺利应用费曼技巧，而不用参考原始资料(讲义.课本等)，就说明已经理解并记住。
形成更深的直觉
练习能帮助理解，但还仅限于表层。
类比、可视化、简化
类比：发现并确认它与某个更易理解的想法之间的重要相似点；
可视化：抽象概念只要能在脑海为构筑画面，即使该画面只是一个更大更多样化想法的不完全表达；
简化：不能用简单语言解释一样东西，说明你还没有完全理解它。简化是一门艺术，它加强了基础概念与复杂想法之间的思维联系。
在有了大致的理解后，下一步就是深入分析，看能不能用以上三种直觉来阐释它。期间，就算是借用已有的意象喻义，也是可以理解的。只要你心里清楚其中的异同，它会变得有用。
这里分享一下类似的方法：
- 知道学的是什么。
- 掌握大致的知识点
- 做基础概念题
- 根据题目查缺补漏
- 完善思考方式和逻辑
- 重新看书，形成逻辑链或知识框架
- 做笔记
- 反馈
如果没有目标，就找出你所在行业的最高端证书，拿下它！！
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异曲同工两小时掌握学英语的秘诀——读书笔记－掌握书面英语《两小时掌握学英语的秘诀》读书笔记－掌握书面英语
掌握这书面英语。这条秘诀在《通天秘笈》中被称为‘灵犀一纸’。”
“很简单，‘灵犀一纸’的要求，就是你将一份报纸复述下来。”
“先选一份英文报纸，最佳方案当然是原版的英文报纸，实在不行，用《21世纪报》（21st Century）也行。然后翻开一个版面，随意挑选出一篇短文或一个段落，长度也就是读起来在两三分钟。接着，你就大声朗读，把自己想象成一位播音员，把那种感觉读出来。当你觉得已经记下所读的内容时，就合上报纸，把刚才所读到的内容复述下来。如果在复述的过程中遇到卡壳的地方，也不要停下来，就按照你自己所想的复述下去，一直到结束。当然，如果你对自己的复述不满意的话，可以再重新来一遍。就这样，一段一段地把整份报纸复述下来，至此，你这‘灵犀一纸’的功夫就连成了。”
当你能够将报纸上的文章一一复述下来的时候，其实在你脑海中生根的是英语语言的逻辑体系。
如果报纸不行，那就选取一份杂志，比如美国发行的《时代周刊》（TIME）、《新闻周刊》（NEWS WEEKLY）等。”
“那如果我遇到生词怎么办呢？”
1、对于英汉类词典，传统的是《新英汉词典》，这本词典凝聚着中国老一辈英语学者的心血，编写得确实非常出色。另一本不错的词典是《英汉多功能词典》，它是由外研社和建宏出版社共同出版，编写体例新颖，适合国人学习英语的特点。
2、对于英汉双解词典，《牛津高阶英汉双解词典》是一个不错的选择，足可满足普通英语学习者的所有要求。
对于是否必须使用英英词典，我认为，如果你学英语的目的只是应付考试，那么就没有必要用英英词典；但如果你的目的是希望学好英语，切实感受英语的魅力，那么在学习时选用英英词典将会令你更容易掌握到一个单词的本义和用法。对于你小米来说，那当然要选取英英词典喽！
在选用英英词典时，你可以根据自己不同的学习阶段选用不同的词典：
1、在初级阶段，可以选用朗文词典、剑桥英语词典、柯林斯词典等，它们的英文注释词一般都在2000词左右，即用最常见的词来解释所有的单词。这种单词注解方式，不仅适合初学者的英语水平，更能使初学者领略到如何用最简单的词来表达所有意思的巧妙手法。要知道，一个外国文盲的词汇量也就2000词左右，但他却可以将所有的意思都表达完，故而能否活用最基本的2000词，这本身也是一个人英语水平高低的体现。
2、在中级阶段，可以选用牛津词典、兰登书屋词典、企鹅英语词典等。这类词典中的注解词一般在5000词左右，看它们的注解会令你感觉一步到位，相当直接。你如果达到这个水平，当然可以选用。
3、在高级阶段，你可以选用美国的韦氏词典，它的注解词大概在一万词左右，而且基本上没有什么例句。你时常会感觉这样的注解比较生涩。有时为了理解一个词的意思，你可能还要把注解词再查一下，甚至一路查下去。
任何一个想成功攻克英语的人都会是一个查词典狂人。只有养成查词典的好习惯，你才会成功。一旦你完全地爱上了查词典，那你的英语水平也就要发生飞越了。”
当你复述报纸时，一定要有一个时限。比如说，你熟读一段话所花费的时间为100秒，那么当你合上报纸复述这段话时，所花费的时间不应该超过120秒，换句话说，你复述时所花费的时间应该不超过熟读时间的20％。
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Study Hacks BlogDECODING PATTERNS OF SUCCESS
Mastering Linear Algebra in 10 Days: Astounding Experiments in Ultra-Learning
October 26th, 2012 · 95 comments
The MIT Challenge
My friend Scott Young recently finished an astounding feat: he completed all 33 courses in MIT’s fabled computer science curriculum, from Linear Algebra to Theory of Computation, in less than one year. More importantly, he did it all on his own, watching the lectures online and evaluating himself using the actual exams. (See Scott’s FAQ page for the details of how he ran this challenge.)
That works out to around 1 course every 1.5 weeks.
As you know, I’m convinced that the ability to master complicated information quickly is crucial for building a remarkable career (see my new book as well as here and here). So, naturally, I had to ask Scott to share his secrets with us. Fortunately, he agreed.
Below is a detailed guest post, written by Scott, that drills down to the exact techniques he used (including specific examples) to pull off his MIT Challenge.
Take it away Scott…
HOW I TAMED MIT’S COMPUTER SCIENCE CURRICULUM, BY SCOTT YOUNG
I’ve always been excited by the prospect of learning faster. Being good at things matters. Expertise and mastery give you the career capital to earn more money and enjoy lifestyle perks. If being good is the goal, learning is how you get there.
Despite the advantages of learning faster, most people seem reluctant to learn how to learn. Maybe it’s because we don’t believe it’s possible, that learning speed is solely the domain of good genes or talent.
While there will always be people with unfair advantages, the research shows the method you use to learn matters a lot. Deeper levels of processing and spaced repetition can, in some cases, double your efficiency. Indeed the research in deliberate practice shows us that without the right method, learning can plateau forever.
Today I want to share the strategy I used to compress the ideas from a 4-year MIT computer science curriculum down to 12 months. This strategy was honed over 33 classes, figuring out what worked and what didn’t in the method for learning faster.
Why Cramming Doesn’t Work
Many student might scoff at the idea of learning a 4-year program in a quarter of the time. After all, couldn’t you just cram for every exam and pass without understanding anything?
Unfortunately this strategy doesn’t work. First, MITs exams rely heavily on problem solving, often with unseen problem types. Second, MIT courses are highly cumulative, even if you could sneak by one exam through memorization, the seventh class in a series would be impossible to follow.
Instead of memorizing, I had to find a way to speed up the process of understanding itself.
Can You Speed Up Understanding?
We’ve all had those, “Aha!” moments when we finally get an idea. The problem is most of us don’t have a systematic way of finding them. The typical process a student goes through in learning is to follow a lectures, read a book and, failing that, grind out practice questions or reread notes.
Without a system, understanding faster seems impossible. After all, the mental mechanisms for generating insights are completely hidden.
Worse, understanding is hardly an on/off switch. It’s like layers of an onion, from very superficial insights to the deep understandings that underpin scientific revolutions. Peeling that onion is often a poorly understood process.
The first step is to demystify the process. Getting insights to deepen your understanding largely amounts to two things:
- Making connections
- Debugging errors
Connections are important because they provide an access point for understanding an idea. I struggled with the Fourier transform until I realized it was turning pressure to pitch or radiation to color. Insights like these are often making connections between something you do understand and the material you don’t.
Debugging errors is also important because often you make mistakes because you’re missing knowledge or have an incorrect picture. A poor understanding is like a buggy software program. If you can debug yourself in an efficient way, you can greatly accelerate the learning process.
Doing these two things, forming accurate connections and debugging errors, is most of creating a deep understanding. Mechanical skill and memorized facts also help, but generally only when they sit upon the foundation of a solid intuition about the subject.
THE DRILLDOWN METHOD: A STRATEGY FOR LEARNING FASTER
During the yearlong pursuit, I perfected a method for peeling those layers of deep understanding faster. I’ve since used it on topics in math, biology, physics, economics and engineering. With just a few modifications, it also works well for practical skills such as programming, design or languages.
Here’s the basic structure of the method:
- Coverage
- Practice
- Insight
I’ll explain each stage and how you can go through them as efficiently as possible, while giving detailed examples of how I used them in actual classes.
Stage One: Coverage
You can’t plan an attack if you don’t have a map of the terrain. Therefore the first step in learning anything deeply, is to get a general sense of what you need to learn.
For a class, this means watching lectures or reading textbooks. For self-learning it might mean reading several books on the topic and doing research.
A mistake students often make is believing this stage is the most important. In many ways this is the least efficient stage because the amount you can learn per unit of time invested is much lower. I often found it useful to speed up this part so that I would have more time to spend on the latter two steps.
If you’re watching video lectures, a great way to do this is to watch them at 1.5x or 2x the speed. This can be done easily by downloading the video and then using the speed-up feature on a player like VLC. I’d watch semester-long courses in two days, via this method.
If you’re reading a book, I would recommend against highlighting. This is processes the information at a low level of depth and is inefficient in the long run. A better method would be to take sparse notes while reading, or do a one-paragraph summary after you read each major section.
Here’s an example of notes I took while doing readings for a class in machine vision.
Stage Two: Practice
Practice problems are huge for boosting your understanding, but there are two main efficiency traps you can get caught in if you’re not careful.
#1 – Not Getting Immediate Feedback
The research is clear: if you want to learn, you need immediate feedback. The best way to do this is to go question-by-question with the solution key in hand. Once you’ve finished a question, check yourself against the provided solutions. Practice without feedback, or with delayed feedback, drastically hinders effectiveness.
#2 – Grinding Problems
Like the students who fall into the trap of believing that most learning occurs in the classroom, some students believe understanding is generated mostly from practice questions. While you can eventually build an understanding simply by grinding through practice, it’s slow and inefficient.
Practice problems should be used to highlight areas you need to develop a better intuition for. Then techniques like the Feynman technique, which I’ll discuss, handle that process much more efficiently.
Non-technical subjects, ones where you mostly need to understand concepts, not solve problems, can often get away with minimal practice problem work. In these subjects, you’re better off spending more time on the third phase, developing insight.
Stage Three: Insight
The goal of coverage and practice questions is to get you to a point where you know what you don’t understand. This isn’t as easy as it sounds. Often you can be mistaken into believing you understand something, but don’t, or you might not feel confident with a general subject, but not see specifically what is missing.
This next technique, which I call the Feynman technique is about narrowing down those gaps even further. Often when you can identify precisely what you don’t understand, that gives you the tools to fill the gap. It’s the large gaps in understanding which are hardest to fill.
The technique also has a dual purpose. Even when you do understand an idea, it provides you opportunities to create more connections, so you can drill down to a deeper understanding.
THE FEYNMAN TECHNIQUE
I first got the idea from this method from the Nobel prize winning physicist, Richard Feynman. In his autobiography, he describes himself struggling with a hard research paper. His solution was to go meticulously through the supporting material until he understood everything that was required to understand the hard idea.
This technique works similarly. By digesting the big hairy idea you don’t understand into small chunks, and learning those chunks, you can eventually fill every gap that would otherwise prevent you from learning it.
For a video tutorial of this technique, watch this short video.
The technique is simple:
- Get a piece of paper
- Write at the top the idea or process you want to understand
- Explain the idea, as if you were teaching it to someone else
What’s crucial is that the third step will likely repeat some areas of the idea you already understand. However, eventually you’ll reach a stopping point where you can’t explain. That’s the precise gap in your understanding that you need to fill.
From that gap, you can research the answer from a textbook, teacher or online. Generally, once you’ve narrowly defined your misunderstanding it becomes much easier to find the precise answer.
I’ve used this technique hundreds of times, and I’ve found it can tackle a wide variety different learning situations. However, since each might be slightly different, it may seem hard to apply as a beginner, so I’ll try to walk through some different examples.
For Ideas You Don’t Get At All
The way I handle this is to go through the technique but have the textbook open to the chapter explaining that concept. Then I go through and meticulously copy both the author’s explanation, but also try to elaborate and clarify it for myself. This “guided” Feynman can be useful when trying to write anything on your own would be impossible.
Here’s an example I used for trying to understand photogrammetry.
For Procedures
You can also use the method to fully understand a process you need to use. Go through all the steps and explain not only what they do, but how they execute it. I would often go through proof techniques by carefully explaining all the steps. I also used it in understanding chemical equations or in organizing the stages of glycolysis in biology.
You can see this example I used when trying to figure out how to implement grid acceleration.
For Formulas
Formulas should be understood, not just memorized. So when you see a formula, but can’t understand how it works, try walking through each part with a Feynman.
Here’s an example I used for the Fourier analysis equation.
For Checking Your Memory
Feynmans also offer a way to self-test your knowledge of the big ideas for non-technical subjects. Being able to finish a Feynman on a topic without referencing the source material means you understand and can remember it.
Here’s one I did for an economics class, recalling the concept of predatory pricing.
DEVELOPING A DEEPER INTUITION
Combined with practice questions, the Feynman technique can peel those first few layers of understanding. But it can also drill deeper if you want to go from not just having an understanding, but to having a deep intuition.
Understanding an idea intuitively isn’t easy. Once again, getting to this point is often seen as a quasi-mystical process. But it doesn’t have to be. Most intuitions about an idea break down into one of the following types:
- Analogies – You understand an idea by correctly recognizing an important similarity between it and an easier-to-understand idea.
- Visualizations – Abstract ideas often become useful intuitions when we can form a mental picture of them. Even if the picture is just an incomplete representation of a larger, and more varied, idea.
- Simplifications – A famous scientist once said that if you couldn’t explain something to your grandmother, you don’t fully understand it. Simplification is the art of strengthening those connections between basic components and complex ideas.
You can use the Feynman technique as a way of encouraging these types of insights. Once you’ve gotten past a basic understanding of the idea, the next step is to go further and see if you can explain it using some combination of the three methods above.
The truth is plagiarism is okay too, and not every insight needs to be unique. Understanding complex numbers as being two dimensional is hardly original, but it allows a useful visualization. DNA replication working like a one-way zipper is not a perfect analogy, but so long as you understand where it overlaps, it becomes a useful one.
The Strategy to Learn Faster
Learning faster doesn’t need to be a trick to work well. It simply means recognizing what is actually going on when we reach a new level of insight and finding tools to help us reach those stages consistently.
In this article I described learning as being three stages: coverage, practice and insight. This gives the false impression that these three occur always in distinct phases and never overlap or repeat.
In truth you may find yourself going between them in a loop as you successfully peel down to deeper layers of understanding. The first time you read a chapter you may get only superficial insights, but after doing practice questions and building intuitions, you may go back and read for deeper understandings.
Applying the Drilldown Method for Non-Students
This process isn’t one you need to be a student to apply. It also works for learning complex skills or building expertise on a topic.
For skills like programming or design, most people follow the first two stages. They read a book teaching them the basics, then they practice with a project. You can extend that process however, and use the Feynman technique to better lock in and articulate the insights you create.
For expertise on a topic, the only difference is that, prior to doing coverage, you need to find a set of material to learn from. That could be research articles or several books on the topic. In either case, once you’ve defined the chunk of knowledge you want to master, you can drill down and learn it deeply.
To find out more about this, join Scott’s newsletter and you’ll get a free copy of his rapid learning ebook (and a set of detailed case studies of how other learners have used these techniques).
(Image by afagen.)
95 thoughts on “Mastering Linear Algebra in 10 Days: Astounding Experiments in Ultra-Learning”
Great article, Scott. I think I have always gravitated to this type of deep understanding my whole life, despite the fact that it really isn’t how we are taught to learn. I remember that we used memorization a *lot* when we were very young (for the alphabet, multiplication tables, etc.).
My problem with memorization is that my memory of things that I memorize is just too tenuous. When I deeply understand it, then I remember it without effort. In fact, if I deeply understand it, it is hard for me to forget.
I definitely will be working on trying to improve my methods for achieving deep understanding, so thanks for the advice.
Same is the case with me, Mark.
Though I am young(and you would think I memorize better), nothing sticks in my mind for long without deep understanding.
There is a saying I used to teach my Spanish students. “Lo que bien se aprende, nunca se pierde.” It means “what is learned well is never lost.” I think it applies very well to what you describe.
This is such a great post. The key takeaway for me is to spend less time on lectures and more time on finding the gaps in my understanding, plugging those gaps, and then building an intuition for the concepts. If only teachers preached this study strategy. Most people, myself included, just watch the lecture, read the chapter, do a few problems and repeat until time is up. What results is partial knowledge of topics that eventually snowballs into higher education. I would love to hear about the mental/emotional challenges you bumped into when doing your year long course. When you felt like giving up, what strategies did you use to keep going?
Great post. I did something similar in my youth, but I wasn’t scientific about it. I like the process that you have developed. Thank you.
Great article as usual, Scott. I have tried some of these things in my earlier years at college and did great in all of my classes. Unfortunately, I’ve fallen down the path of learning through repetition again. Reading your learning advice once more motivates me to study like I used to