Here’s how to become a math whiz:
Keep working on your problem set after you get stuck.
以下是關於如何成為一名數學高手的建議:
即使感到困惑也持續專注於你的問題。
Don’t just sit and stare at it: think hard; until you’re exhausted; then come back the next day and try again. This will be uncomfortable, but that discomfort is the feeling of your brain stretching to accommodate new abilities.
不要只是坐在桌前盯著問題：仔細想一想；直到你筋疲力盡為止；隔天起床後還要繼續。這可能使你感到“不爽”，但是這種“不爽”是因為你的頭腦正在舒展以便容納新的能力。
This advice came to mind recently when I received an e-mail from a high school senior. “Yesterday, I was accepted to MIT,” he began. “I’m ecstatic, but on the other hand, I’m a little nervous…I was hoping you could give me some tips.”
這個建議是我最近收到某個高中生的郵件時才想起，“昨天，我被MIT錄取了，”他開頭寫道，“我很高興，但另一方面，我又有些緊張，我希望您能給我些意見。”
I explained that I had been studying theoretical computer science and mathematics at a high-level for the past decade, much of it spent right here at MIT. Over these years, one conclusion has become increasingly clear: the more hard focus you dedicate to a technical subject — be it computer science, chemistry, or physics — the better you get.
在過去十年我學過高等理論電腦科學和數學，大部分時間正是在MIT。過去的這些年，有個結論越來越清晰，即你在技術學科如電腦科學、化學或物理方面研究的越多，你將得到的也越多。
Junior graduate students think senior graduate students are smarter, but they’re not: they simply have more practice.
Senior graduate students think junior professors are smarter, but they’re not: they simply have more practice.
And so on.
初三的學生認為高中生們很聰明，但實際上不是的，他們只是練習了很多。
高中生們認為大學教授很牛，實際上也不是，他們只是學的很多。
等等。
When I arrived at Dartmouth, to name another example, I didn’t consider myself good at math. I had taken AB calculus during high school (not BC), and had scored a 4 on the AP exam (not a 5). By my sophomore year of college, however, I had made a name for myself by snagging the highest grade out of 70 students in an advanced discrete mathematics class. What happened in between? A lot of hard focus.
還有個例子，當我來到達達特茅斯時，我不認為我的數學很好。我在高中學的是數學微積分AB（不是BC），我得了4分在數學微積分考試AP中（不是5分）。然而到我大學二年級的時候，我已經在有70名學生的高等離散數學課中脫穎而出了。這期間發生了什麼？很多艱苦的努力。
Eventually, this all becomes clear, but for an incoming freshman, it’s not intuitive. When you struggle with a calculus problem set while a classmate knocks it out in an hour, it’s easy to start to thinking that you’re just not a “math person.”
最後，一切都清楚了，但對於一個新生，這還不太能憑直覺感知。當你還在鑽研一個微積分問題時，你的同學已經在一個小時內解決了它，你很容易就認為自己就不是那種“math person”。
But this isn’t about natural aptitude, it’s about practice. That other student has more practice. You can catch-up, but you have to put in the hours, which brings me back to my original advice: keep working even after you get stuck.
That’s where you make up ground.
但是這無關天資，這需要練習。別的學生做了很多聯繫。你可以趕上來，但你需要付出很多時間，這也正是我最初的建議：即使感到困惑也持續專注你的問題。
這就是你需要做的。
