确定学习高等数学的具体方向

1. 但我遇到过很多人认为他们想学习高等数学，直到他们意识到其中 95% 的内容几乎没有应用到现实世界。

In either of these cases, the number one takeaway should be: TALK TO YOUR ADVISOR/MENTOR. They can get you in contact with people from the Mathematics department at your school and they can also explain how relevant advanced maths is to your area of study. Another point of concern is this: do you really know what advanced mathematics is about? Unless you have a firm understanding of what kinds of problems Analysis, Topology, and Abstract Algebra attempt to solve, then I would question your motivation for studying advanced maths. Considering you don’t know derivatives yet, there’s almost no way you can understand what Analysis is all about, because it focuses entirely on proving results from Calculus. So, unless you have an extreme interest in the fields of Topology or Abstract Algebra, I would research the problems in these fields and re-evaluate whether it’s something you really are interested in. I don’t say any of this because I think you don’t have an interest in these areas, but I have come across many people who THINK they want to study advanced maths, until they realize that 95% of it has almost no applications to the real world. If you want to study it because you think it will be beneficial to your skills in other areas, I can almost guarantee you that it will not be.
在任何一种情况下，首要任务应该是：与您的顾问/导师交谈。他们可以让您与学校数学系的人员取得联系，他们还可以解释高等数学与您的学习领域的相关性。
还有一点值得关注的是：你真的了解高等数学吗？除非你对分析、拓扑和抽象代数试图解决的问题类型有深刻的理解，否则我会质疑你学习高等数学的动机。考虑到您还不知道导数，您几乎无法理解分析的全部内容，因为它完全侧重于证明微积分的结果。所以，除非你对拓扑学或抽象代数领域有极大的兴趣，否则我会研究这些领域的问题并重新评估它是否是你真正感兴趣的东西。
我不说这些是因为我认为你对这些领域不感兴趣，但我遇到过很多人认为他们想学习高等数学，直到他们意识到其中 95% 的内容几乎没有应用到现实世界。如果你想学习它是因为你认为它对你在其他领域的技能有益，我几乎可以向你保证它不会。

All that being said, if you want to learn something, learn it. Just make sure you’re learning it for the right reasons, and make sure that you can afford to learn it.
话虽这么说，如果你想学点什么，就去学吧。只要确保您出于正确的原因学习它，并确保您有能力学习它。

2. 在课上`明白` `是的，这很有意义`，与能用`直觉`解决问题之间存在巨大差异。

Learning is an iterative process. You're right, there is a massive difference between "getting it"/ "yeah, that makes sense" in class to being able to leverage that intuition to solve problems.
学习是一个迭代(反复)的过程。你是对的，在课堂上“明白了”/“是的，这很有意义”与能够利用直觉解决问题之间存在巨大差异。

If it's a relatively unfamiliar area, then there's no way around struggling with the material before you see the light. Here are things that made a lot of difference to me:
如果它是一个相对陌生的领域，那么在你看到曙光之前就没有办法在材料上挣扎。以下是对我产生很大影响的事情：

Read before you go to class. You should go in with a decent idea of what's going to be taught. When the instructor goes over the material, you are no longer just going through the material linearly, you're making connections in your head with the stuff you already know. This helps tremendously.
上课前阅读。你应该对要教什么有一个体面的想法。当讲师复习材料时，您不再只是线性地浏览材料，而是在头脑中与您已经知道的东西建立联系。这有很大帮助。
If you're not able to solve problems on the problem sets, do not start with them. When you self study, solve baby problems. Trivial ones you make up for yourself to make sure you got what you just read, or perhaps even the examples (no peeking!).
如果你无法解决问题集中的问题，请不要从它们开始。当你自学时，解决宝宝的问题。 您自己编造一些琐碎的内容，以确保您得到了刚刚阅读的内容，甚至可能是示例（不要偷看！）。
Examples are very useful. Never, ever just read them. You lose a valuable opportunity to have applied what has been freshly learnt. Preferably, you've worked on the examples before you go to class.
示例非常有用。永远不要只读它们。你失去了一个宝贵的机会来应用刚刚学到的东西。 最好在上课之前就已经研究过这些示例。
About working in groups: It can be extremely useful to study in groups. If you simply talk about the material with your friends, you're making connections, remember learning is iterative. Only go to a group after you've tried to solve the problems all by yourself. If you get boosts from friends while practicing, it will cost you dearly during an exam.
关于分组工作：分组学习非常有用。如果您只是与朋友谈论这些材料，那么您就是在建立联系，记住学习是反复进行的。只有在尝试自己解决所有问题后，才去一个小组。如果你在练习时从朋友那里得到提升，那么在考试期间你会付出高昂的代价。