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I am a sophomore in high school and have discovered the joys of mathematics. However,due to sheer irony, at times I find the subject difficult. I am what one could label "lopsided" in my faculties, excelling in languages but always floundering with numbers. This may be biological, for I was psychologically tested and found that my linguistic skills were abnormally high but my spatial reasoning...wasn't (I was diagnosed with Asperger's Syndrome). I do not want this to deter me from pursuing a degree in physics/math and finding work in the field. On the other hand, what if I just don't have the ability and fail? I see the mathematics geniuses at my school (or other students who are just better than me) and I feel horribly discouraged. Recently, I took a math test and felt completely lost, even though I studied diligently and received high scores on other assignments, which may be due to the anxiety I felt as I received the exam. The mere thought of that test kills me, as if it is a reminder that I shall never succeed in the subject no matter how hard I try (I was given a B last semester). I was wondering if those in advanced mathematics have had the same experiences, and somehow overcame their struggles. Is it optimal I should just study a humanities subject even though this is what I like, or is there some way I could find a path in the field? Also, as an aside, what gives you your passion for math? What is the best thing you find about this subject? Perhaps that can force my motivation to the point where I become proficient...

P.S. If you could point me in the direction of some math/physics related extra-curricular/summer activities and programs, it'd be much appreciated.

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marked as duplicate by Sami Ben Romdhane, Claude Leibovici, Ayman Hourieh, Did, M Turgeon Feb 2 at 19:06

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My advice: Do what you enjoy! Don't worry about your skill level. Keep doing math until it stops being fun, and if it never stops being fun, then congratulations! You're a mathematician. –  Jair Taylor Feb 2 at 6:20
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Don't be so afraid of failure. What if you do fail a test or even a class? The world won't end. I have often felt lost during a college math class lecture or during a test, and later ended up with an A. That lost feeling could mean you're way over your head, or it could just mean that you're doing something hard. Worst case scenario: you try to major in math, figure out you can't, then switch majors. Switching majors is not a disgrace, it's a change. –  Michael Shaw Feb 2 at 8:53
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See: Does one have to be a genius to do maths?. –  user93957 Feb 2 at 10:32
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I love it how one guy opens his heart and another says "possible duplicate" :-) –  PatrickT Feb 2 at 18:02
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@adobe: Tao's advice is dead-on. But nevertheless this dead-on advice would somehow be more convincing if its author were not so obviously a genius. –  Pete L. Clark Feb 3 at 3:43
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The title of your question presents an interesting dichotomy. It is of course true that a necessary condition to succeed as a mathematician is that one be "strong at mathematics"...but exactly what that means is less clear-cut. And whatever "strong at mathematics" means, it is something that can be developed (or not) over time: it is not something that either inheres in one's soul or doesn't like some sort of Calvinist election.

This may be biological, for I was psychologically tested and found that my linguistic skills were abnormally high but my spatial reasoning...wasn't (I was diagnosed with Asperger's Syndrome).

Wondering as a teenager what one may or may not be "biologically capable of" is not something that I would recommend: you are placing too much stock in biology and testing. (Frankly, I am skeptical even of the Asperger's diagnosis. I am by no means professionally trained in psychology, but in my opinion there is a faddishness in the diagnosis of such "syndromes". I believe that Asperger's diagnoses are based on measurable cognitive differences, but worry that the implications of these differences may be exaggerated. To quote from the wikipedia article on Asperger's: "Some researchers have argued that AS can be viewed as a different cognitive style, not a disorder or a disability, and that it should be removed from the standard Diagnostic and Statistical Manual, much as homosexuality was removed.") Also there is much, much more to mathematics than "spatial reasoning": I am a research mathematician with some kind of international reputation, but my inherent spatial reasoning abilities are no better than average. I bought a bookcase a few months ago, but delayed assembling it and seem to have lost the instructions: without them, I think I would rather buy a new bookcase than worry about how to assemble the one I already have! When it comes to applying spatial reasoning to mathematics, I can see that I have to work harder than many of my peers but I can still do it: for instance, when I discuss finding volumes of revolution in calculus I can always identify a few students who can visualize the regions more easily than I can...but overall I am still better at these problems than almost any calculus student I have ever taught because (i) I have so much more experience with these problems than they do and (ii) there are aspects of solving these problems other than spatial reasoning. I think that if you want to work in certain aspects of low-dimensional topology then it is important to have strong spatial reasoning skills -- sometimes I have seen talks or read papers in which proofs in these areas are done with the aid of pictures that look very confusing to me -- but even here I think that other skills are just as important or more as naked spatial reasoning. (In fact I did undergraduate research in low-dimensional topology, really enjoyed it, and would be happy to do more of it some day.)

I do not want this to deter me from pursuing a degree in physics/math and finding work in the field.

It is good to keep an eye on the future, but if you try to deal with the future as if it were the present then you're bringing a lot of unnecessary stress on yourself....we simply don't know so well what the future will bring. Having a goal of getting an undergraduate degree in physics or math is a good level of future planning for a high school sophomore. You can also start looking into the various careers that make use of this degree: there are many, and most of them would not go under the name "mathematician". The best way to stay on track towards this goal is simply to take challenging math classes, do your best to get what you can out of them, and also feed your interest in mathematics by reading up on whatever outside of class mathematical material interests you.

On the other hand, what if I just don't have the ability and fail?

We all worry about that. I still worry about that (maybe my definitions of "ability" and "failure" have been adjusted, but the emotional effect is much the same). I feel very confident that the world's leading mathematicians have these thoughts as well. They're helpful up to a point -- the point where they drive us to improve ourselves and have lofty goals -- and then they stop being helpful (at the point where we uselessly worry about our innate abilities or future success rather than try to figure out what to do next).

Recently, I took a math test and felt completely lost, even though I studied diligently and received high scores on other assignments, which may be due to the anxiety I felt as I received the exam. The mere thought of that test kills me, as if it is a reminder that I shall never succeed in the subject no matter how hard I try (I was given a B last semester).

It happens -- none of this is proof positive that you are not cut out for a mathematical career. When I was in school I usually did very well in my math classes but not always: I remember one "honors precalculus" course I took: the teacher was just very intense and had a way of making things tricky and complicated. I think he was probably a very good teacher, but his high ambitions and intensity did not translate well for me: I remember when we started a unit on linear equations, a subject which I had seen in several previous courses and knew was easy to understand. But somehow he made that material complicated as well: he discussed at least four (!!) different forms of equations of lines, and the exam managed to contain questions on linear equations that were really difficult! For one of the four quarters I got a $B$ in this course, and it was a bit discouraging. The next year I took calculus and it was both easier and more interesting than the precalculus course I had taken the year before: but I remember that though I found that most everything in the calculus course came very quickly, I also spent more than two hours a night solving calculus exercises...which must have been much more time than I had put into the "harder" precalculus course.

I was wondering if those in advanced mathematics have had the same experiences, and somehow overcame their struggles

In the first graduate level mathematics course I took, I studied "the wrong version of the Radon-Nikodym Theorem" for the final exam and was unable to do anything with one out of the three questions. I got a $B$ in that course and later found out that there had been some real question as to whether I should be allowed to continue with the graduate analysis sequence. In the end I got an award for being one of the best undergraduate math majors. Being "strong at mathematics" doesn't mean that you are inhumanly perfect: virtually everyone has at least a few lack-of-success stories to tell.

Is it optimal I should just study a humanities subject even though this is what I like, or is there some way I could find a path in the field? Also, as an aside, what gives you your passion for math? What is the best thing you find about this subject? Perhaps that can force my motivation to the point where I become proficient...

What I find interesting about your question is that while you sound very confident about your skills in language and the humanities, you seem very intent on pursuing a mathematical field....but you don't really say why you like mathematics better than humanities. (I'm not blaming you: such things can be hard to explain.) I can identify with this, because all throughout my pre-collegiate education I had skills in the humanities which were as strong as those I had in mathematics and more consistent: I never had a day where e.g. I tried to read Shakespeare and failed! Perhaps I had the sense that mathematics was deeper and more challenging than the humanities and was drawn to the idea that there was more clear room for improvement. As a high school student I learned a bit about calculus and saw that there were interesting things that lay far beyond -- like number theory, which I have been interested in since my sophomore year of high school -- and I'm not sure what is the equivalent in the humanities of "gunning for number theory". On the other hand I took creative writing as a high school sophomore, enjoyed that immensely and felt approximately the same infinite challenge as I did (and still do) with mathematics. I am somewhat regretful that I gave that up...although, given that I have more than 2000 pages of "extra" mathematical writing on my webpage, perhaps I have not really given it up so completely. So I wonder:

What is the right path for a high school student who is extremely talented and ambitious in the humanities?

I didn't know the answer to that question then, and I still don't now...but I wish I did. You might want to look into it. If you feel much more confident in your abilities in the humanities, shouldn't you at least think about going into the humanities?

Let me also say that the phrase about "forcing my motivation until I become proficient" worries me a bit. You also ask what gives mathematicians their passion for math and what they like about it. To me this sounds a little like someone who is thinking of becoming engaged asking an older married friend exactly what it was about her spouse that made her decide to get married. If you have to ask, then maybe you should be dating someone else! My passion for mathematics comes from the fact that I love it...it is not really something that can be further analyzed or explained. If you love mathematics, spend more time with it and develop your knowledge and skills. Otherwise keep taking mathematics courses at least until you get to college, but keep your mind open to finding the true object of your affections. It will be out there somewhere...

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When I was a TA of calculus 2 for mechanical engineering students, all of them had better spatial reasoning than me. And the teachers insisted to teach them how to understand integration via visualization. So I was always in a rut, and always had to try hard to get them to understand what I'm saying and explain it to me... Thank goodness that set theory has none of that sort of spatial reasoning (although I do seem to visualize these weird sets that I'm so interested in much better than I can visualize in my head simple bodies in $\Bbb R^3$...) –  Asaf Karagila Feb 2 at 7:32
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@Asaf: I agree with you that there are probably other scientific fields in which spatial reasoning is much more important than in (most branches of) mathematics. Mechanical engineering may well be one of those fields. (On the other hand, I taught multivariable calculus once to entirely engineering students, and their lack of ability to visualize three-dimensional objects was disappointing to me. In retrospect they were probably having difficulty with aspects other than the pure visualization, and my way of thinking and explaining things was too different from theirs to be very helpful...) –  Pete L. Clark Feb 2 at 7:43
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This is truly inspired writing. I relate to your early struggles and share your sentiments to such a great deal that this comment is really not necessary except to say "ditto" and "I wholeheartedly concur". What I really want to say is that if you ever write a book, I will buy it no matter what the subject. I love reading your writings. –  J. W. Perry Feb 2 at 9:40
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Amazing response! However, I'd argue that anything that is a ""a different cognitive style" is a disability, or, more precisely, an extreme disadvantage. This can cause severe problems fitting into the one size-fits-all education methodology that we use (at least in the US), which, in turn, limits the benefit that the student receives. That can cause a student to fail where, had they received education in the way best suited for their "different cognitive style," they could have flourished. –  David Lively Feb 2 at 16:15
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@David: Almost any significant difference between an individual and the majority of members of their group is going to be a source of stress for that individual, and -- at least in the short run -- often a disadvantage. That does not make every difference a syndrome, i.e., a kind of disease. Psychology has learned this lesson with respect to things like sexual orientation; it is plausible to think that it may make a similar change of stance on cognitive deviations. –  Pete L. Clark Feb 2 at 17:29
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Don't let a test tell you how good you are or are going to be at math. Math requires many dimensions of the human intellect, not just spatial reasoning. You may also be wrong about math not being your forte. In high-school, I thought I was bad at math and I thought I would never study it again. In college, I gradually found my way into the mathematical sciences, and when I began studying pure math, I discovered I was a great deal stronger than my classmates. Now I'm not only very happy doing math, but also very confident in my abilities. So give it time.

You shouldn't worry about "not being good enough". For one, becoming good at mathematics (or anything) requires, above all, great practice and diligence and passion. Natural talent can help and make it easier, but it is by no means the only way to get there. Notably, many people with a great deal of natural talent end up never succeeding because they are lazy. So don't worry about being talented or not. Worry about putting in the necessary work.

Secondly, how good at math do you think you have to be to do it seriously? For the vast majority of us, there's (almost) always a "better mathematician" who makes us and our work look small. So don't worry about feeling insignificant. Many mathematicians, some of them very good, share the feeling.

It's important that you're not incapable of doing math. You're passing your classes: you can do it. You might not be able to do it as easily as the others, but you can do it. And however far you want to go in math, if you really try, you'll get there. Determination is extremely important. Passion is what creates truly great work. So long as you really want to, you can go very far.

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I'm going to give you some real world advice. Don't listen to people here saying feel good stuff like follow your dreams or desires. Nobody wants to stifle a kid's dreams but at the same time, they are doing the kid a disservice by not telling him/her the truth. At this age, you have no idea what real math is. Once it gets really hard, you may feel that you do not still enjoy it. I enjoyed physics too when I was in high school. It was at a time when it was easy and you got to learn all the fun stuff. The ideas and insights in the hard sciences are very interesting BUT they are extremely difficult. You must be able to pass the hard part to get to the good part. It starts out easy and fun, then extremely hard, then fun again at the end. You don't know what real math is until you've done the extremely hard part. So I suggest to you to not simply say, I like math in high school and I will major in mathematics. Even calculus in college is not real math. Unless you LOVE that calculus course and put all your time into thinking about math and all of its insights and implications, do not devote your life to mathematics. There's no money in it, only love. You may find that you don't love it after all. That's the real world.

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Well you don't really stifle that dream at all, do you? The main point you make is saying that one has to have a look at "real" math to know if it fits. (I can't emphasize enough how much I agree with that!) So your answer to the question in the title is: Yes, it's possible. (But still can go wrong - obviously...) –  Piwi Feb 2 at 17:38
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As a teenager you probably never had a chance to see the kind of mathematics that mathematics is all about, you've probably seen mostly accounting.

When I wanted to explain to my parents and grandparents why I wanted to go into mathematics and what mathematics is about, I told them to read Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis so that's the advice I'd like to give to you too.

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As someone who is mathematically inclined, with an IQ tested as over 140 from a very early age, and one class away from my BS in mathematics, I can confidently say that mathematics is more about work than talent.

It's like kung fu or breakdancing; talent is a nice perk, but work is what gets you there. Seriously, just work at it. The more you do the better you'll be.

Also, math is huge. Linguistic talent could translate to mathematical talent in a great number of mathematical subjects. I've had brilliant professors, doctors of math, who are terrible at arithmetic. Seriously, math will eventually underlie every other field; so just develop the mathematics of linguistics if you don't want to think too much on what to do.

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Start with recreational mathematics, mainly puzzles. My all time favorite is

Recreations in the Theory of Numbers by Albert Beiler

It is a paperback book from Dover and it will give you hours and hours of pleasure.

http://www.amazon.com/Recreations-Theory-Numbers-Dover-Recreational/dp/0486210960

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Anyone who likes math should get this book. –  neofoxmulder Feb 2 at 10:43
    
+1, This is my most-thumbed book. I have had this copy since 1966 and it is high-lighted, wrinkled, and crinkled, but because it is Dover it is good for another 40+ years. –  Fred Kline Feb 2 at 12:51
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Mine too. It is in tatters and I recently bought an extra copy but I still prefer the old one with all my marks and notes! –  user44197 Feb 2 at 18:07
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I was diagnosed with Asperger's Syndrome

Hmm... Before coming over to this site, I spent a lot of my time on the Wikipedia Math Reference Desk for the better part of last year, where I met a user that was very good at answering questions concerning topology and category theory. That user has Asperger's as well. Perhaps you two could talk ? I'm sure his insights might prove to be valuable, especially since, like you, he also struggled with “spatial reasoning”, such as is required in classical geometry, for instance.

What gives you your passion for math?

I guess it's genetic, since my father and his siblings are into mathematics as well... But a book that sparked my interest for maths at a fairly early age was C.W. Trigg, Mathematical Quickies: $270$ Stimulating Problems with Solutions.

I see the mathematics geniuses at my school (or other students who are just better than me) and I feel horribly discouraged

I've spent (wasted) the better part of last year re-discovering things that have already been known for centuries... Then I came over to this site and saw people solving impossible integrals and other outrageously hard math problems, which I would have never been capable of approaching on my own. And the odd part is that I was one of those people with a special ability and inclination for math compared to my other colleagues from school and college. So ultimately everybody has its limits, and everyone looks up to someone else: This should not be a deterrent in and of itself...

My linguistic skills were abnormally high

I understand five languages, including my own: That did not prevent me from being reasonably good at math. :-)

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your question kinda resonated with me. When I was in high school, I was very dedicated and took a bunch of AP classes starting junior year. By the end of senior year I had taken 10, and AP Calculus AB among them. I graduated with a 4.0 and started college with almost a year of college "credits". My first math class in college was Calculus II, which some people liked to call "pre-business" since this class would truly filter out the men from the boys when it came to math. I passed it, not with an A though, since freshman year distractions occurred... Anyhow, I continued towards a math degree but as I took harder and harder classes I realized that though I was pretty decent at math, I was getting very bored by it. The higher the math the more theory and abstraction you get, and the more bored I got... I begun to realize I could not possibly do this for a living - I need more hands-on, more concrete/applied stuff, stuff I could build/tinker with...

I also had started taking some Computer Science classes and I noticed I was extremely good at them. My math background and logical thinking gave me a great edge, and I loved building programs and solving programming "puzzles." Mid-way thru my college career I switch my major and I couldn't have been happier I did!

The moral of the story is go after what you enjoy and where your strengths are! I've meet some people what were horrible at math, did not enjoy it, and yet had dreams of being Math PhDs -- I called these people masochists. It was worse than the kid whose parents were forcing him to become a doctor, even though he wanted to become a linguist (or whatever). If you go against your strengths and what you enjoy doing you will be miserable.

Now I've also meet people (now my best friend actually) who were pretty bad at math in high school, was a slacker, used drugs, etc. But in college he really applied himself and thru hard work basically ended up being better at math than I ever was! He bought all my math textbooks when I switched majors! My friend went on to become president of the Math Club as an undergrad, and he is now going for a PhD in math.

Now I commend you for starting to plan so early! My advise to you is to take the hardest possible math classes your high school offers (AP or IB level) and get as much college credit for these math classes as you can. These will somewhat prepare you for college-level math. But be warned that in college professors are hit or miss. Some know how to teach, other are clueless and are more or less there because they have to teach x hours while they get paid mostly for doing their research.

In college you learn to learn. You learn what you enjoy learning and what you don't. Often, you'll have to be able to read thru your textbook and pretty much teach yourself and then go to class with questions for the professor/lecturer.

Best of luck!

Ps: I'd say that if you pass Calculus II with an A as a freshman, you're pretty well cut out for being a math major, if you truly enjoy it!

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Mathematics is a beautiful subject, and you are very intelligent for perceiving that. You can certainly appreciate mathematics regardless of whether or not you pursue it as a professional discipline.

Please continue to investigate the subject. It doesn't really matter whether or not you become a professional mathematician. Just keep enjoying and learning about it.

Everyone here that understands anything about the subject wishes you the best, I'm sure.

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I am a PhD student in game theory which is at the intersection of Economics and Mathematics. Throughout highschool I HATED math, and was terrible at it as well(I got a D in pre-calculus). Little be known, once I got to college and had to take a calculus course for the first time I fell in love with mathematics and ended up graduating with a B.A. in Math with all but one A in my major.

Long story short, at this stage of your life what ever you really want to do you can do it. No one is born with the skills to be great at anything, we learn them as we get older, and the more time you spend the better you accumulate those skills. The more you love what you do, the more time you want to spend on it, the better you get at it, the more you love it, etc. It is a continuing cycle. Some people may be a bit faster at your age but if you stick with it, and it is really what you want to do, then you can succeed.

I am not sure what type of math you are learning at school, but I assure you it only gets more and more beautiful as you continue your studies. Also, keep in mind that there are many mathematicians in the world who are brilliant at math, but terrible at arithmetic! You should learn the differences and maybe you might see some of your strengths start to shine.

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And game theory can nicely be combined with humanities... –  Michael Greinecker Feb 2 at 12:23
    
Ditto @Thomas. I too was terribe at maths in secondary school but in 2 short years, at my mother's insistence that I take additional mathematics classes, maths went from my worst to my best subject. The only one I scored the hightest possible grade in. I now have a PhD in computer science and I still love mathematics and frankly regret that I studied engineering and not maths for my BSc. –  Olumide Feb 2 at 13:41
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Do not despair. The fact that you've fallen in love with the wonder and beauty of mathematics will help you overcome any initial feelings of inadequacy. Do lots of reading of good books about the practice of mathematics:

http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ref=sr_1_1?ie=UTF8&qid=1391320699&sr=8-1&keywords=what+is+mathematics would be a good start.

Paul Graham has some good advice for high school students at http://www.paulgraham.com/hs.html

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I am mathematician, I cherish mathematics and they are basically what defines me. Nonetheless, they still look strange, surprising, difficult and challenging to me every day of my life. So, from my experience, you may not be "very good" with maths and still try to go on and become a mathematician. Even a good one.

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Been there, done that. Let me explain...

At high school (I'm from Australia so I don't know what all this sophomore/freshman stuff is.. Senior years here) I just barely scraped through my maths classes. At uni I did bachelors of education and science majoring in mathematics. I did not do very well in the math classes. Most of them I just managed to pull passes or the odd credit ( 4 or 5 from a max of 7 with 4 being the pass). However, the education classes I was getting 6 and 7's for. What was different? The assessment. For the education classes most assessment was essay based; the maths was exam based. I have a very bad short term memory. I cannot multiply or often even add two 2 digit numbers in my head. Not because I don't know how, but because by the time I get to the second number I've already forgotten the first. This problem showed in my work at uni; I understood the content and could do it, but when it came to the exams I really struggled. Especially since most of the course's assessment was 80 - 90% in one final 3 hour exam at the end of the semester.

To cut a long story short, I decided that I liked the maths better than teaching and so started looking for a job in that field after I finished. Now I work as a developer/mathematician on an agricultural simulator for a large research organisation. Now, I'll be the first to admit I'm not a pure mathematician. A lot of what I do is actually data analysis, but as you probably know that includes a lot of mathematical and statistical work. I am quite fluent in the language of mathematics but I still have a desk reference of the meaning of math symbols because I can't always remember them. I can do a fourier or solve a system of differential equations but I need a reference manual to help me. Mostly because I do them so rarely I need to jog my memory.

And therein lies my main point. Many jobs in mathematics (excluding academia of course) don't actually require you to do maths. They require you to understand it. All those algebra equations and integrals and matrices they have you do? After you graduate you will probably never do one by hand again. We have computers to do the heavy lifting now. But you need to understand how the equations and integrals, etc work. You need to be able to set up the problem for the computer and understand the output that comes back. You need to be able to communicate your results to others. That is primarily what (in an ideal world) your classes are setting you up for. The exams where you have to do the calculations yourself from memory are to show that you have that understanding.

Now, having said all that. Do your best to do as well in your classes as possible. While it sucks that two people with the same ability can get wildly differing grades simply due to the form of assessment, that's the world we live in. And up to a point, especially when you're a recent graduate, employers will judge you on the basis of those grades simply because you don't have any work experience behind you yet that you can point to. But in the end, realise that your grades do not have to reflect your ability and as long you understand the content, don't let lower grades discourage you from a career in mathematics. The field is much larger and diverse than most people realise, which is probably another topic in and of itself, really.

So the tl;dr version: I sucked pretty hard at maths through school and university. I didn't expect to get a career in it but I did and I love it. Final grades are not always an indicator of mathematical success. If you understand what you're being taught but have trouble when it comes to assessment that can be for reasons other than a lack of ability. If you enjoy, keep going and you will be successful.

Best of luck, Justin

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If you love maths, then you probably already are a mathematician in reality. I didn't like math at school, and was not good at it as I found it hard and boring. When I went to university on computing, there were all sorts of undreamt of maths areas, which I found absolutely natural. At your age especially, the teacher can make a very big difference. If you enjoy it, keep doing it until it stops being fun.

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