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I love math and I feel happy to learn it. But in universities, I think there's too much to learn. For example, in my university, there are mathematical analysis(I,II,III), linear algebra(I,II), ODE, topology, differential geometry,..., 24 courses altogether. In addition, we don't attend courses in the 4th year, so I have to finish 24$\div$3$\div$2=4 math courses per semester. That's too much. In such a hurry, it's almost impossible for me to digest the big bulk of knowledge, and to gain deep understanding of the concepts. It's painful.

So my problem is: Are universities teaching too fast? Or it's unnecessary to try to fully understand what I've learnt in a rush, because it cannot be achieved until I gain more experience at job?

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closed as primarily opinion-based by Thomas, Antonio Vargas, Cameron Buie, Daniel Robert-Nicoud, Arthur Fischer Oct 24 '13 at 17:31

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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I would view the learning as a process rather than something you do once. It is useful to get exposure to many different ideas and see how they weave together. –  copper.hat Oct 24 '13 at 7:37
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Pfft. When I was an undergraduate, our terms were only 8 weeks long, and we had more than 4 courses per term (on average). –  Zhen Lin Oct 24 '13 at 7:48
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Like @copper.hat said, learning never ends. In fact most of my learning I did after my graduation, outside of school/university. But you do need to learn how to learn and teach yourself. The rewards are then endless. –  Fixed Point Oct 24 '13 at 7:49
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Agree with what has been said. I feel that I am learning maths more now than during my degree (I learnt a lot, but didn't have the time to enjoy some deep arguments there). –  Duronman Oct 24 '13 at 9:01
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@copper.hat But what if I have to get exposure to these ideas in a not-long-enough time and I can't understand them all? –  AaronS Oct 24 '13 at 11:56

2 Answers 2

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In my opinion, mathematics is a very poor fit for the standard university model of lectures, presented linearly, over the course of a quarter/semester.

If you fall a little behind in a math course, it is painful, if not impossible, to catch up. This is because it is not enough to learn the material—you have to practice it, and in doing so you might discover very deep issues in your understanding of the background ideas. Yet the course speeds along, giving you no time at all to polish what needs polishing.

We, the teachers, handle such situations very badly. We tell ourselves that these students weren't ready for the course, that they should have taken something else first, or, worst of all, we think that they aren't "true mathematicians". But this is all wrong. Becoming great at mathematics involves all sorts of false starts and reworking of basic techniques. The people who are "ready" for the course are often the people who get the least out of it; they could as well have read the book!

Math should be taught in a way that allows students to explore their weaknesses more deeply. This is impossible to do if every student must be learning the same material at the same time.

I was lucky enough to have access to an at-my-own-pace style of math education throughout high school. In college, I felt like I was a great student, always on top of my math classes. Looking back, the classes were far too slow for where I was, and I was just a terrible student with the right background. I wasn't being challenged, and so I learned a lot of bad habits, which made grad school very, very painful for me until I remembered how to push myself and focus on my weaknesses again.

So I don't think that math goes too fast exactly. I don't think that math should have any pace at all, but be driven by the needs and abilities of the student. This is not an easy ideal to achieve, of course, but I think that everyone in mathematics too easily accepts the myths surrounding the system that now exists. Those at the top become arrogant, and everyone else becomes anxious and fearful. You are right. We can do much, much better.

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While there is certainly some truth in this, I think that you’re over-generalizing from your own experience. –  Brian M. Scott Oct 24 '13 at 9:32
    
@BrianM.Scott I was only bringing up my own experience to contrast with the poster's assertion that math is too fast. My justification for my opinions rests far more heavily on my experience with my students than on my own time as a student, which I readily admit to be highly atypical. –  Slade Oct 24 '13 at 9:35
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But there are still things that I think are overstated. I don’t think that ‘great’ is really relevant, but becoming very good at mathematics does not necessarily involve all sorts of false starts and reworking of basic techniques. People who are ‘ready’ for a course are often people who get the most out of it, even though many of them could have learned much of it on their own from the book. And some students really aren’t ready some courses. Note that I am not saying that one size fits all: it doesn’t. Nor am I saying that the traditional structure works well for all students: it doesn’t. –  Brian M. Scott Oct 24 '13 at 9:41
    
@BrianM.Scott You're right; it's almost as though this is the perspective of a single person, and should have started with "In my opinion". –  Slade Oct 24 '13 at 9:44
    
@user33433 By saying "too fast", I don't mean "math goes too fast", but mean "university goes too fast", that is, are they unnecessarily setting too many courses that I have no time to digest? –  AaronS Oct 24 '13 at 12:56

I am a graduate student from Applied Mathematical &Physical Sciences.I had 59 courses,about 10 for Physics and 49 for Maths.I remember and use about 10-12 of them.
But i don't regret studying there. It was hard but i did not feel that we were going ''too fast''. It is better to learn as much as you can when you are young,and then keep only what is interesting and necessary for yourself.
P.S. I don't use anymore measyre theory but learning that some sets have measure $0$ was really exciting. I believe that for someone who loves mathematics we can find many more examples like this.

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