How to study maths quickly and efficiently?

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In summary, the conversation discusses the speaker's first semester taking distance courses and their struggle with managing time while also wanting to fully understand the material without rushing through it. They mention placing time limits on proofs and the importance of patient contemplation in understanding mathematics. The speaker also mentions their courses in Algebra and Galois Theory and Analysis, and their unique approach to learning math by starting with problems and then referencing the necessary sections for understanding. The conversation ends with advice on the importance of thoroughly understanding concepts and not just memorizing methods.
  • #1
QIsReluctant
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This is my first semester taking courses by distance, and although I'm not a fan of cramming, managing time has been an issue. I have two tests in the near, but not super-near, future and I'd like to get through these chapters as quickly as possible whilst actually learning and not just speed-reading. The problem is that I do not like to take anything in math for granted; if I skip a proof or proceed through a section without a very clear understanding it bothers me -- and with good reason.

One way that I've sped things up is by placing a time limit on proving corollaries/theorems; if I can't figure out the proof in the allotted time, I will pretend that "This proof is beyond the scope of this course" is written where the proof should be and content myself with "someone has proven it somewhere." After all, it's not like I'm breaking new ground.

I did some planning today and I have to cover pages at about 20 minutes per page. Can I do it, or is my "math problem" a riddle even Oedipus, with Euclid's help, couldn't solve?
 
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  • #2
It depends on your ability to grasp abstract concepts and arguments, your ability to formulate these independently, the level of your course and the density of your text.

For example, I have two books in front of me. One is an introductory abstract algebra book. The other is an advanced book about topology or representation theory. I will probably spend less than 10 minutes per page on the first book, but may need a day or two to swallow one page of the second.

You cannot rush mathematical understanding. It comes from patient contemplation.
 
  • #3
One typically does not study Mathematics quickly. Twenty minutes per page as you stated seems about right. Study-time can easily reach about 15 hours per week. One should ideally study every day for about 2 hours per day, maybe more if you can deal with the added time per day.

Math course by "distance learning": What course is yours exactly?
 
  • #4
It is completely dependent on what course your taking, page by page analysis of a math textbook has always been a waste of time for me.

So what math is it?
 
  • #5
There are two courses, both graduate-level. One is Algebra and Galois Theory, and the other is Analysis (right now, specifically, measure theory and topology).

"Distance" = I do not attend courses; instead, I self-study with the course materials and show up for the final. I picked it beca
 
  • #6
Sorry, overactive Return key. To continue: because of my afternoon job and because I'm a little behind and need more time to digest the material.
 
  • #7
JonDrew said:
It is completely dependent on what course your taking, page by page analysis of a math textbook has always been a waste of time for me.

:confused: A careful analysis of the textbook is the only way to learn math. How do you learn math? Just look at the formulas and memorize them??

QIsReluctant said:
There are two courses, both graduate-level. One is Algebra and Galois Theory, and the other is Analysis (right now, specifically, measure theory and topology).

But the proofs in topology, analysis and galois theory are awesome and insanely interesting. Why would anybody want to skip them? Furthermore, they contain important ideas and techniques that are useful.
 
  • #8
But the proofs in topology, analysis and galois theory are awesome and insanely interesting. Why would anybody want to skip them? Furthermore, they contain important ideas and techniques that are useful.

Ah, but micromass -- you've returned to the reason for my original question.
 
  • #9
JonDrew said:
It is completely dependent on what course your taking, page by page analysis of a math textbook has always been a waste of time for me.

So what math is it?

This sounds like me. I normally just study axioms and methods. I also look at examples from notes/books.
 
  • #10
micromass said:
:confused: A careful analysis of the textbook is the only way to learn math. How do you learn math? Just look at the formulas and memorize them??

No, I just start a problem not really having a clue of what I am actually doing, then look at the sections I need to look at in order to figure out the problem. I don't think I have met many people who do it the sit and read method though but then again I am not a graduate student.

I read Physics books, but never math books.
 
  • #11
JonDrew said:
No, I just start a problem not really having a clue of what I am actually doing, then look at the sections I need to look at in order to figure out the problem.
So you just skip over all the proofs and examples? I have never seen anyone use this method before and probably for good reason.
 
  • #12
JonDrew said:
No, I just start a problem not really having a clue of what I am actually doing, then look at the sections I need to look at in order to figure out the problem. I don't think I have met many people who do it the sit and read method though but then again I am not a graduate student.

I read Physics books, but never math books.

Some constructive advice: that is an awful method to learn math. I suppose engineers can get away with just reading the methods, but you're a math major!
 
  • #13
JonDrew said:
No, I just start a problem not really having a clue of what I am actually doing, then look at the sections I need to look at in order to figure out the problem.

For what math courses have you done this?
 
  • #14
George Jones said:
For what math courses have you done this?

Everything up through Differential equations, just to add, I have never even taken a trig or Algebra 2 class. I think the formal introduction to applicable concepts is far over rated, for higher level proofs this might not be the case but for undergraduate math an under, I show as proof math average right now is about a B+ and that is only because I had to take Calc II over the summer, it was crammed into five weeks.
 
  • #15
WannabeNewton said:
So you just skip over all the proofs and examples? I have never seen anyone use this method before and probably for good reason.

Most of the time I don't even take a peek.
 
  • #16
JonDrew said:
Everything up through Differential equations

I don't know what this means. Humour me; please give an explicit list of courss.
 
  • #17
JonDrew said:
Most of the time I don't even take a peek.

Well, if you just want to apply the concepts to real-life situations, then I'm sure this approach could work. But this is the complete wrong mentality for a math major. Don't you feel the slightest curiosity as to why properties are true? For example, the fundamental theorem of calculus:

[tex]\int_a^b f(x)dx=F(b)-F(a)[/tex]

Aren't you interested why this extraordinary theorem holds true?
Or the rules of derivatives, things like [itex]\frac{dx^2}{dx}=2x[/itex]. Do you just memorize these facts and accept them without justification??

If so, I think you should quit your math major.
 
  • #18
Consider the analogy of tourism. A walking tourist will learn a lot about a little. A flying tourist will learn a little about a lot. You need both. For an overview perhaps you need to read about the history of your mathematical topic, before slowing down and looking at the detail. So, a variable speed approach is best?
 
  • #19
JonDrew said:
Most of the time I don't even take a peek.

That may have worked for regular calc 1 - 3 and DiffyQs because those classes tend to be quite easy but it won't work for the upper division classes. Actually if you had taken honors calc 1 or 2 (the spivak or apostol courses) you would have noticed your method failing there as well.
 
  • #20
Basic addition, Basic division, Basic multiplication, Long division... you asked of humor... more seriously, Calc I, Calc II (some multivariable but not much), Analytical Statistics and Dif. Eq.
 
  • #21
micromass said:
Some constructive advice: that is an awful method to learn math. I suppose engineers can get away with just reading the methods, but you're a math major!

Applied math, more stats and stuff. Often I ask my self "self, what is the difference between applied math and physics?" then I answer "self, I don't know yet."

I still have to go through ACalc, which is the only thing that frightens me, from my understanding the teacher at my University isn't very good. Would you happen to know any ACalc OCW's?

And what is the difference between a quantum physics and quantum mechanics class? My guess is field theory, Am I correct? Anyone?
 
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  • #22
JonDrew said:
Basic addition, Basic division, Basic multiplication, Long division... you asked of humor... more seriously, Calc I, Calc II (some multivariable but not much), Analytical Statistics and Dif. Eq.

So you haven't taken an actual math class yet?? And you find yourself capable enough to tell others how to study math?
 
  • #23
micromass said:
So you haven't taken an actual math class yet?? And you find yourself capable enough to tell others how to study math?

Why yes, I have taken a math class, I don't know what sort of connotations you're associating with the term actual math class but whatever.

And yes I find myself rather capable of telling others how to study math, weather my advice is good or not is a point for a subjectivist but I really feel that my ability to do so has been rather obvious.

Maybe you could look into some actual sociological classes, because the generation gap is very clear. My generation simply needs to read less in order to get information effectively, we simply don't need to sit down like military children and drill readings into our head. Rather, a lot of us are like me and let technology supplement many of the things we don't like to do.

Just to drive home your appall-o-meter, I can also tell you there is a direct correlation between the textbooks I can buy on ebook and my respective classes grade point average because I simply don't read textbooks anymore, I have my computer read them to me. So, yeah, I hardly look at my math textbook other than problem sets and solutions, not a big deal, the old methods of learning are starting to die anyway, I don't feel like picking them up as they go out the door.
 
  • #24
JonDrew said:
Why yes, I have taken a math class, I don't know what sort of connotations you're associating with the term actual math class but whatever.

You have taken math classes in which you can get by with using recipes. You have not taken math classes like real analysis, abstract algebra, point-set topology, functional analysis, etc.
JonDrew said:
And yes I find myself rather capable of telling others how to study math, weather my advice is good or not is a point for a subjectivist but I really feel that my ability to do so has been rather obvious.

You have demonstrated repeatedly on Physics Forums a clear lack of ability to give useful advice.
Maybe you could look into some actual sociological classes, because the generation gap is very clear. My generation simply needs to read less in order to get information effectively, we simply don't need to sit down like military children and drill readings into our head. Rather, a lot of us are like me and let technology supplement many of the things we don't like to do.

Just to drive home your appall-o-meter, I can also tell you there is a direct correlation between the textbooks I can buy on ebook and my respective classes grade point average because I simply don't read textbooks anymore, I have my computer read them to me. So, yeah, I hardly look at my math textbook other than problem sets and solutions, not a big deal, the old methods of learning are starting to die anyway, I don't feel like picking them up as they go out the door.
Stop being so immature; put your big-person pants on.
 
  • #25
JonDrew said:
No, I just start a problem not really having a clue of what I am actually doing, then look at the sections I need to look at in order to figure out the problem. I don't think I have met many people who do it the sit and read method though but then again I am not a graduate student.

I read Physics books, but never math books.


i love it. mr drew you have just described the absolute most clueless losers way to attack a math course, always with disastrous results, unless the course is a joke.

pardon me but to us this is funny, if it weren't sadly the same way so many of our hapless students proceed.
 
  • #26
JonDrew said:
I show as proof math average right now is about a B+ and that is only because I had to take Calc II over the summer, it was crammed into five weeks.

Proof that your method does not work as well as you think it does?
 
  • #27
JonDrew said:
Why yes, I have taken a math class, I don't know what sort of connotations you're associating with the term actual math class but whatever.

And yes I find myself rather capable of telling others how to study math, weather my advice is good or not is a point for a subjectivist but I really feel that my ability to do so has been rather obvious.

Maybe you could look into some actual sociological classes, because the generation gap is very clear. My generation simply needs to read less in order to get information effectively, we simply don't need to sit down like military children and drill readings into our head. Rather, a lot of us are like me and let technology supplement many of the things we don't like to do.

Just to drive home your appall-o-meter, I can also tell you there is a direct correlation between the textbooks I can buy on ebook and my respective classes grade point average because I simply don't read textbooks anymore, I have my computer read them to me. So, yeah, I hardly look at my math textbook other than problem sets and solutions, not a big deal, the old methods of learning are starting to die anyway, I don't feel like picking them up as they go out the door.

Speak for yourself. I am only perhaps 5 years older than you and therefore in your generation. If you are a shining example of our generations learning in action, we may be doomed after all.
 
  • #28
To be fair, sometimes math books are far too tersely written (whether unintentional or intentional is unknown to me) and yes, reading proofs of theorems is more or less a waste of time for me, because I simply can't understand what the proof is trying to say.

To give an example, I can't prove that a subspace of a vector space is closed under addition and closed under scalar multiplication. I have given up on the prospect of trying to understand it, so I just accept it as truth and use it so solve the questions asking whether something is a subspace or not. I am doing engineering though, so perhaps this is more acceptable to me? I'm not too interested in math theories anymore, I simply don't follow the logic well enough (this is bad on my part, not their fault I can't follow them).
 
  • #29
Woopydalan said:
To be fair, sometimes math books are far too tersely written (whether unintentional or intentional is unknown to me) and yes, reading proofs of theorems is more or less a waste of time for me, because I simply can't understand what the proof is trying to say.

To give an example, I can't prove that a subspace of a vector space is closed under addition and closed under scalar multiplication. I have given up on the prospect of trying to understand it, so I just accept it as truth and use it so solve the questions asking whether something is a subspace or not. I am doing engineering though, so perhaps this is more acceptable to me? I'm not too interested in math theories anymore, I simply don't follow the logic well enough (this is bad on my part, not their fault I can't follow them).

Less people are interested in math for this reason, while the PF mentor up there jumps up and down if your don't use the Atypical methods which have failed students for generations. I hated math until I discovered youtube, yes, I discovered it, my family and I lived in an area with terrible internet service for a while, once I moved to a place would good internet service I love math.

And to the other guy, your not in my generation you didn't grow up with KhanAcademy and other methods of the like in your back pocket.
 
  • #30
mathwonk said:
i love it. mr drew you have just described the absolute most clueless losers way to attack a math course, always with disastrous results, unless the course is a joke.

pardon me but to us this is funny, if it weren't sadly the same way so many of our hapless students proceed.

There is no reason to go over proofs you already know, and you find your gaps more effectively by trying a math problem. Ever read a math text, tried the example problem, then realized you still don't have a clue of what your doing. Everyone knows you've done it, we've all done it, in this case you may hove just wasted a half an hour or maybe even an hour learning nothing. I understand math often works this way I find my method more effective, problems first, it finds your gaps in knowledge, then you fill them in.
 
  • #31
JonDrew said:
And to the other guy, your not in my generation you didn't grow up with KhanAcademy and other methods of the like in your back pocket.

LOL Khan Academy...yeah good luck using Khan Academy when you start taking actual math classes.
 
  • #32
JonDrew said:
Less people are interested in math for this reason, while the PF mentor up there jumps up and down if your don't use the Atypical methods which have failed students for generations. I hated math until I discovered youtube, yes, I discovered it, my family and I lived in an area with terrible internet service for a while, once I moved to a place would good internet service I love math.

And to the other guy, your not in my generation you didn't grow up with KhanAcademy and other methods of the like in your back pocket.

JonDrew said:
There is no reason to go over proofs you already know, and you find your gaps more effectively by trying a math problem. Ever read a math text, tried the example problem, then realized you still don't have a clue of what your doing. Everyone knows you've done it, we've all done it, in this case you may hove just wasted a half an hour or maybe even an hour learning nothing. I understand math often works this way I find my method more effective, problems first, it finds your gaps in knowledge, then you fill them in.

I know you talk about being a new generation and things like that (even though I'm just a few years older than you), but trust me: your approach is not the right one. There are many people on this forum who struggled through physics and math and who have much experience in the field. Why do you refuse to listen to them?

How well do you really know your mathematics? I'm sure you can solve the exercises in Stewart and ace your exams, but that doesn't say much. If you just memorize derivative and integral rules without knowing where they come from, then I'm afraid you are not much more than a parrot who just shouts whatever he was taught.

To be fair, there is something interesting in an "exercises first" approach. The idea is to first give a few exercises which are easy to understand but difficult to solve. After an attempt of solving it, you read the chapter and learn how the material in the chapter helps you with the practical problems. If you have a good professor/tutor, then I think this approach can yield very good results. However, it is absolutely not the idea to skip the proofs.
 
  • #33
JonDrew said:
And to the other guy, your not in my generation you didn't grow up with KhanAcademy and other methods of the like in your back pocket.

Please, please do not only rely on khan academy or internet videos. These give you a false feeling of knowing the material.
One of the troubles with khan academy is that it is easy. Too easy. Once you get to more difficult courses such as QM or GR or real analysis or else, then there are no more easy videos available. You will have to start reading the textbook to understand the material. If you only watched khan academy videos, then you are not prepared for this step. The idea of textbooks is to slowly increase the difficulty and to get you used to abstract arguments. So when you start real analysis, you will find the step less steep (even though it is still quite difficult).

Khan academy is an excellent secondary resource. It should never replace a textbook. If you do allow it to replace your textbook, then you will feel the consequences later. And I also doubt that you will be able to solve more complex problems in physics and mathematics.
 
  • #34
micromass said:
And I also doubt that you will be able to solve more complex problems in physics and mathematics.

Especially when just about every exercise asks you to prove a theorem or derive a relation not mentioned in the main text.
 
  • #35
micromass said:
I know you talk about being a new generation and things like that (even though I'm just a few years older than you), but trust me: your approach is not the right one. There are many people on this forum who struggled through physics and math and who have much experience in the field. Why do you refuse to listen to them?

How well do you really know your mathematics? I'm sure you can solve the exercises in Stewart and ace your exams, but that doesn't say much. If you just memorize derivative and integral rules without knowing where they come from, then I'm afraid you are not much more than a parrot who just shouts whatever he was taught.

To be fair, there is something interesting in an "exercises first" approach. The idea is to first give a few exercises which are easy to understand but difficult to solve. After an attempt of solving it, you read the chapter and learn how the material in the chapter helps you with the practical problems. If you have a good professor/tutor, then I think this approach can yield very good results. However, it is absolutely not the idea to skip the proofs.

Which proofs are you referring to? The proof for the derivative with Reman-sums (or however you spell it)? I haven't run into many proofs that couldn't be explained in a ten minute video with the exception of Eulers formula/identity, and I agree they help a heck of a lot intuitionally.

Linear algebra proofs are the only things I've skipped because I am in Diff. Eq. and am supposed to have already taken Linear A, there just to complex to try. Anyway still if you guys really think these proofs are so important in the upper math courses which don't have video proofs, I would really appreciate it if you made the videos yourselves, there is a whole generation coming who have never learned a single math concept from a textbook. Screen cast are extremely easy to make especially with a PC, look it up. All you need is a basic editing software and a drawing program called smooth draw its free. Bring out your inner philanthropist, and put these up for the world to see, you know the methods which you learned these things sucked so make it easy for the next generation.

And to the other guy Khan is just the poster child and innovator, there are many who have theorem proofs in much higher level math and physics ranging from string to measure theory. Really, I think you guys are pretty ignorant of this stuff. this is why I made the https://www.physicsforums.com/showthread.php?t=644282 post.

P.S. to make the video you also need a drawing tool, but you can get one used for cheap.
 
<h2>1. How can I improve my speed in solving math problems?</h2><p>One effective way to improve your speed in solving math problems is to practice regularly. Set aside a specific amount of time each day to work on math problems and increase the difficulty level as you progress. Additionally, try to identify and eliminate any distractions that may be slowing you down.</p><h2>2. What are some tips for retaining information while studying math?</h2><p>One helpful tip for retaining information while studying math is to break down complex concepts into smaller, more manageable parts. This will make it easier to understand and remember the material. Additionally, try to actively engage with the material by asking yourself questions and explaining the concepts to someone else.</p><h2>3. How can I stay motivated while studying math?</h2><p>Staying motivated while studying math can be challenging, but setting specific goals and rewarding yourself for achieving them can help. It can also be helpful to take breaks and switch up your study methods to keep things interesting. Lastly, try to remind yourself of the importance and relevance of math in your daily life.</p><h2>4. What are some effective study techniques for understanding complex math concepts?</h2><p>One effective study technique for understanding complex math concepts is to break them down into smaller, more manageable parts and then work on each part separately. Another helpful technique is to use visual aids such as diagrams or graphs to better understand the concepts. Additionally, practicing and applying the concepts in real-life situations can also improve understanding.</p><h2>5. How can I make the most of my study time for math?</h2><p>To make the most of your study time for math, it is important to have a designated study space that is free from distractions. Prioritize your study topics and focus on the most challenging ones first. It can also be helpful to use mnemonic devices or create study guides to aid in memorization. Lastly, take breaks and don't forget to review previously learned material to reinforce your understanding.</p>

1. How can I improve my speed in solving math problems?

One effective way to improve your speed in solving math problems is to practice regularly. Set aside a specific amount of time each day to work on math problems and increase the difficulty level as you progress. Additionally, try to identify and eliminate any distractions that may be slowing you down.

2. What are some tips for retaining information while studying math?

One helpful tip for retaining information while studying math is to break down complex concepts into smaller, more manageable parts. This will make it easier to understand and remember the material. Additionally, try to actively engage with the material by asking yourself questions and explaining the concepts to someone else.

3. How can I stay motivated while studying math?

Staying motivated while studying math can be challenging, but setting specific goals and rewarding yourself for achieving them can help. It can also be helpful to take breaks and switch up your study methods to keep things interesting. Lastly, try to remind yourself of the importance and relevance of math in your daily life.

4. What are some effective study techniques for understanding complex math concepts?

One effective study technique for understanding complex math concepts is to break them down into smaller, more manageable parts and then work on each part separately. Another helpful technique is to use visual aids such as diagrams or graphs to better understand the concepts. Additionally, practicing and applying the concepts in real-life situations can also improve understanding.

5. How can I make the most of my study time for math?

To make the most of your study time for math, it is important to have a designated study space that is free from distractions. Prioritize your study topics and focus on the most challenging ones first. It can also be helpful to use mnemonic devices or create study guides to aid in memorization. Lastly, take breaks and don't forget to review previously learned material to reinforce your understanding.

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