Started university mathematics and struggling

[converting1]

I'm from the UK and began my bachelors in mathematics around a month ago, with the hopes of going on to further study afterwards. My university is one of the best for mathematics and so far I've aced all my previous exams so this is quite a big shock to me.

I'm struggling with near enough everything pure, and even some applied courses. For instance, in my real analysis class we get 1 problem sheet a week and it takes me a good few days to work through it. The first problem sheet I got I spent the whole weekend on it and only managed to do one question, after researching it a lot online.

For some reason, I just cannot seem to grasp concepts as I used to be able to so easily. It takes me quite a long time to get my head around it and I'm having troubles formalising written proofs so I end up spending 3-4 days to complete the 4 problem sheets we're given, while I have a friend who manages to do them in 3 hours or so. I'm also finding myself giving up too easily on problems. For instance, I would spend around 30 minutes on a problem, then if I can't solve it I would look it up online and worse case scenario I would ask on this forum or other forums and don't feel like I really solved the problem myself at all.

As far as my grades are looking I'm still getting 80%+ in all the problem sheets but this is after many hours work, while there are students who spend time partying and a little studying who are getting the same results as me.

I also can't seem to find time to read ahead - I used to always read ahead in my high schools class book and was always on top of my work, now - I'm just average. I can't even find the time to do the "extra" problem sheets for those who want to go on to a phd.

My question is, is this normal? Or should I think about switching degree now (I have no idea what else I want to do, I've always wanted to do mathematics). In all honesty I think I can stick it out for 3 years and still end up with a good classification of my degree, but whether or not I would have learned something useful is another story.

 

[pasmith]

I'm from the UK and began my bachelors in mathematics around a month ago, with the hopes of going on to further study afterwards. My university is one of the best for mathematics and so far I've aced all my previous exams so this is quite a big shock to me.

May I ask which university?

I'm struggling with near enough everything pure, and even some applied courses. For instance, in my real analysis class we get 1 problem sheet a week and it takes me a good few days to work through it. The first problem sheet I got I spent the whole weekend on it and only managed to do one question, after researching it a lot online.

For some reason, I just cannot seem to grasp concepts as I used to be able to so easily. It takes me quite a long time to get my head around it and I'm having troubles formalising written proofs so I end up spending 3-4 days to complete the 4 problem sheets we're given, while I have a friend who manages to do them in 3 hours or so. I'm also finding myself giving up too easily on problems. For instance, I would spend around 30 minutes on a problem, then if I can't solve it I would look it up online and worse case scenario I would ask on this forum or other forums and don't feel like I really solved the problem myself at all.

As far as my grades are looking I'm still getting 80%+ in all the problem sheets but this is after many hours work, while there are students who spend time partying and a little studying who are getting the same results as me.

I also can't seem to find time to read ahead - I used to always read ahead in my high schools class book and was always on top of my work, now - I'm just average. I can't even find the time to do the "extra" problem sheets for those who want to go on to a phd.

My question is, is this normal?

This is normal.

I think a very common attitude to this is that doing mathematics (that is, thinking of proofs) is something that you can’t really be taught directly: instead, you read your notes and do lots of carefully designed questions and find that proving results is a skill that you develop with practice, especially if you were born with a mysterious quality called mathematical ability. And undoubtedly there is some truth in the previous sentence — the method described is the method by which pretty well all mathematicians working today have learned how to do maths. But there is a significant downside to this method, which is that there are also many people for whom it does not work. They go to university full of enthusiasm for mathematics and find that the subject at university level is much harder than they expected, and that they don’t know how to go about developing the skills that I’ve just been talking about. Gradually as the course proceeds, they fall further and further behind, while some of their contemporaries seem not to.

...

Mathematics becomes hard. Every mathematician will be able to tell you rather precisely when it was that they found that mathematics had stopped being an easy subject that they could understand with very little effort and became a difficult subject that they had to struggle with if they wanted to get anywhere. It isn’t necessarily an advantage if this happens to you later rather than sooner.

The above comes from Prof Gowers's Welcome to the Cambridge Mathematical Tripos, some of which is relevant for non-Cambridge mathematics. It is followed by a series of posts on basic mathematical logic, which may be worth reading if you have trouble constructing proofs.

There are also many books designed to assist with the transition from school to university mathematics, such as this or this.

Or should I think about switching degree now (I have no idea what else I want to do, I've always wanted to do mathematics).

It's far too early to start thinking about switching degrees.

 

[verty]

I have never studied math, what math I have learned in school was really elementary. So take what I say in the correct context, but I think you may be trying to reinvent what others have already discovered. No one expects you to prove what people like Gauss proved before, without help. No one expects you to ace everything the first time you see it. I think those friends that are partying looked at their learning in this light: what have others already discovered/proved about this? I don't want to reinvent the wheel, so let me get to the cutting edge as quickly and efficiently as possible.

Especially when you say you aren't reading ahead, this is almost certainly a mistake. It's true what people say, you should already know why something is true before you try to prove it, and the only way to find out why something is true is to read the arguments/reasoning of geniuses who came before. Working it all out for yourself just won't be worth it.

 

[converting1]

May I ask which university?



This is normal.



The above comes from Prof Gowers's Welcome to the Cambridge Mathematical Tripos, some of which is relevant for non-Cambridge mathematics. It is followed by a series of posts on basic mathematical logic, which may be worth reading if you have trouble constructing proofs.

There are also many books designed to assist with the transition from school to university mathematics, such as this or this.



It's far too early to start thinking about switching degrees.

It's Cambridge.

I think it's a little too late to start reading books on closing the gap between high school maths and university maths. Plus, I'd see it as more beneficial to read a relevant text in analysis or something.

I have never studied math, what math I have learned in school was really elementary. So take what I say in the correct context, but I think you may be trying to reinvent what others have already discovered. No one expects you to prove what people like Gauss proved before, without help. No one expects you to ace everything the first time you see it. I think those friends that are partying looked at their learning in this light: what have others already discovered/proved about this? I don't want to reinvent the wheel, so let me get to the cutting edge as quickly and efficiently as possible.

Especially when you say you aren't reading ahead, this is almost certainly a mistake. It's true what people say, you should already know why something is true before you try to prove it, and the only way to find out why something is true is to read the arguments/reasoning of geniuses who came before. Working it all out for yourself just won't be worth it.

I understand no one expects me to ace everything the first time, but there are students around me who do seem to be doing that and it's really scary. They are so unbelievably lazy, and are managing to get my scores on problem sheets with minimal effort - if I had their talent I'd surely make much more use of it, but unfortunately I don't. I've always been a believer that if I work hard enough, everything will click and eventually I will just "get" it and improve and become a good mathematician, but now I don't think so any more. I will continue working hard in my degree, but that's because I enjoy mathematics and want to get a degree in it. I don't think I can have a career in mathematics any more, and that's rather depressing.

I think you misunderstood me about not reading ahead. I meant to say was I simply don't have the time to read ahead and finish every problem (compulsory and non-compulsory ones). I'm too slow and need to think about stuff too much. I also go off tangents when working and end up getting frustrated at small trivial points - I've always had this problem.

I'd also like to state how much I'm struggling exactly: We get our problem sheets at the end of the week, during the weekend I will usually do 2-3 hours Saturday and 8 hours on Sunday and get through (usually) all of the applied ones but have the pure ones left to do. From the beginning of the week I go around asking my supervisors and tutors for help with the problems, or fellow classmates. The day before I then write up my solutions in my own wording.

 

[AlephZero]

Well, it's good to know the Cambridge math BA is still pretty much like it was when I took it (a long time ago). You start by being thrown in the deep end, and if you survive that, it only gets deeper

But since this is Cambridge, you have two great resources you can use. One is your supervisors. Don't be afraid to "throw the problems sheet out of the window" and ask for the help YOU need. If your supervision partner doesn't like that, get your supervisions reorganized.. The other resource is your college director of studies (or whatever he/she is called these days). His/her job is to help you succeed, not to get you kicked out as a failure. But if you only see him/her for a quick glass of sherry once a term, he/she doesn't know what's going on - so get talking!

I would guess that if you came from a fairly "ordinary" UK state school, you had very little experience of "real math" in the sense of formal proofs starting from axioms. Getting your head around that is a challenge in itself, quite apart from keeping up with the particular courses you are doing.

And don't bother about the fact that some people seem to be acing everything with no effort. Eventually, everybody has to learn that they are not the smartest kid on the block any more. That can come as a shock, if you were one of the smartest kids in your "school-sized" block, and suddenly you are in a class with the smartest from the whole of the UK - not to mention the overseas students!

Final comment: to get on the course at all, you have been through a tough selection procedure. If the university didn't think you have what it takes to succeed, you wouldn't even be there. That doesn't mean "succeeding" is going to be a stroll in the park, but giving up after half a term is WAY too soon.

 

[Group_Complex]

converting1, you should consider the fact that Cambridge is one of the top universities in the world for mathematics, by getting 80% you are certainly working at a level above most humans mathematically speaking.

Seeing as you are are at Cambridge it is no wonder that some of your fellow students are able to achieve amazing things with less effort than you. These are the next leaders in mathematics or physics, some of them might even be future fields medalists or nobel prize winners.

You should not be too hard on yourself, some people are better at grasping mathematical ideas than others and some people are just more intelligent than others. There is no point in comparing yourself to these people, as it mostly comes down to innate abilities which you cannot change. Not everyone is cut out to be a mathematician or physicist and even if you did become a mathematician you would spend most of your career trying to catch up to these naturals, probably achieving half as much. It may not be the politically correct answer, but careers in mathematics are best left for those who will do great things, there is no need for second rate minds.

I advise you to investigate other careers you could use your degree in: finance, engineering and computer science are some options. However focus on finishing your degree first.

 

[Student100]

converting1, you should consider the fact that Cambridge is one of the top universities in the world for mathematics, by getting 80% you are certainly working at a level above most humans mathematically speaking.

Seeing as you are are at Cambridge it is no wonder that some of your fellow students are able to achieve amazing things with less effort than you. These are the next leaders in mathematics or physics, some of them might even be future fields medalists or nobel prize winners.

You should not be too hard on yourself, some people are better at grasping mathematical ideas than others and some people are just more intelligent than others. There is no point in comparing yourself to these people, as it mostly comes down to innate abilities which you cannot change. Not everyone is cut out to be a mathematician or physicist and even if you did become a mathematician you would spend most of your career trying to catch up to these naturals, probably achieving half as much. It may not be the politically correct answer, but careers in mathematics are best left for those who will do great things, there is no need for second rate minds.

I advise you to investigate other careers you could use your degree in: finance, engineering and computer science are some options. However focus on finishing your degree first.

This such a load of bull, average students at Cambridge are more then capable of making contributions. Second rate minds? Haha.

All of those next leaders in mathematics you mention will invariably stumble at some point in their classes, and if their anecdotal poor work ethic is true, may not even finish. Further, don't believe everything someone tells you, just because they say it takes three hours for them, doesn't mean it's true. Some people just have an over inflated ego.

Hell, don't even think of yourself as average, because if you were you wouldn't be there. You already learned an important skill, a great work ethic. Alpha has the best advice, by far.

 

[Eagle's Wings]

your no doubt smarter than me, I do not go to a high end/ popularly known college but when I hit a road block, I just take a moment to stop and take the whole thing one step at a time. and I agree completely with AlphZero. Don't be afraid to ask people. you may feel dumb asking questions that other peers already know, but it would be better for you to feel embarrassed than to not know what you are doing. its your career that's important, not your reputation.

 

[converting1]

Thanks for your replies, I'd like to stress that it's not because it's Cambridge that I'm struggling. I originally thought that also, so I went searching through the internet and found questions on the same topics we do at Cambridge but at other universities. For instance, I took a look at UCL Analysis 1 homework problems and they are equally as hard so far so I struggle with their papers too. I also took a look at Imperials/Warwicks and still, it's hard. It's not Cambridge that's hard, it's just maths.

 

[Seydlitz]

Thanks for your replies, I'd like to stress that it's not because it's Cambridge that I'm struggling. I originally thought that also, so I went searching through the internet and found questions on the same topics we do at Cambridge but at other universities. For instance, I took a look at UCL Analysis 1 homework problems and they are equally as hard so far so I struggle with their papers too. I also took a look at Imperials/Warwicks and still, it's hard. It's not Cambridge that's hard, it's just maths.

I think you are very fortunate to be in Cambridge even though you might have some difficulty. Just hold on, do your best, stay optimistic, and things will get better eventually. Like the other said, use the supervision system to maximum effect, the supervisors are there to help you. Looking from your study habits, I also think you are very diligent. University math is indeed a little bit different from applied math because it requires you to think differently other than to move symbols around, and moreover you are in Cambridge. Do I need to repeat that? The entrance tutors consider you to be capable, don't think otherwise. It's not everyday you can study knowing that Stephen Hawking is one of the professors.

 

[WannabeNewton]

This such a load of bull, average students at Cambridge are more then capable of making contributions. Second rate minds? Haha.

You have to learn to ignore Group_Complex's posts I'm afraid. He/she basically puts other people down because of his/her own inferiority complex.

 

[Group_Complex]

You have to learn to ignore Group_Complex's posts I'm afraid. He/she basically puts other people down because of his/her own inferiority complex.

I think my advice is sound, if you do not like it point out why, please don't postulate that I have an inferiority complex on the basis of a few forum posts.
There are many second rate minds in academia who will have you believe that mathematics is the result of many small contributions, but I am afraid this is false. Most mathematical progress is made by a few mathematical giants, while lesser mathematicians attempt to use the deep insights generated by these mathematical greats to resolve small issues which greater minds have little time to squander their time on.

 

[jgens]

Most mathematical progress is made by a few mathematical giants, while lesser mathematicians attempt to use the deep insights generated by these mathematical greats to resolve small issues which greater minds have little time to squander their time on.

Citation please.

 

[Crake]

I think this link might be usefull to both the OP and group_complex.

I'm not sure if that kind of advice means much, coming from Terence Tao, but well, I think it would mean less coming from a non medalist.