How to be a successful math undergraduate student? [closed]

I am an undergratuate student in my first year of combined bachelor of electrical engineering and bachelor of mathematics. For my mathematics degree, this year I am supposed to take two math courses and for both the required textbook is Stewart Calculus.

I was wondering if you have any advice for a first year undergraduate student to be successful both at math and at university? What exercises from the textbook should I do? Should I try every single exercise in the textbook? How can I improve my math skills? when I can not solve a problem, what should I do?

I completed two bachelor degrees in mathematics and physics in Vienna. I don't know how it compares to an non-European bachelor degree, but I think my experience may be of help. I can't give advice on (calculus) textbooks but maybe I can give you some general advice about studying mathematics in university:

• Be precise: precision and adherence to the definitions are core elements in mathematical thinking. When I was pondering some mathematical statements and got confused, the main reason for the confusion was that I mixed up my intuitive notion of an (mathematical) object and its exact definition or did not see the difference between statements that looked very similar but were not identical. Check for differences (for example, where are the quantors placed in a theorem: "for all ... exists ..." is not the same as "there is ... such that for all ...") and check the direction of implications in a theorem (is a A necessary for B or sufficient or even equivalent?). Ask yourself what the crucial assumptions of a theorem are and what their role in the proof is.
• Grasp the concepts: Behind a lot of mathematical entities and algorithms there is a rough idea. Try to understand the idea behind it and why it is used. It makes it easier to memorize and connect different topics. But be aware that although a definition of a mathematical entity can be based on an intuitive idea, it can have unintuitive consequences or realizations. (For example, the discrete metric.)
• Exercise and repetition: yea, it's necessary in order to memorize the stuff, get used to the formalism, and apply it.
• Fiddling around: Sometimes when I have difficulties in understanding something (even very simple things) I fiddle around with it. It helps to discern the topic one is thinking about from similar looking concepts, theorems, etc. and to close in on the crucial, difficult-to-understand points and to finally resolve them. Exercising and fiddling around can take an awful amount of time, but when you resolve a problem and have an "Aha!" moment, you realize it was worth it.
• Discuss: When you have problems with exercise sheets or content from a lecture, discuss them with peers! Others can provide some enlightening insight or crucial ideas. And it helps to see that others are struggling sometimes too! Stick around diligent students and be one yourself as it helps to motivate yourself and others! This does not mean that one should be dragged along by better students and copy from them all the time. When I work on a problem or topic I need some time for myself to think about it, get and try some ideas and see where the problems are. Then I'm ready to discuss it with my fellow students. Sometimes I can contribute a complete solution or at least a substantial part to the work of someone else and sometimes somebody else has to explain an exercise to me.
• Teach: Teach what you've learnt! In order to explain something to others you first must have understood it yourself; i.e. you must know where the important and difficult points are in the topic or mathematical problem. When you try to get them across you may get some further insight into the topic. Furthermore, it helps to memorize the stuff.
• Don't be afraid to ask: Don't be afraid to ask the professor questions during or after a lecture, even when the questions seem simple and you may think you look stupid in the eyes of your fellow students. Swallow your pride. In most cases the other students are relieved that somebody asks a question which they were themselves too afraid to ask. When I prepare for an exam I sometimes ask a professor if she can arrange a time (1-2 hours) where she can answer several of my questions and explain things in more detail. Oh, and when you don't have the opportunity to discuss something with colleagues or the professor, there is still MathStackExchange ;-)

Apart from all the good answers that the other guys provided, I have one suggestion:

Use pen and paper!

In other words, do the exercises (or new concepts) instead of studying them. Imagine a day in the future when you are reading a question or studying a new concept. Then you start a conversation like this with yourself while looking at the textbook: "Nah! I know this, let's check the solution."

You go to solutions and you see your answer was actually wrong. Conversation continues: "Okay, I knew the answer, I just made a little mistake. We should go on..."

From the moment you start that internal conversation to the rest of your academic (or professional) career you are going to have lots of bad days. You can however, avoid all those bad days by using a pen and piece of paper, solving the problem instead of thinking about the solution!

Train you brain.