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Because this book is aimed at students, it is written in the style of a friendly, readable (though challenging and thought-provoking) self-help book.
Finally, this book is dedicated to my teacher George Sutcliff, who allowed me to find out how well I could think.
A mathematics major should be challenging—if it were easy, everyone would have one.
Note that I do not aim to teach the mathematics—that is the job of your professors and instructors. So you will not find that the book contains a lot of mathematical content. What it contains instead is information on how to interact with the content. It thus includes detailed illustrations, but not exercises.
They know that independent learning is expected in college, but some think this means that they will have to study alone with no help or support. This is far from true,
Before each lecture, you could read through a roughly appropriate part of the notes, review any earlier concepts that seem to be needed, and highlight anything you don’t understand so that you can pay particular attention when it comes up. This is part of what is meant by independent study—taking control of what you need to know and finding ways to seek it out.
You might come across other problems, and the important thing is to avoid getting carried away with blaming someone else—as I’ve said, you’ll be expected to take responsibility for your own learning.
Speaking of taking responsibility for your own learning, another thing to be ready for is that you do not necessarily have to attend all your lectures. If you are working hard but finding a particular course very frustrating, and you think you might learn more if you just downloaded the notes and studied them independently, then maybe you should consider trying that for a week. You should do so with great caution and planning, however, and you should only consider this if you know you are a disciplined person who will do the work. There’s more on why in the next section.
One thing research is pretty clear on is that college students who consistently go to their lectures do better than students who don’t.
Don’t use lectures you don’t like as an excuse to slack off, and don’t let missing a couple of lectures push you onto a slippery slope towards doing less and less work.
learning in lectures is easier and more enjoyable if you are reasonably well prepared.
If you don’t study, you’re not asserting your independence, you’re just creating trouble for yourself. People do fail, and when they fail they have to do inconvenient and possibly expensive things like resitting courses. And your college and department do care about you. They would very much like you to succeed. But you’re an adult now and, while they consider it their responsibility to provide high-quality teaching, they don’t consider it their responsibility to make you study. That’s down to you.
In particular, people are often late. I think this happens for two reasons. The first is that students often live very close to the location of their lectures. That sounds paradoxical but it isn’t really—if you know it usually takes you 45 minutes to get somewhere, you tend to allow an hour, but if you know it only takes you five minutes, you tend to allow only five minutes.
So pretend that the lecture starts five minutes earlier if that will get you there on time.
More importantly, though, please take the opportunity to write constructive comments. I stress, constructive please. There really isn’t much point in responding to a question like “What did you like about this course?” by writing “Nothing.” This comment contains no information that would help a professor to improve. Try to be specific and, if you have a complaint, give a constructive suggestion about what could usefully be changed.
if you enjoy someone’s lectures and learn a lot from them, do find a way to let them know that. Professors often put a lot of work into their courses and it’s nice to have this acknowledged.
At college it might not be obvious that you have access to authoritative people. In high school, your teachers probably offered you help on a daily basis. They talked to you individually, noticed when you were having trouble getting started on a problem, that kind of thing. In college, classes can be bigger, so your professors probably won’t do that. In a class of 100 people, it just isn’t practical for a professor to catch someone and say “So, Nina, how did you get on with yesterday’s work?” Because of this, it’s easy to feel distanced from your professors, and to get the impression that
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Professors tend to look very authoritative and hyper-intelligent, and some people find it intimidating to talk to them. But they are just people, and they love mathematics, and the vast majority of the time they will be delighted to talk to you about it.
You shouldn’t knock on their door the minute you can’t do something—obviously you’re expected to give it some proper thought first. But, if you do decide you want to see them, you might find it a surprisingly pleasant experience. As a student, I found this out much later than would have been ideal, when I went to see a professor toward the end of a masters course. He was very happy that someone was taking an interest in his subject, and he was willing to talk to me for ages.
One thing that really puts off employers of all kinds is a person who doesn’t know where to put their apostrophes or who makes other glaring grammatical errors.
Don’t let the name make you think that learning centers are only for people who are really struggling—usually they are open to people who are doing well too.
at some point, you (or a friend) will have a bad experience when asking for help. This is very, very rare, but I do occasionally hear that an instructor or tutor has made a student feel bad by telling them they should already know how to do something, or has acted surprised at their lack of knowledge. In the unlikely event that this happens to you, the thing to remember is that this person has behaved insensitively, and that you don’t want to let one insensitive person (or perhaps an ordinarily sensitive person who’s just having a really bad day for some reason) put you off.
my view is that the benefits of doing a project generally outweigh the risks.
On the one hand, having an agreement to get together and work on something can make sure you actually do it, a bit like having a buddy for going to the gym. On the other hand, if a bunch of you get together with the aim of working on mathematics, but actually you just distract each other, that’s probably not very useful.
If I were you, though, I would think about getting the right balance, and make sure you’re not working in seclusion just because you think that’s how genius mathematicians do it. Genius mathematicians do sometimes work alone, but they also sometimes work in research groups, and either way they certainly engage with wider networks of mathematicians by going to conferences and seminars and debating whether a proof is valid or whether an idea will really generalize to a new type of object.
I guarantee that at some point in your studies, some idiot will say to you, “Ah, come on, you can retake the course/get your GPA up later—you don’t need to work so hard!” This person has failed to grasp the basic principle that actions have consequences.
In our culture, many people think that mathematical ability is particularly indicative of intelligence, and that it is based on inherent talent rather than hard work. This view is questionable but very pervasive. As a result, those who have always been good at mathematics tend to be thought somewhat special, and this gets built into their sense of why they are worthwhile human beings. So, if they no longer stand out, it can shake their sense of their place in the world.
The fact that you don’t understand everything in lectures doesn’t mean that you are not good at mathematics any more, it just means that the mathematics is harder and the pace is faster. To keep up with it, you will have to do more (or better) work. But so will everyone else. And that’s as it should be (if studying for a mathematics major was easy, everyone would do it).
Lots of people think that being good at mathematics means being fast at it. This belief seems to be associated with several errors in reasoning, so let’s unpick the logic and work out how the two actually fit together.
Mathematicians don’t want students to end up with encyclopedic knowledge but dodgy underlying understanding. They’re more impressed by deep understanding, and, at least in upper-level courses, they tend to test for this by making at least some of the questions on their exams require original thinking or novel applications of the mathematical ideas.
Obviously understanding everything is a fine aim, and you should certainly begin every course with the intention of grasping all the main ideas and principles. You probably won’t manage it every time, but that’s okay. What you shouldn’t do is take this as evidence that you’re not very good any more, or as an indication that you should give up. Take it as an occasion to rise to the challenge, instead.
no matter how much talent this person has, the mathematics doesn’t just jump into their brain unbidden.
human beings don’t exist in isolation, and a great amount of what you learn in life you will learn from other people. If you are willing to share your knowledge and understanding, and to do so in a way that is friendly, equitable, and allows everyone to participate, you will find that others are willing to return the favor.
Undergraduate mathematics is more difficult than high school mathematics and is presented at a faster pace; finding this a challenge is normal, and does not mean that you are not good enough any more.
Being good at mathematics is not necessarily the same as being fast at it. Taking the time to understand things properly is worthwhile.
It is normal not to understand everything by the time of an exam; knowing this, you should think about how to distribute your effort in order to do well.
Many professors spend a large amount of time doing research, which in the case of mathematicians means developing new mathematics. This sometimes surprises new undergraduates who, if they’ve thought about it at all, tend to assume that mathematics is already “finished.
