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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?

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don't go to parties –  Vic Feb 27 at 20:36
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@Vic I disagree - spending all your time working on a pasty complexion doesn't make you a better mathematician, and of course the opposite is also true - partying doesn't make you a bad mathematician! The trick is to get the balance right. You can party, but make sure you do your work first... (Also, employer's don't just look at your degree classification. If all you got from uni was a good degree then why should I employ you? If you got a half-decent degree whilst producing anarchist left-wing plays then you are an interesting person whom I would like to meet and perhaps even employ...) –  user1729 Feb 28 at 13:15
    
(So long as you leave your politics at the door!) –  user1729 Feb 28 at 13:18
    
partying is no big problem. copy pasting instead of trying stuff on your is a huge problem and makes people fail. –  Max Feb 28 at 19:22
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I think part of this has to do with what your goals are? Do you just want to enjoy some maths, have a pretty good GPA, and get on with your life? Or do you want to be a research mathematician (or somewhere in between)? The biggest problem with parties is drunkenness, which can significantly lower math abilities for an entire day. –  Jacob Wakem Feb 28 at 19:22

10 Answers 10

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 about some mathematical statements and got confused, the main reason for the confusion was because 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. (Example: 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 point, which is hard to understand, and to resolve it. Exercising and fiddling around can take an awful amount of time, but when you resolved a problem and have an "Aha!" effect it was worth it.
  • Discuss: When you have problems with exercise sheets and or content from a lecture, then discuss it 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 know where the important, crucial 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 ;-)
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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.

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Halfway through your answer I thought you were suggesting to be crazy by talking to yourself, lol. +1 –  John Odom Feb 27 at 16:18
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Solving the problem instead of thinking about the solution. I can't tell you how many times I say "Oh, I can just do this problem", and then two hours later I'm like "Fine, I'll get out some paper" and then I'm done in twenty minutes. –  Eric Stucky Feb 28 at 19:10
    
I used to be all pen and paper, following my dad's advice, who would not help me if I hadn't filled a page of nonsense first (because what else would it be if you didn't understand the problem?). Years later: Genius mathematician friend of mine was doing everything in his head. I started trying it, closing my eyes for long periods of time, visualizing the problem, solving equations, etc. in my head: it took me a while to improve my ability to retain and develop thoughts in my head without paper, but I believe that's now the best way. Make sure you stay awake while your eyes are closed. ;-) –  PatrickT Mar 1 at 9:16
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@PatrickT, your point is valid to some extents. Actually I was going to end my answer by saying, there will come a time when you can solve everything in your mind, but it is not now. However, I didn't. Not because the OP is less than a genius, but because your genius friend's and your ability to tackle problem with closed eyes, is based on a pen and paper foundation. At least for most use! –  Pouya Mar 1 at 9:28

I recently completed second year undergrad down at Melbourne Uni, and although we had prescribed texts, I used Stewarts calc. It's important to realise that although people stress that repetition (eg doing question after question) results in success, I disagree. Understanding is the key, persevere to understand the thinking behind the mathematics and you'll succeed. For instance, instead of understand the rules for partial differentiation, understand what it means, and everything will fall into place. Sometimes it takes forever, well for me it did, but understanding the concepts fully rather than simply knowing how to utilize the mathematics, is far more effective, especially if mathematics is your focus.

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Also, this website is awesome for those question which really have you stumped. –  M. Knight Feb 27 at 12:28
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I venture that repetition can form part of the path to understanding. Using repetition, you can gain familiarity, recognize commonalities and free up your working memory to focus on new concepts. –  J W Feb 28 at 7:57

The best advice I have ever gotten: Do the exercises without looking at the suggested solutions. This will force you to think critically. In the beginning this will probably be very time consuming, but do not give up, because after a while the payoff will be huge.

If you really get stuck, discuss the problem with someone else. In this way, you will have to identify the reason(s) why you are stuck. (you could for instance ask for hints at this site).

I would also advice you to read (a lot) about how professional mathematicans approach mathematics. Have a look at the following pages:

Terence Tao's blog contains a section on Career Advice, which I really appreciate. Terence Tao and Timothy Gowers have several informal discussions about mathematical topics on their homepages. I really benefited a lot from them.

Also, on the philsophical level, do not get too obsessed with trying to understand the intrinsic meaning of mathematical notions. Mathematical objects are important because of what they do not what they are.

To quote what John Von Nuemann once said:

"Young man, in mathematics you don't understand things. You just get used to them"

(A reply to Felix Smith who said he was afraid he did not understand the method of characteristics.)

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Okay I'm not all that good with the advice. But I will tell you this. I am not a fan of Stewart's book. Trust me if you really want to grasp the roots such as Mitch Knight suggests above I strongly urge you to consider a few other texts as supplements. These are my recommendations. You won't go through them all, obviously. But if you can properly grasp a good chunk of one or two of them your exams should be a piece of cake.

  1. Calculus by Michael Spivak
  2. Calculus I, II and III by Jerrold E. Marsden and Alan Weinstein
  3. Calculus - Volume I and Volume II by Tom Apostol

All these books introduce calculus in a much more intuitive way than Stewart's more mechanical methodology. You have a way better shot at understanding the roots behind calculus and that is a great foundation for a future course in Analysis

And since this was marked online resources you might also want to look at the calculus courses on MIT Open courseware. The lecture notes, videos and exams are all free and of exceptional quality.

Forgot this one - Oxford Mathematics Course Materials.

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Good advice. Although I never purchased it, Calculus I,II,III by Marsden and Weinstein were recommended to me numerous times. –  M. Knight Feb 27 at 12:42
    
@MitchKnight:I have to confess I haven't gone through it all and have promptly transferred now to Analysis books. But when I had trouble understanding limits and developing integrals Marsden came in extremely handy. Very rigorous and extremely descriptive. Perfect for studying alone. Lots of pictorial aids. Pretty good. –  Ishfaaq Feb 27 at 12:44
    
Looking at the link you supplied, looks like a good one to buy, especially to fill in the gaps that have developed in my knowledge. –  M. Knight Feb 27 at 12:47
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@MitchKnight: Oh absolutely. Complete book in every sense. Single variable, multiple variable and lots of preliminaries you'd be surprised you didn't know. I know what you mean. This is great for patchwork. –  Ishfaaq Feb 27 at 12:49
    
Where's Courant's book? –  metacompactness Feb 27 at 21:33

Get some old test questions from previous years for the same professor. Hate to say it, but after wondering why the dumb frat guys were scoring so well in tests, I came upon this 'secret'. Turns out professors are lazy and recycle old questions. You can beat yourself over the head doing hundreds of example questions from books, but there's nothing that works better than redoing old test questions (that's how SAT test prep works).

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This may be useful advice if your goal is to get good grades (assuming old questions are frequently recycled), but not so much if your goal is to learn the material. (Sure it has some value, but for learning, studying old test questions is no better than studying other, equivalent questions that wouldn't be used on a test.) –  David Z Feb 28 at 7:29
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I wouldn't go so far as to say that professors are lazy when it comes to recycling old questions, assuming they do. Often they are simply very busy and need to prioritize what they spend their time on. –  J W Feb 28 at 8:03
    
@DavidZ, as the past quesions should releate well to what is being covered, as there not a bad way of leaning whatever the goal is. –  Ian Mar 1 at 17:15
    
@IanRingrose right, but the "frat boys' secret" that ThomasEdison is talking about is where people make a point to use the same questions that will be on the test so they recognize those questions specifically. They may even memorize the question and its answer, or a formula to get the answer, without ever actually solving it. That's not useful for learning. –  David Z Mar 1 at 22:11
    
Actually I had a CS professor who would randomly generate his exams via software he wrote. This makes your little 'secret', a ridiculous mistake in todays day and age. –  Gary Drocella Jun 13 at 16:59

The best advice I got while an undergrad Math major was to take programming/software courses. After picking up a minor in Computer Science (and then a graduate degree in the same), I'm now a full-time software developer and loving it!

Once you get calculus out of the way and have time for a few electives, try your hand at discrete mathematics: Combinatorics, Graph Theory, Geometry, etc. (Cuz if you enjoy them, you should probably be in computer science.)

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I agree that it's usually sensible to learn at least the basics of programming. Courses in data structures and algorithms, supported by discrete mathematics, build nicely on this foundation. I wouldn't say that Geometry as a whole is a subfield of discrete mathematics, but parts of it certainly have close connections with discrete math. However, don't be too quick to leave calculus behind. It leads to many beautiful and useful subjects. Finally, don't forget linear algebra - it's the bread and butter of a great deal of mathematics, both pure and applied. –  J W Feb 28 at 21:40
    
Calculus is usually the first required classes for an undergraduate Math degree. In this day and age, I'd like to see the discrete maths getting more attention. (I wouldn't mind if a little software programming/theory became a requirement at every level of school.) –  EthanB Mar 1 at 21:39
    
In theory, you could start with much of discrete maths, before or in parallel with calculus, but that's a whole different discussion. Personally, I like graph theory and discrete/computational geometry. However, I also see the value of calculus, leading to differential equations and dynamical systems, numerical analysis, real and complex analysis, topology, differential geometry and beyond. Along the way, there is often a beautiful interplay between discrete and continuous, supported by the structures of algebra. –  J W Mar 2 at 7:18

I can't comment or even upvote, so I'll put this as an answer. Several people already told you this, but it can't be stressed enough: Solve a lot of problems! After that, solve even more. Solve problems from the textbooks, try to solve problems which you've heard about from your professors. Try to solve the famous ( P = NP ? ) problem - in this case you will not be able to look into the solution :) Regarding the ( P = NP ? ) suggestion, find a problem which really excites you, no necessary this one.

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Don't go it alone.

Don't be afraid to talk to your professor/TAs when you're having trouble understanding something. (Actually, this is a good rule for any class.)

Don't be afraid to collaborate with your classmates, either -- doing so is silently discouraged in most high school classes, where homework is a big part of your grade, but it should be encouraged in college (and your future career). Consider your homework to be practice for quizzes and tests, and learn to work the problems out with your peers.

Finally, be prepared for frustration -- calculus and onward demand a different way of thinking and problem-solving than plain algebra. You'll probably bang your head against the wall a few times before things really click. You won't be alone.

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I am speaking as a former math major/CS minor who's now working full-time as a web developer. Post-calc math was fundamentally different from anything I'd done before, and I did terribly because I stubbornly tried to work it out by myself. In my Computer Science classes, "machine problems" (assignments) were explicitly supposed to be done with another classmate, and it made both homework and learning far easier. –  Blazemonger Feb 28 at 22:10

Alert .. highly iconoclastic answer follows :)

.

I believe you have to just have the "gift" for it. You either do or don't. It's like music. Unfortunately I think that's the truth.

(Consider too Douglas Hofstadter's writings on how he had a gift for math -- BUT -- only to a certain level. Very astute!)

In terms of just getting through, it's straightforward and obvious:

(1) take up running, run every day

(2) eat an anthropologically correct (see: Dr Wolfgang Lutz) diet. you literally are what you eat. if you think for a living, what you eat is everything. the only input variable affecting your moment-to-moment mental clarity is dietary intake.

(3) have sex constantly: as many times a day, every day, as possible. the most creative human endeavours, that is to say the largest to paradigm-space outcome milieu, are (in order) programming, then math, then art.

since math is the 2nd most creative human endeavour, you need sex constantly, pretty much say four times a day (if you're typical student age).

(4) other than those activities, study math and do math exercises literally every waking minute. Literally every minute. If you have any ideas like "I'll allocate N hours a day ..." just forget it and find some other lifestyle.

Other than the essentials (sex, running, and eating): every waking minute is math.

{Regarding "parties", utterly idiotic waste of time. 10 year olds go to parties.}

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