How to study maths quickly and efficiently?

[QIsReluctant]

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?

 

[espen180]

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.

 

[symbolipoint]

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?

 

[JonDrew]

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?

 

[QIsReluctant]

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

 

[QIsReluctant]

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.

 

[micromass]

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.

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??

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.

 

[QIsReluctant]

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.

 

[Best Pokemon]

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.

 

[JonDrew]

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.

 

[WannabeNewton]

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.

 

[micromass]

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!

 

[George Jones]

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?

 

[JonDrew]

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.

 

[JonDrew]

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.

 

[George Jones]

Everything up through Differential equations

I don't know what this means. Humour me; please give an explicit list of courss.

 

[micromass]

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:

$$\int_a^b f(x)dx=F(b)-F(a)$$

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

If so, I think you should quit your math major.

 

[pongo38]

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?

 

[WannabeNewton]

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.

 

[JonDrew]

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.

 

[JonDrew]

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?

 

[micromass]

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?

 

[JonDrew]

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.

 

[George Jones]

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.

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.

 

[mathwonk]

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.

 

[dustbin]

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?

 

[ZombieFeynman]

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.

 

[member 392791]

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).

 

[JonDrew]

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.

 

[JonDrew]

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.