Is it normal to feel overwhelmed by the abstraction of undergrad Math at first

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Im still a Sixth Form student (senior high school equivalent in the US) but have finished my school syllabus so Ive recently bought and started working on an introductory analysis book which is supposed to assume no more than A Level Maths (Calculus in US).

One main problem Im finding with higher Maths is the huge increase in abstraction which makes lines of reasoning often way more convoluted and generally difficult to follow than I have been used to. With A Level I could work through many chapters in a few hours and understand everything but now I sometimes find myself spending up to an hour trying to understand a single theorem, often having to try more concrete examples etc to try help me actually understand whats going on with the longer ones.

Is this "normal" and are you supposed to get used to it and be able to handle it eventually because at this stage it is just disheartening...
 

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  • #2
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Its normal for me anyway. I'm a visual and practical thinker/learner so I have trouble with abstraction especially when I can't see any practical application for it. If you're like me then you will find this site insanely helpful:
http://www.khanacademy.org/#calculus
Its normal to struggle with mathematical concepts before you gain a grasp of them. If you're new to calculus, give it time. Calculus concepts take longer to learn than any other maths fields I've studied.
 
  • #3
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I still find math to be overwhelmingly abstract even after completing a math degree; which is why I take a specific approach in my learning. I could easily stare at a theorem for days and still not understand anything at all. The reason for this is because learning a theorem before learning bits and pieces of the material is equivalent to having a helicopter drop you off at the top of a mountain and then ask how did we get up here by hiking?

I try to find different resources that will give me the little problems that lead up to the theorem. After I start finding out more bits of the idea I can usually get a good idea of what the theorem will say. I find that the beauty of math is the ability to generalize, but the generalization doesn't happen quickly it's only after many "experiments." I can't stress enough that you need to develop experimenting with the math. This view of math isn't usually taught at all by most mathematicians which is an absolute shame because this is the way math is actually discovered. Math is an experimental science, don't let anyone tell you otherwise.
 
  • #4
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I remember it once took me four days of work to understand a proof of Zorn's Lemma. Even the proof of the "binomial theorem" took me 3 days to understand. If you keep going things get better. For example I was able to do the proofs of a few major theorems in linear algebra without looking them up.
 
  • #5
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It's the same for me. I'm in my first year of a degree and taking Algebra and Calculus I, I find Algebra by far the hardest thing I have to study at the moment purely because it's so abstract. It's so different from what I've done until now, and even the 'explanations' can be just as abstract and hard to follow
 
  • #6
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I'm a math major and feel the same way actually, I think it gets easier though, especially once you adjust to the flow of things.
 
  • #7
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It gets easier once you realize it all actually means something. Or at least was meant to.
 
  • #8
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When I changed my major from engineering to math, it was actually refreshing, and, in some ways, easier. I found it more congenial to my way of thought. But I am not typical. One thing that helped was that I took a sort of easy set theory class when I started college, so it was not the first time I had seen proofs. That class was nice because you could get accustomed to proofs without having really difficult material to worry about. I had a lot of good math profs in undergrad, which also helps.

However, in grad school, things got cranked up a notch. Even I had difficulty with the level of abstraction sometimes.


I try to find different resources that will give me the little problems that lead up to the theorem. After I start finding out more bits of the idea I can usually get a good idea of what the theorem will say. I find that the beauty of math is the ability to generalize, but the generalization doesn't happen quickly it's only after many "experiments." I can't stress enough that you need to develop experimenting with the math. This view of math isn't usually taught at all by most mathematicians which is an absolute shame because this is the way math is actually discovered. Math is an experimental science, don't let anyone tell you otherwise.
I agree. It's not that mathematicians have it completely wrong all the time, but in many cases, they are WAY too formal, heavy-handed, and there is not enough motivation.
 
  • #9
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Mathematics will always be difficult, no matter how 'smart' you are or how advanced you are. Isn't that part of why we enjoy it?

Anyway, here is a book that might help you make the transition to abstract maths:
http://www.amazon.co.uk/How-Prove-S...ref=sr_1_4?ie=UTF8&qid=1321132070&sr=8-4&tag=
Basically it introduces the logic behind maths and the set theory which is the language, and some other fundamental things. A large part of why you feel overwhelmed may be because you don't have this background. I strongly recommend looking at this book.
 
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  • #10
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I agree. It's not that mathematicians have it completely wrong all the time, but in many cases, they are WAY too formal, heavy-handed, and there is not enough motivation.
This is indeed, unfortunately, true. There are many reasons for this. But the basic story is that a math degree is really a journey in discovering how YOU learn best. You really have to make it personal to do well. (This is the trick that I've found all the kids who "got it" really quickly in class had learned. I wish someone would've told me this!)

There are many threads that talk about personal methods and helping you find your own.

For algebra, this is the BEST book to get started with. Even if you are experienced with it, he gives so much history, motivation, and insight that it is like reading a novel! And the exercises are scaled and work you up to a very rigorous level. (actually, it's not SO much motivation, but it's more than usual which in contrast makes it seem as such!).

Here ya go: https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20&tag=pfamazon01-20
 
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  • #11
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When I knock on the door for Calculus to come out, she always tells me, "I'm still dressing". I have the knowledge in there, but, its not all "dressed up".
 
  • #12
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'tis pretty normal, yes. The best thing you can do for yourself right now is make sure you master all the precalculus courses shoved at you in high school, and any introductory calculus courses you might've taken as well. Do the exercises, practice, and you'll probably be fine. I've known some people who got good grades at math in high school so they never bothered to do the exercises. Without exception, these people (*cough-including-me-cough*) got in trouble when they entered college (insofar as they majored in math or physics, that is - I don't know all of 'em anyway).
 
  • #13
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holy ****, this topic so hard.

i feel this way myself, as ive taken calc 1-3, diffeq, and have started intro to analysis topics.
 
  • #14
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What book are you studying? Some writers are just incredibly bad at explaining things!
 
  • #15
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Thanks for the replies guys, I think one thing that makes this even worse is the standard style used by Mathematicians in writing rigorous Mathematics. I mean it seems like the standard style of writing rigorous Mathematics is to write it as succinctly as possible which makes it difficult to guess what mightve been the thoughts of the Mathematician when doing a proof, i.e. some of the steps in a proof seem quite random etc.

The book on Analysis Im studying is "Introduction to Real Analysis" by Bartle and Sherbert, I figured it might be better to study a book on proofs first so Ive ordered "Numbers and Proofs" by Allenby (recommended by my older brother whos a Mathematician).
 
  • #16
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SophusLies: Dead true. My strongest maths field by far is algebra and I only gained the intuition I have in it by doing my own haphazard experimentation with it. Though playing around with my calculator and satisfying all those spur of the moment curiosities that popped into my head, I've gained good intuition of how proportions work and ultimately, algebra is all about proportions. Its not abstract at all when you look at it like that. I should add also that I'm a programmer. The concept of variables and functions isn't anything new to me. Anyone who has programmed is familiar with many practical applications of algebra and I'm guessing thats why I became so good at a so called highly abstract field when I'm actually a predominantly visual thinker.

OP: You can't rely on a single book to teach you, especially not when it comes to mathematical fields. I got a book on calculus but noticed I could barely understand 20% of what they were talking about in it. My technique instead is to good the concepts, open up at least 5 web pages on them, look up youtube videos and find practice problems with worked solutions. In my case, this approach works well. I shoulda emphasized the youtube videos part there. You can find plenty of videos on youtube of people working through the problems step by step. The good ones explain things every step of the way. khanacademy is brilliant but there are some other good video series that cover calculus too. Look at khanacademy though, that guy covers everything.
 
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  • #17
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Thanks for the replies guys, I think one thing that makes this even worse is the standard style used by Mathematicians in writing rigorous Mathematics. I mean it seems like the standard style of writing rigorous Mathematics is to write it as succinctly as possible which makes it difficult to guess what mightve been the thoughts of the Mathematician when doing a proof, i.e. some of the steps in a proof seem quite random etc.

The book on Analysis Im studying is "Introduction to Real Analysis" by Bartle and Sherbert, I figured it might be better to study a book on proofs first so Ive ordered "Numbers and Proofs" by Allenby (recommended by my older brother whos a Mathematician).
Try

"Understanding Analysis" by Abbott

It is written for students like you who try to read their first analysis text and who struggle with it. It's an excellent book, and I think you should try this instead of Bartle.
 
  • #18
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Try

"Understanding Analysis" by Abbott

It is written for students like you who try to read their first analysis text and who struggle with it. It's an excellent book, and I think you should try this instead of Bartle.
Thanks, will do.
 
  • #19
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So far i haven't found math to be very abstract at all. At first i thought it was all about the graphs and trying to understand the graphs. But then i learnt about functions and mappings. From there, i realized it was a well defined area and the graphs don't matter at all. It's not abstract at all. IT'S MEGA LOGICAL. I'll give mathematics that.
 
  • #20
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So far i haven't found math to be very abstract at all.
Then, with all due respect, you must not have gotten very far in it.


At first i thought it was all about the graphs and trying to understand the graphs. But then i learnt about functions and mappings. From there, i realized it was a well defined area and the graphs don't matter at all.
Of course, the graphs matter. They may not be the whole story, but they do matter.


It's not abstract at all. IT'S MEGA LOGICAL. I'll give mathematics that.
It's very abstract at times. Try some homological algebra or non-commutative ring theory. If that's not abstract, what is?
 

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