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Stack-Exchange, I am a freshman in Mathematics Department.

I always learn mathematics the hard way. Everytime I learn mathematics on my own by reading from Mathematics textbooks, I would read the definition, try to figure out what it really means, stick new definitions which concepts that I already know, prove theorem without ever reading given proof.

But recently, I find this method cumbersome. In order to get a deep level of understanding those concept and solve a good amount of problems, I trade it off with a lot of my time, and my energy.

My seniors tell me that I should learn from different source of material to get more insight view, and separate the processes of learning theory and problem solving. I should read all the theory before trying to apply them to save times. But I'm used to the way of Pólya's mouse, so I am worried that I won't be able to catch up with all my classes like this. I want to try the way my seniors told me; that might be faster. But I want to know if the method would come with a loss of depth in understanding.

So I am asking for help. Also, please give me some tips to boost up the performance and the quality of learning mathematics all by myself. Thank you all very much.

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closed as too broad by Matthew Towers, reuns, Morgan Rodgers, Watson, user91500 Dec 7 '16 at 6:17

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Nice question, however, not typical in this community. $\endgroup$ – Babak Dec 4 '16 at 13:16
  • $\begingroup$ Thank you, and I am so sorry that I might have asked in a wrong place. $\endgroup$ – Michael D Nguyen Dec 4 '16 at 14:14
  • $\begingroup$ What is your maths level ? $\endgroup$ – reuns Dec 4 '16 at 15:10
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    $\begingroup$ I'd try asking this type of question in Academia@CS. There are many mathematicians who hang out there, and, honestly, you are getting pretty useless answers here. $\endgroup$ – Sasho Nikolov Dec 4 '16 at 16:44
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    $\begingroup$ It takes all of us a very long time. You will be okay. I promise. $\endgroup$ – The Count Dec 4 '16 at 19:32
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I will try to give an idea of what I did when I was in your situation (It worked for me, and I hope it will work for you too).

I had a teacher who used to say self learning is best learning. I was one of the strong believer of his philosophy. He once said, being a teacher I can only do one things for you which is Introducing the beauty of subject to you. The main thing in learning is the efforts you give.

Now since you are studying such a great subject on your own, You should have two things.

$1$. Interest in the subject

$2$.Patience.

Mathematics is a language that attract at first, then repel and then stick to you forever. You just have to bear the period of repulsion.

The period of repulsion comes when you can't solve problems, the formulas go over your head and at that time you just start reading the solutions of problems and start convincing yourself that "Yes I am learning (Though you are not)".

The basic idea is to stick to a topic until you have a complete belief that you can do any question of that particular topic. The mathematics means solving Problems and if you can't solve problems then you can't do mathematics.

I agree with your seniors but to a small extent. You should do problems from different books but avoid use of too many books as that leads to confusion.

So, I will sum up all the discussion regarding self learning in points as:

$1$. Bring your interest in mathematics.

$2$. Be patient and don't lose confidence.

$3$. Do problems as much as possible.

$4$. Avoid use of so many books until you have completed one.

Many other people can disagree with my opinions but that doesn't matter. I respect opinion of everyone that is why if you think I m wrong somewhere leave comment.Hope it helps.

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    $\begingroup$ @THELONEWOLF, on the contrary, I only spend time on the theory. Yes, problem solving is necessary, but often the degree of originality in mathematics itself can not be substituted for exercise problems. Frequently I have encountered exercise problems which don't have any importance outside of the problem itself, and required a clever 'trick' to solve. I like your advice for patience and interest, but I don't quite get this one. $\endgroup$ – codetalker Jan 3 '17 at 4:58
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    $\begingroup$ @Chill2Macht Your comment is confusing to me. It's likely there are alien civilizations that have solved much of the maths we're working on. So, then would it be "pointless" to get better at maths devising tricks even in research? Results undiscovered by a particular student are like mini-research problems. The upside is that the student can actually get the solution if they aren't able to devise the trick. Just because someone in history has gotten abs before doesn't mean all of humanity has washboard abs. Each student learns by their own efforts. $\endgroup$ – JDG Jan 22 at 14:31
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    $\begingroup$ @JDG You are right. I get that the point of such problems is to prepare students for careers in mathematics research, but not everyone who wants to learn some math wants to become a math professor. Such problems aren't necessary to learn how to use the math, and therefore don't really suit the needs of the broader audience who wants to apply the topic, rather than research it. Also, in practice, unless the person who wrote the book beforehand created solutions for all of the problems and published them online, it's often difficult to find the solution if the student can't devise the trick. $\endgroup$ – Chill2Macht Jan 30 at 18:12
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    $\begingroup$ @Chill2Macht You're right: Most students don't need the tricks. But then we should also design classes around student needs. A pre-med student doesn't need so much calculus. Rather, it might be better to issue a few one or two credit courses that highlight the cultural impact of calculus, as well as giving students a taste for simple derivatives, integrals, diff eqs: Focus on appreciation and piquing interest. Of course, that's in an ideal world. $\endgroup$ – JDG Feb 8 at 16:58
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    $\begingroup$ @Chill2Macht As for solutions, imo it's a crime that textbooks don't publish full solutions online. Many strong students I know benefit tremendously from Chegg on the daily---it's almost like 24/7 tutor access. With self-restraint so a student gives themself a chance to struggle, full solutions are invaluable: compared with going into school for office hours/tutoring, it saves countless hours, countless dollars, multiplies the efficacy of a textbook (fully worked solutions >>> answer summaries for odds), and so on. $\endgroup$ – JDG Feb 8 at 17:03
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I don't know why, but after reading the question, remembered this saying:

Theory is when you know everything and nothing works. Practice is when everything works and nobody knows why. In this office we combine theory and practice: nothing works and nobody knows why.

Summing up. I agree with THE LONE WOLF. And do problems as much as possible.

I would add: if you have doubts, ASK.

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  • $\begingroup$ Yes, thank you so much :) $\endgroup$ – Michael D Nguyen Dec 4 '16 at 14:44
  • $\begingroup$ You are reminding me of Fraunhofer diffraction, which is mathematically wrong but works anyway. $\endgroup$ – richard1941 Dec 9 '16 at 4:15
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I wish I had you for a teacher or a student. I am long past the age when I can take classes in school, and everything I do is driven by curiosity. Recently, I became interested in retro-tech and began with the table of four-place logarithms we were given in high school. (Before computers or calculators log tables were the one of the instruments by which human computing power could be amplified.) Wow! What a bunch of errors in that thing, but I am finding ways to weed out the worst mistakes and improve the accuracy.

I think I have discovered things about log tables that nobody else knows. It is an empowering feeling. And some day, when some conference needs some kind of paper, I will have one pre-generated. Most likely this would be the HHC 2017 conference dealing with HP calculators.

It is a good way to learn. Never quit!

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  • $\begingroup$ Thank you so much, that is so nice of you :) $\endgroup$ – Michael D Nguyen Dec 7 '16 at 5:52
  • $\begingroup$ But if you continue this way, you need to keep it OUT of your resume. Human Resource resume screeners are very intolerant of learning outside a prescribed college course. $\endgroup$ – richard1941 Dec 9 '16 at 4:18

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