During med school, I remember going through massive QBanks on UWorld for big exams like the MCAT or the USMLE.

在醫學院就讀期間，我曾在 UWorld 上瀏覽過大量的 QBank，以應對 MCAT 或 USMLE 等大型考試。

Massive, like literally thousands of practice problems.

大量的練習題，多達數千個。

But I've also taken classes where I was only given a few past papers, and I still made it work.

但我也上過只給我幾張過去試卷的課，但我還是成功了。

And so I realized it doesn't matter if you have 10 practice problems or 10,000.

於是我意識到，有 10 道練習題還是 10,000 道練習題並不重要。

If you're not using them properly, then it's going to be a waste of time.

如果使用不當，就會浪費時間。

So how is this done?

那麼，如何做到這一點呢？

The most important idea to understand here is confidence.

在這裡，最重要的一點就是要有信心。

You know when you use flashcard apps like Anki or RemNote?

你知道，當你使用 Anki 或 RemNote 等閃存卡應用程序時？

After you answer the flashcard in your head and reveal the answer, it asks you to rate how well you know it.

當您在腦海中回答閃卡上的問題並揭曉答案後，它會要求您對自己的知識掌握程度進行評分。

That's the important part.

這才是最重要的部分。

It's asking, how confident were you with that decision?

這是在問，你對這個決定有多大信心？

I would argue that it's smarter to rank your confidence before even checking the answer.

我認為，更明智的做法是，在核對答案之前，先對自己的信心進行排序。

Because it's not enough to just get the answers right, you also need to be confident that you know the answer and that you'll get it right again.

因為僅僅答對答案是不夠的，你還需要對自己知道答案並能再次答對答案充滿信心。

More confident that we know something, but it turns out that we were wrong, that piece of information is way more likely to stick in our memory.

如果我們更自信地認為自己知道某些事情，但事實證明我們錯了，那麼這條資訊就更有可能留在我們的記憶中。

In psychology, this is called the hypercorrection effect.

在心理學中，這被稱為超矯正效應。

Like I remember once I got into it with my friend that the main villain in Lord of the Rings was Saruman.

就像我記得有一次我和朋友討論《指環王》中的大反派是薩魯曼。

And you know, we got this heated argument about it.

我們為此爭論不休。

Then it turns out I was wrong and it's Sauron, not Saruman.

後來發現我錯了，是索倫，不是薩魯曼。

Lost 50 bucks for it.

為此損失了 50 塊錢。

And so yeah, I'm never going to forget that again.

所以，是的，我再也不會忘記了。

So here's a three-step framework to problems.

是以，這裡有一個解決問題的三步框架。

Step one is to always make sure you know why the right answer is right and why the wrong answers are wrong.

第一步是始終確保自己知道為什麼正確的答案是對的，為什麼錯誤的答案是錯的。

That's why it's always helpful to have an answer key to check your answers.

這就是為什麼有一個答案密鑰來核對答案總是很有幫助的原因。

And you want the answer key to be a detailed explanation.

你希望答案要點是詳細的解釋。

So it explains the thought process.

這就解釋了思考的過程。

Step two is to change the variables of the answer choices or change the variables of the question itself.

第二步是改變答案選項的變量或問題本身的變量。

So for example, if you know why the wrong answers are wrong, how would you change the wrong answers to then make them right?

例如，如果你知道為什麼錯誤的答案是錯誤的，那麼你將如何改變錯誤的答案，使之成為正確的答案？

Or how would you change the question itself so that the start manipulating the questions and the answers, you'll start to think like an exam writer.

或者，你會如何改變問題本身，從而開始操縱問題和答案，開始像考試命題人一樣思考。

I would say, try to go even deeper and ask yourself, how could this question be asked differently on the test?

我想說的是，試著更深入地問自己，這個問題在考試中可以有什麼不同的問法？

Or ask, how could the teacher ask a curve ball question or combine multiple variables into the same question?

或者問，老師怎麼會提出一個曲線問題，或者把多個變量合併到同一個問題中？

So now I'm going to use tips number one and two on some practice problems to see how this all works.

現在，我將在一些練習題中使用提示一和提示二，看看這一切是如何實現的。

And because I love triangles, let's do some trick.

因為我喜歡三角形，所以我們來變個戲法。

So let's draw a right triangle here like that.

是以，讓我們在這裡畫一個這樣的直角三角形。

It's actually pretty good.

其實還不錯。

And let's say this side here is a four, this one over here is six.

假設這邊是 4，這邊是 6。

And we'll be solving for this length here, which is x.

我們將求解這裡的長度，即 x。

And I'll put some answer choices from this problem set here.

我會把這套試題中的一些答案選項放在這裡。

So A is going to be 12.

所以 A 將是 12。

B is going to be 7.2.

B 將為 7.2。

And C is 17.2.

而 C 是 17.2。

So for my long lost memory of trigonometry, and from googling it like 20 minutes ago, I recall that to find the hypotenuse of a right triangle, the long side that's opposite from that right angle can be found using the Pythagorean theorem.

是以，根據我久違的三角學記憶，以及 20 分鐘前在谷歌上搜索的結果，我記得要找到直角三角形的斜邊，可以使用勾股定理找到與該直角相對的長邊。

So the Pythagorean theorem, I think I spelled that right, which is also equal to A squared plus B squared equals C squared.

所以勾股定理，我想我拼對了，也等於 A 的平方加上 B 的平方等於 C 的平方。

C being the length of the hypotenuse, the long side.

C 是斜邊（即長邊）的長度。

So with some simple plug and chug, we get four squared plus six squared equals C squared.

是以，通過一些簡單的插拔操作，我們可以得到 4 的平方加上 6 的平方等於 C 的平方。

Simplifying this down, we're going to get 16 plus 36 equals C squared. 16 plus 36, or is that 52, equals C squared.

簡化後，我們可以得到 16 加 36 等於 C 的平方。16 加 36，還是 52，等於 C 的平方。

And finally, we just take the square root of each side because this one has A squared over here, plug that into your calculator or Wolfram Alpha or whatever, and we'll get that C is equal to 7.2.

最後，我們只需取每邊的平方根，因為這裡有 A 的平方，將其輸入計算器或 Wolfram Alpha 或其他工具，就能得出 C 等於 7.2。

And that makes B the correct answer choice.

是以，B 是正確的答案選項。

Now, what if this one practice problem was all you had to study for your upcoming quiz?

現在，如果這道練習題就是你為即將到來的測驗而學習的全部內容呢？

We can use tips one and two to learn a lot more from this problem.

我們可以利用提示一和提示二，從這個問題中學到更多。

So tip number one is to know why all the wrong answers are incorrect and why the right answer is correct, right?

所以，第一條建議就是要知道為什麼所有錯誤的答案都是不正確的，為什麼正確的答案是正確的，對嗎？

For math, this is much more black and white.

對於數學來說，這就更加黑白分明瞭。

Like obviously I can just google square root of 52 and I'll know that it equals 7.2, not 12 or 17.2, right?

很明顯，我只要谷歌一下 52 的平方根，就知道它等於 7.2，而不是 12 或 17.2，對嗎？

But I can also stop and think more carefully about why answers A and C were included at all.

但是，我也可以停下來仔細思考一下，為什麼要把答案 A 和 C 包括在內。

Like why is 12 and why is 17.2 possible answers that it could have gotten given this situation?

比如，在這種情況下，為什麼 12 和 17.2 是可能得到的答案？

So from the limited amount of information that was given from this problem, what else could I actually solve for, right?

那麼，從這個問題給出的有限資訊中，我還能求解出什麼呢，對嗎？

Well, I also know that I can find the area of a right triangle if I have the height and base of the triangle.

嗯，我還知道，如果有了直角三角形的高和底，就可以求出直角三角形的面積。

And those are two variables that I am given.

這就是我得到的兩個變量。

So the area of a triangle is equal to the height times base divided by 2.

是以，三角形的面積等於高乘以底除以 2。

Let's do some simple plug and chug here.

讓我們在這裡做一些簡單的插拔操作。

We have the height of 6 times the base of 4 divided by 2.

6 的高乘以 4 的底除以 2。

That is going to equal, doing simple math in my head, 12.

我在腦子裡簡單算了算，等於 12。

Okay, cool.

好吧

So answer choice A was solving for the area of a triangle, whereas B was testing for my knowledge of the Pythagorean theorem.

是以，答案選項 A 是求解三角形的面積，而答案選項 B 則是測試我對勾股定理的瞭解程度。

What about C?

C 呢？

I see a 0.2 there, that decimal, and answer B, which we just solved for, which was the hypotenuse, was 7.2.

我看到這裡有一個 0.2，即小數點，而答案 B，也就是我們剛剛求出的斜邊，是 7.2。

So the only other variable that I can think of that would be equal to 17.2 would be the perimeter of a triangle, right?

是以，我能想到的唯一等於 17.2 的其他變量就是三角形的周長，對嗎？

Just adding each side length together, the perimeter of a triangle.

將每條邊長相加，就是三角形的周長。

It's equal to A plus B plus C.

等於 A 加 B 加 C。

Plugging all of those in, we would also get 4 plus 6 plus 7.2.

把所有這些都算進去，我們還可以得到 4 加 6 加 7.2。

Since we just solved for it in the Pythagorean theorem, this is going to equal 17.2.

由於我們剛剛用勾股定理求出了它，是以它將等於 17.2。

So C is solving for the perimeter of a triangle.

所以 C 是在求解三角形的周長。

So now I know why answer choices A and C were wrong.

現在我知道為什麼答案選項 A 和 C 是錯誤的了。

They were using different formulas, one for the area of a triangle, one for the perimeter.

他們使用了不同的公式，一個計算三角形的面積，一個計算三角形的周長。

And I know why B was correct because we used the Pythagorean theorem, which was the right formula for that equation.

我知道為什麼 B 是正確的，因為我們使用了勾股定理，而勾股定理是該方程的正確公式。

Now let's move on to tip number two.

現在，讓我們來看看第二條建議。

How can I actually change the conditions so that I would solve for a different variable of this equation overall?

怎樣才能真正改變條件，從而求出這個方程的不同變量？

What would it look like if instead of getting this here, I was actually given 7.2 and asked to solve for this variable right here?

如果我得到的不是這個，而是 7.2，並要求我求解這裡的變量，結果會怎樣？

How would that change the way that I applied the Pythagorean theorem?

這會如何改變我應用勾股定理的方式？

So we can just kind of do the same plug and chug as we just did before, but this time it would look like this.

是以，我們可以像之前一樣進行插拔操作，但這次看起來會是這樣。

We would have A squared plus 6 squared equals to 7.2 squared.

A 的平方加上 6 的平方等於 7.2 的平方。

And that would give me this final answer of A equals to 4, right?

最後得到的答案是 A 等於 4，對嗎？

All right, what if, for example, this angle right here was unknown?

好吧，如果，比如說，這裡的這個角度是未知的呢？

It wasn't right angle.

角度不對

That means that the Pythagorean theorem wouldn't apply, right?

這意味著勾股定理不適用，對嗎？

I would have to use a different formula, right?

我必須使用不同的公式，對嗎？

The law of cosines is equal to C is equal to the square root of A squared plus B squared minus 2AB cosine of that angle.

餘弦定理等於 C 等於 A 的平方根加 B 的平方根減去 2AB 的餘弦。

So you see, I can just keep manipulating this one practice problem to generate a whole different set of problems to solve for.

所以，你看，我可以通過不斷處理這道練習題，來生成一整套不同的問題來求解。

Applying different constraints to solve for different variables, like how would I solve for this angle instead?

應用不同的約束條件來求解不同的變量，比如我如何求解這個角度？

How would I solve for this angle if it was a right triangle?

如果是直角三角形，如何求解這個角？

What if this was a completely different shape, like it had another triangle over here?

如果這是一個完全不同的形狀，比如這裡還有一個三角形呢？

What if it was three-dimensional, you know, and it had this shape like that?

如果它是三維的，你知道，它有這樣的形狀會怎麼樣？

How would that change solving for different angles?

不同角度的解法會有什麼變化？

How would this change the way that I approach this problem?

這將如何改變我處理這個問題的方式？

So effectively, I can turn this one practice problem into like 10 problems or more.

是以，我可以有效地將這道練習題變成 10 道或更多的題。

This is such an underrated way to learn because it emphasizes making connections between topics and really challenging yourself to think more deeply.

這是一種被低估了的學習方法，因為它強調在主題之間建立聯繫，並真正挑戰自己進行更深入的思考。

Connecting ideas like this differentiates them and makes them more applicable and usable in different situations.

將這些想法聯繫起來，就能區分它們，使它們在不同情況下更加適用和可用。

And remember that this is what your teachers are doing when they write exam questions.

請記住，你們的老師在出考題時就是這樣做的。

They're testing your ability to manipulate and distinguish between different kinds of concepts.

他們在測試你操作和區分不同概念的能力。

All right, so let's get back to the framework.

好了，讓我們回到框架上來。

The last tip actually happens after you finish reviewing the problems for the day, and it's a really important step.

最後一個提示實際上是在你複習完當天的問題之後出現的，這一步非常重要。

You have to track your confidence for every topic you study.

您必須跟蹤自己對每個學習主題的信心。

Otherwise, you'll end up wasting precious time reviewing past papers or even entire chapters that didn't need to be reviewed.

否則，你就會浪費寶貴的時間去複習過去的試卷，甚至是不需要複習的整個章節。

There are different ways to do this from the old algorithms, but let's be real.

與舊算法相比，有不同的方法可以做到這一點，但我們還是要實事求是。

The simpler, the better.

越簡單越好。

Our preferred way is something we call the GROW method.

我們的首選方法是我們稱之為 GROW 的方法。

Not only does it keep track of how well I know each topic, but it also serves as a study schedule that recommends which topics to study at any given moment, which is a huge time saver because I don't need to worry about planning a schedule ahead of time.

它不僅能記錄我對每個題目的掌握程度，還能作為學習計劃表，推薦我在任何時候學習哪些題目，這大大節省了時間，因為我不必擔心提前計劃時間表。

But I go way more into detail with a step-by-step walkthrough on the GROW method in this video right over here, and you won't want to miss it.

不過，我將在這段視頻中詳細介紹 GROW 方法的步驟，你一定不會錯過。

All right, bye.

好的，再見