費曼技巧】——1年自學完MIT 33門課，10天內掌握線性代數
斯考特·楊在12個月內自學完成了4年麻省理工學院計算機科學的33門課程，並通過了MIT的實際測試。 平均算來，楊修完每門課程大概只需要一個半星期。訣竅在於，他有一套加速學習的策略，而且這套策略，並不只是天才們的專利。最快的學習方法就是——【費曼技巧】 斯考特·楊（Scott Young）博客
——
————————————————————————————————
——————————————————————————————————如何在十天內掌握線性代數
最近，我的朋友斯考特·楊（Scott Young）
成就了一個驚人的壯舉：他在一年之內，完成了傳說中的MIT計算機科學課程表的全部33門課，從線性代數到計算理論。最重要的是，他是自學的，觀看在線教程講座，並用實際的考試作自我評估。（到斯考特的FAQ頁面，看看他如何完成這個挑戰）
按照他的進度，讀完一門課程大概只需要1.5個星期。我堅信，能快速掌握複雜信息，對成就卓越事業至關重要。因此，我很自然地問起斯考特，讓他給我們分享他的學習奧祕。所幸他答應了。接下來是一份斯考特的詳細解說稿，深入剖析他的學習技巧（包括具體例子），展示他如何拿下這MIT挑戰。以下時間交給斯考特……
看我怎麼駕馭MIT計算機科學的課程
我老想着學快一點，再快一點，併爲此興奮不已。掌握那些重要的學問吧，專業知識與嫺熟技藝將是你的職業資本，幫你賺取金錢與享受生活。如果過得好是你的目標，學問能引你到嚮往之地。
儘管學得更快有很多好處，但大多數人並不願意學習“如何學習”。大概是因爲我們不肯相信有這種好事，在我們看來，學習的速度只取決於好基因與天賦。確實總有些人身懷天賦本錢，但研究表明你的學習方法也很重要。更深層次的知識加工，與時而反覆的溫故知新，在某些情況下會加倍你的學習效率
。是的，“刻意練習”方面的研究表明，沒有正確的方法，學習將永遠停滯。
今天，我想分享一下學習策略，看看我如何在12個月內完成
4年MIT計算機科學的課程。這套策略歷經33門課的錘鍊，試圖弄清楚學得更快的竅門，哪些方法有用，哪些沒用。
爲什麼臨時抱佛腳沒用？
很多學生可能嘲笑我，妄想只花1年的時間學會4年的課程。畢竟，我總可以臨時抱佛腳，什麼都不懂還能順利通過考試，不是嗎？ 很可惜，這個策略在MIT行不通。首先，MIT的考試苛求解決問題的技巧，還經常出些沒見過的題型。其次，MIT的課程講究循序漸進，就算你能死記硬背僥倖通過一次考試，同系列課程的第七課可能就跟不上了。除了死記硬背，我不得不另闢蹊徑，加速理解過程。
你能加速理解嗎？
“啊哈！”當我們終於想通了，都曾經這樣恍然大悟地歡呼過。問題是，大多數人都沒有系統地思考。經典的學生求學之路，就是聽講座，讀書；如果還不懂，只好枯燥地做大量習題（題海）或重看筆記。沒有系統的方法，想更快地理解似乎是天方夜譚。畢竟，頓悟的心理機制，還全然不知。
更糟的是，理解本身，很難稱得上是一種開關。它像洋蔥的層層表皮，從最膚淺的領會到深層次的理解，逐層鞏固對科學革命的認知。給這樣的洋蔥剝皮，則是常人知之甚少、易被忽略的理解過程。
加速學習的第一步，就是揭祕這個過程。如何洞悉問題，加深你的理解，取決於兩個因素：
- 建立知識聯繫；
- 自我調試排錯。
知識聯繫很重要，因爲它們是瞭解一個想法的接入點。我曾糾結於傅里葉變換，直至我意識到它將壓強轉化爲音高、或將輻射轉化爲顏色。這些見解，常在你懂的和你不懂的之間建立聯繫。調試排錯也同樣重要，因爲你常常犯錯，這些錯誤究根到底，還是知識殘缺，胸無成竹。貧瘠的理解，恰似一個錯漏百出的軟件程序。如果你能高效地自我調試，必將大大提速學習進程。建立準確的知識聯繫與調試排錯，就足夠形成了深刻的問題見解。而機械化技能與死記硬背，通常也只在你對問題的本質有了肯定的直覺以後，纔有所裨益。
鑽研（The Drilldown Method）：你學得更快
經年累月，我完善了一個方法，可以加速逐層增進理解的過程。這個方法至今已被我用於各科目的課題，包括數學、生物學、物理學、經濟學與工程學。只需些許修改，它對掌握實用技能也效果很好，比如編程、設計或語言。這個方法的基本結構是：知識面、練習、自省。我將解釋每個階段，讓你瞭解如何儘可能有效率地執行它們，同時給出詳細的例子，展示我是怎麼應用在實際課程的。
第一階段：知識面覆蓋
你不可能組織一場進攻，如果你連一張地形圖都沒有。因此，深入研習的第一步，就是對你需要學習的內容有個大致印象。若在課堂上，這意味着你要看講義或讀課本；若是自學，你可能要多讀幾本同主題的書，相互考證。
學生們常犯的一個錯誤，就是認爲這個階段是最重要的。從很多方面來講，這個階段卻是效率最低的，因爲你每單位時間的投入只換來了最少量的知識回報。我常常加速完成這個階段，很有好處，這樣，我就可以投入更多時間到後面兩個階段。
如果你在看課程講座的視頻，最好是調到1.5x或2x倍速快進。這很容易做到，只要你下載好視頻，然後使用播放器（如VLC
）的“調速”功能。我用這法子兩天內看完了一學期的課程視頻。如果你在讀一本書，我建議你不要花時間去高亮文本。這樣只會讓你的知識理解停留在低層次，而從長遠來看，也使學習效率低下。更好的方法是，閱讀時只偶爾做做筆記，或在讀過每個主要章節後寫一段落的總結。
這裏
有個例子，是我上機器視覺這門課時的筆記。
第二階段：練習
做練習題，能極大地促進你的知識理解。但是，如果你不小心，可能會落入兩個效率陷阱：
- 沒有獲得即時的反饋：研究表明，如果你想更好地學習，你需要即時的反饋。因此，做題時最好是答案在手，天下我有，每做完一題就對答案，自我審查。沒有反饋或反饋遲來的練習，只會嚴重牽制學習效率；
- 題海戰術：正如有人以爲學習是始於教室終於教室，一些學生也認爲大多數的知識理解產自練習題。是的，你總能通過題海戰術最終搭起知識框架，但過程緩慢、效率低下。
練習題，應該能凸顯你需要建立更好直覺的知識領域。一些技巧，比如我將會談到的費曼技巧（the Feynman technique），對此則相當有效。對於非技術類學科，它更多的是要求你掌握概念而不是解決問題，所以，你常常只需要完成最少量的習題。對這些科目，你最好花更多的時間在第三階段，形成學科的洞察力。
第三階段：自省
知識面覆蓋，與做練習題，是爲了讓你知道你還有什麼不懂。這並不像聽上去那麼容易，畢竟知之爲知之，不知爲不知，難矣。你以爲你都懂了，其實不是，所以老犯錯；或者，你對某綜合性學科心裏沒底，但又看不確切還有哪裏不懂。
接下來的技巧，我稱之爲“費曼技巧”，將幫助你查漏補缺，在求知路上走得更遠。當你能準確識別出你不懂的知識點時，這個技巧助你填補知識的缺口，尤其是那些最難以填補的巨大缺口。這個技巧還能兩用。即使你真的理解了某個想法，它也能讓你關聯更多的想法，於是，你可以繼續鑽研，深化理解。
費曼技巧（The Feynman Technique）
這個技巧的靈感，源於諾貝爾物理獎獲得者，理查德·費曼（Richard Feynman）。在他的自傳
裏，他提到曾糾結於某篇艱深的研究論文。他的辦法是，仔細審閱這篇論文的輔助材料（supporting material），直到他掌握了相關的知識基礎、足以理解其中的艱深想法爲止。
費曼技巧，亦同此理。對付一個知識枝節繁雜如髮絲、富有內涵的想法，應該分而化之，切成小知識塊，再逐個對付，你最終能填補所有的知識缺口，否則，這些缺口將阻撓你理解這個想法。對此，請看這個簡短的教程視頻
。
費曼技巧很簡單：
- 拿張白紙；
- 在白紙頂部寫上你想理解的某想法或某過程；
- 用你自己的話解釋它，就像你在教給別人這個想法。
最要緊的是，對一個想法分而化之，雖然可能重複解釋某些已經弄懂的知識點。但你最終會到達一個臨界點，無法再解釋清楚。那裏正是你需要填補的知識缺口。爲了填補這個缺口，你可以查課本、問老師、或到互聯網搜尋答案。通常來說，一旦你精準地定義了你的不解或誤解，找到確切的答案則相對而言更輕鬆。
我已經使用過這個費曼技巧有數百次，確信它能應付各種各樣的學習情境。然而，由於學習情境各有特點，它需要靈活變通，似乎顯得難以入門，所以，我將嘗試舉些不同的例子。
對付你完全摸不着頭腦的概念
對此，我仍堅持使用費曼技巧，但翻開課本，找到解釋這個概念的章節。我先瀏覽一遍作者的解釋，然後仔細地摹仿它，並也試着用自己的思維詳述和闡明它。如此一來，當你不能用自己的話寫下任何解釋時，“引導式”費曼技巧很有用處。這裏
有個例子，展示我如何理解攝影測量學。
對付各種過程
你也能通過費曼技巧去了解一個你需要用到的過程。審視所有的步驟，不光解釋每一步在幹什麼，還要清楚它是怎麼執行的。我常這樣理解數學的證明過程、化學的方程式、與生物學的糖酵解過程。這裏
有個例子，展示我如何想到怎麼實現網格加速。
對付各種公式
公式，應該被理解，而不只是死記硬背。因此，當你看到一個公式，卻無法理解它的運作機理時，試着用費曼技巧分而化之。這裏
有個例子，展示我如何理解傅里葉分析方程。
對付需要記憶的內容
費曼技巧，也可以幫你自查是否掌握非技術類學科那些博大精深的知識概念。對於某個主題，如果你能順利應用費曼技巧，而無需參考原始材料（講義、課本等），就證明你已經理解和記住它。這裏
有個例子，展示我如何回憶起經濟學中的掠奪性定價概念。
形成更深刻的直覺（Deeper Intuition）
結合做習題，費曼技巧能幫你剝開知識理解的淺層表皮。但它也能幫你鑽研下去，走得更遠，不只是淺層的理解，而是形成深刻的知識直覺。直觀地理解一個想法，並非易事。它看似有些許神祕，但這不是它的本相。一個想法的多數直覺，可作以下歸類：
類比、可視化、簡化
類比：你理解一個想法，是通過確認它與某個更易理解的想法之間的重要相似點；可視化：抽象概念也常成爲有用的直覺，只要我們能在腦海爲它們構築畫面，即使這個畫面只是一個更大更多樣化想法的不完全表達；簡化：一位著名的科學家曾說過，如果你不能給你的祖母解釋一樣東西，說明你還沒有完全理解它。簡化是一門藝術，它加強了基礎概念與複雜想法之間的思維聯繫。
你可以用費曼技巧去激發這些直覺。對於某個想法，一旦你有了大致的理解，下一步就是深入分析，看能不能用以上三種直覺來闡釋它。期間，就算是借用已有的意象喻義，也是情有可原的。例如，把複數放到二維空間裏理解，很難稱得上是新穎的，但它能讓你很好地可視化這個概念，讓概念在腦海中構圖成型。DNA複製，被想象成拉開一條單向拉鍊，這也不是一個完美的類比，但只要你心裏清楚其中的異同，它會變得有用。
學得更快的策略
在這篇文章裏，我描述了學習的三個階段：知識面、練習、與自省。但這可能讓你誤解，錯以爲它們總在不同的時期被各自執行，從不重疊或反覆。實際上，隨着不斷地深入理解知識，你可能會周而復始地經歷這些階段。你剛開始讀一個章節，只能有個大概的膚淺印象，但做過練習題和建立了直覺以後，你再回過來重新閱讀，又會有更深刻的理解，即溫故而知新。
鑽研吧，即便你不是學生
這個過程不只是適用於學生，也同樣有助於學習複雜技能或積累某話題的專業知識。學習像編程或設計的技能，大多數人遵循前兩個階段。他們閱讀一本相關的基礎書籍，然後在一個項目裏歷練。然而，你能運用費曼技巧更進一步，更好地鎖定與清晰表述你的深刻見解。積累某話題的專業知識，亦同此理；唯一的差別是，你在建立知識面以前，需要蒐集一些學習材料，包括相關的研究文章、書籍等。無論如何，只要你弄清楚了想掌握的知識領域，你就鑽研下去，深入學習它。
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
費曼技巧——高速學習的方法
這個技巧是我幾個月前在譯言上看到的，作者非常NB，用了一年完成了MIT課程，且還是自學完成的。他總結了自己的學習方法，這裏我也複製簡化成文。[進入作者的博客]
[三個階段]
一 知識面覆蓋
- 瞭解學習的大致內容：看目錄
- 看講義
- 讀課本
- 閱讀同主題書
Notice：加速完成這個階段，把書過一遍。不必畫重點，也就偶爾做做筆記，或在讀過每個主要章節後寫一段落總結。利用指讀來加速閱讀速度。不會同時閱讀兩本書。
二 練習
做練習能加速知識的理解。
Notice：兩個效率陷阱
- 沒有及時獲得反饋。(做完題馬上對答案。)
- 題海戰術
三 自省
上面兩個階段是爲了讓你明白你還有什麼不懂的。
然後有目的的查缺補漏。
[費曼技巧]
分化和梳理
一旦你精準地定義了你的不解或誤解，找到確切的答案則相對而言更輕鬆。
“費曼技巧”的要點正在於，不需要對教科書照本宣科，而是按自己的需要重構一個理論系統。
- 拿張白紙；
- 在白紙頂部寫上你想理解的某想法或某過程；
- 用你自己的話解釋它，就像你在教給別人這個想法。
下面是作者根據自己的經驗，針對不同學習內容的具體應用方法：
概念
翻開課本，他首先會找到解釋這個概念的章節，瀏覽一遍作者的解釋，然後仔細地摹仿它，並試着用自己的思維詳述和闡明。
過程
審視所有過程和步驟，不光解釋每步在幹什麼，還要清楚如何執行。
公式
公式應該被理解，而不是死記硬背。若看到公式卻無法理解其運動機理或其過程。請用費曼技巧分而化之。
需記憶的內容
針對某主題，如果你能順利應用費曼技巧，而不用參考原始資料(講義.課本等)，就說明已經理解並記住。
形成更深的直覺
練習能幫助理解，但還僅限於表層。
類比、可視化、簡化
類比：發現並確認它與某個更易理解的想法之間的重要相似點；
可視化：抽象概念只要能在腦海爲構築畫面，即使該畫面只是一個更大更多樣化想法的不完全表達；
簡化：不能用簡單語言解釋一樣東西，說明你還沒有完全理解它。簡化是一門藝術，它加強了基礎概念與複雜想法之間的思維聯繫。
在有了大致的理解後，下一步就是深入分析，看能不能用以上三種直覺來闡釋它。期間，就算是借用已有的意象喻義，也是可以理解的。只要你心裏清楚其中的異同，它會變得有用。
這裏分享一下類似的方法：
- 知道學的是什麼。
- 掌握大致的知識點
- 做基礎概念題
- 根據題目查缺補漏
- 完善思考方式和邏輯
- 重新看書，形成邏輯鏈或知識框架
- 做筆記
- 反饋
如果沒有目標，就找出你所在行業的最高端證書，拿下它！！
__________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________
異曲同工兩小時掌握學英語的祕訣——讀書筆記－掌握書面英語《兩小時掌握學英語的祕訣》讀書筆記－掌握書面英語
掌握這書面英語。這條祕訣在《通天祕笈》中被稱爲‘靈犀一紙’。”
“很簡單，‘靈犀一紙’的要求，就是你將一份報紙複述下來。”
“先選一份英文報紙，最佳方案當然是原版的英文報紙，實在不行，用《21世紀報》（21st Century）也行。然後翻開一個版面，隨意挑選出一篇短文或一個段落，長度也就是讀起來在兩三分鐘。接着，你就大聲朗讀，把自己想象成一位播音員，把那種感覺讀出來。當你覺得已經記下所讀的內容時，就合上報紙，把剛纔所讀到的內容複述下來。如果在複述的過程中遇到卡殼的地方，也不要停下來，就按照你自己所想的複述下去，一直到結束。當然，如果你對自己的複述不滿意的話，可以再重新來一遍。就這樣，一段一段地把整份報紙複述下來，至此，你這‘靈犀一紙’的功夫就連成了。”
當你能夠將報紙上的文章一一複述下來的時候，其實在你腦海中生根的是英語語言的邏輯體系。
如果報紙不行，那就選取一份雜誌，比如美國發行的《時代週刊》（TIME）、《新聞週刊》（NEWS WEEKLY）等。”
“那如果我遇到生詞怎麼辦呢？”
1、對於英漢類詞典，傳統的是《新英漢詞典》，這本詞典凝聚着中國老一輩英語學者的心血，編寫得確實非常出色。另一本不錯的詞典是《英漢多功能詞典》，它是由外研社和建宏出版社共同出版，編寫體例新穎，適合國人學習英語的特點。
2、對於英漢雙解詞典，《牛津高階英漢雙解詞典》是一個不錯的選擇，足可滿足普通英語學習者的所有要求。
對於是否必須使用英英詞典，我認爲，如果你學英語的目的只是應付考試，那麼就沒有必要用英英詞典；但如果你的目的是希望學好英語，切實感受英語的魅力，那麼在學習時選用英英詞典將會令你更容易掌握到一個單詞的本義和用法。對於你小米來說，那當然要選取英英詞典嘍！
在選用英英詞典時，你可以根據自己不同的學習階段選用不同的詞典：
1、在初級階段，可以選用朗文詞典、劍橋英語詞典、柯林斯詞典等，它們的英文註釋詞一般都在2000詞左右，即用最常見的詞來解釋所有的單詞。這種單詞註解方式，不僅適合初學者的英語水平，更能使初學者領略到如何用最簡單的詞來表達所有意思的巧妙手法。要知道，一個外國文盲的詞彙量也就2000詞左右，但他卻可以將所有的意思都表達完，故而能否活用最基本的2000詞，這本身也是一個人英語水平高低的體現。
2、在中級階段，可以選用牛津詞典、蘭登書屋詞典、企鵝英語詞典等。這類詞典中的註解詞一般在5000詞左右，看它們的註解會令你感覺一步到位，相當直接。你如果達到這個水平，當然可以選用。
3、在高級階段，你可以選用美國的韋氏詞典，它的註解詞大概在一萬詞左右，而且基本上沒有什麼例句。你時常會感覺這樣的註解比較生澀。有時爲了理解一個詞的意思，你可能還要把註解詞再查一下，甚至一路查下去。
任何一個想成功攻克英語的人都會是一個查詞典狂人。只有養成查詞典的好習慣，你纔會成功。一旦你完全地愛上了查詞典，那你的英語水平也就要發生飛越了。”
當你複述報紙時，一定要有一個時限。比如說，你熟讀一段話所花費的時間爲100秒，那麼當你合上報紙複述這段話時，所花費的時間不應該超過120秒，換句話說，你複述時所花費的時間應該不超過熟讀時間的20％。
__________________________________________________________________________________________________________________________________________________________________________________________________
_____________________________________________________________________________________________
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.)
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