Studying Share self-studying mathematics tips

[Hellmut1956]

i can only reemphasize what I have written earlier! Go to MITOPENCOURSEWARE, here the link to the first calculus course. You not only have an extremely helpful offer, but you also start to see how real courses at a university take place! Research just the courses offered for free in 18.xx, focus on the ones made for self study and dig into it.!

 

[Dr. Courtney]

A great way to know if you are really ready for self-study (or other study) of college level math is to sign up for a month of ALEKS and take the pre-calculus assessment. If you can't complete the pre-calc material in a couple of weeks, you are not really ready for calculus.

 

[Akorys]

I have two questions related to self study, so instead of starting a new thread I assume it is ok to simply ask them here!

First of all, I am a bit confused about when "Advanced Calculus" by Loomis and Sternberg would be best studied. Is it a first text in multivariable calculus? I saw posts on pf recommending that one first studies the subject with a different book. Is it rather an introduction to real analysis?

Second of all, is it necessary to often review (and to avoid misunderstanding, by review I mean rewriting and understanding theorems, some proofs, doing harder problems etc.) chapters one has already studied, or is it better to use one's time learning new subjects entirely?

 

[Hellmut1956]

Akorys, I am starting to believe I am talking and writing in chinese! This is due to the fact that I see reading this thread I see the members sticking to search for books for self study! Sticking to books means throwing away the benefits from attending a course at a university of your choice.Now imagine that the university of choice you take is the MIT! Expensive? Not at all! Just go to their OpenCourseware offering and there you find videos of the lectures of the professor. You find the video of the assignment sessions, you find the notes of the lecture, you find the proper book, all for free!
But what makes OCW even better then a presence at a course at the MIT is that the professor in the video is available 24/7 and he keeps repeating it,if desired every single word until you grasp what is meant. if you learn about a person, a topic or a term, just hit pause and investigate in the Internet! You do not like the style of teaching of a certain professor, no problem, you will find iin the Internet another Professor better suited to your preferences! You want to join a study group? No problem yo get the link to where you can join others studying the same course!
So why are you and others sticking to mere books for self study?

 

[micromass]

First of all, I am a bit confused about when "Advanced Calculus" by Loomis and Sternberg would be best studied. Is it a first text in multivariable calculus? I saw posts on pf recommending that one first studies the subject with a different book. Is it rather an introduction to real analysis?

It is a very advanced book. It is certainly NOT a first text in multivariable calculus. The title "calculus" is pretty misleading. Maybe after you had a decent course in analysis, you can think of tackling this book.

Second of all, is it necessary to often review (and to avoid misunderstanding, by review I mean rewriting and understanding theorems, some proofs, doing harder problems etc.) chapters one has already studied, or is it better to use one's time learning new subjects entirely?

Yes, it is necessary to review. The superficial reason for this is that you won't forget essential things later on. But the deeper reason is that you mature constantly. So coming back to a chapter will often reveal new information and new points of view. There will be things that you thought you understood or that you ignored because it seemed unimportant, but that you now realize are pretty essential. This is a very pleasant experience to go through since you can feel yourself growing. Not every book will induce such experience, but the better (and usually the more rigorous) books will have this a lot.

 

[Akorys]

Hellmut1956, I have read your advice and I both appreciate and make use of it! I have watched almost all of the lectures for single variable calculus, barring those for series and sequences which I would like to first study through my text. I find MIT resources very helpful, and often refer to their notes and assignments. In general, internet resources I find are extremely useful as supplementary material for clarification, but I find that a good textbook can introduce a subject in an enthralling way! Also, I dislike reading books from a computer when I can instead hold a book in my hands, which I realize may be a disadvantage.

I stick mostly to books as a primary resource as they have been written by knowledgeable people, and when other people who are well educated about math, for ex., agree that these books are very good, as can be seen on these forums in numerous places, I trust that I will be exposed to a subject in a great way. I then use internet resources (MIT OCW, Physics Forums, etc) to help me understand things that my book may not present in a way I understand. I assume that most people who stick to textbooks think in a similar way.

It is a very advanced book. It is certainly NOT a first text in multivariable calculus. The title "calculus" is pretty misleading. Maybe after you had a decent course in analysis, you can think of tackling this book.

Thank you for clarifying this! Perhaps after an analysis course I will look into this.

Yes, it is necessary to review. The superficial reason for this is that you won't forget essential things later on. But the deeper reason is that you mature constantly. So coming back to a chapter will often reveal new information and new points of view. There will be things that you thought you understood or that you ignored because it seemed unimportant, but that you now realize are pretty essential. This is a very pleasant experience to go through since you can feel yourself growing. Not every book will induce such experience, but the better (and usually the more rigorous) books will have this a lot.

I was inclined to ask because I am experiencing something similar to what you describe. I decided in the book I'm reading on calculus that, after about halfway through, I had shaky understanding of the first several chapters despite working through them. The second time through things seemed much clearer, as you state they would. However, this does lead into the question of: when have I studied this enough? I can imagine that one may be stuck on one subject for an excessive period of time and never seem to move to a new area.

 

[micromass]

I was inclined to ask because I am experiencing something similar to what you describe. I decided in the book I'm reading on calculus that, after about halfway through, I had shaky understanding of the first several chapters despite working through them. The second time through things seemed much clearer, as you state they would. However, this does lead into the question of: when have I studied this enough? I can imagine that one may be stuck on one subject for an excessive period of time and never seem to move to a new area.

You need to find a balance of course. You need to study a specific chapter a good amount of time, but you shouldn't overdo it. If you are reading a chapter for the first time, then there's only so much you'll get out of it. You don't yet have the bigger picture that you will have when you finished more material. So while it is important to study a chapter well and to make sure you understand everything, but you should move on rather quickly. It is much more productive to move on and come back to things later when you have more perspective. So when do you know when you have studied it enough. I think that if you understand all the specific steps in the book, if you can solve the problems and see the big picture in the chapter, then you have done enough for now. Reviewing the chapter later on is much more important that spending a long time on one page.

 

[brunopinto90]

Good afternoon. I am planning on studying computer science or a math major, haven´t decided yet. I am passionate about programming, mathematics, pysichs and logic. I struggled at mathematics (3s and 4s out of 20, yes that bad!) because i didn´t see the beauty of it and now after becoming passionate, i am quite satisfied with my skills (got 16 out 20 in the national high school exam), but i could do much better. By the way i didn´t made any Math subject, so my exam performance was my final grade. I learned all the math by my self using Khan Academy, Explicamat (Portuguese website).

I am passionate about math, i took the liberty to dig deep and create insights, which most schools don´t do, the main reason, students fail miserably in the national exam, which tests students logical and analytical skills. I did so much better, despite self-learning, because i understood the concepts, didn´t just memorize formulas.

Since i am taking an engineer course quite similar to computer science or even a math major, i will be taking integral and differential calculus, complex analysis, discrete mathematics, linear algebra and calculus-based pysichs, i really need a deep understanding of the material covered in high school. I feel like i can to much better, so i am devising a plan to cover high school math material with more rigour, proofs included, so to speak, increasing my math maturity.

Why i am doing this? I don´t want to faill those math classes in the first year already. I want to be the best, i am willing to work to achieve such massive goal and for that i need the basics well developed just like a building a house.

I was thining of reading Basic Mathematics by Serge Lang. I don´t want some silly plug and chug exercises ( i had enough), i am looking for problem-solving exercises, word problems, proofs, logic, foundations, etc.. Will that book provide me such needs?

Short story: I want to develop a mathematics mind set and the foundations necessary to study harder subjects. What do you recommend me?

Thanks in advance.

 

[micromass]

I was thining of reading Basic Mathematics by Serge Lang. I don´t want some silly plug and chug exercises ( i had enough), i am looking for problem-solving exercises, word problems, proofs, logic, foundations, etc.. Will that book provide me such needs?

Yes, basic mathematics is a book that covers high school mathematics, but in a very mathematical way. The book of course offers plug and chug exercises (because they are always important), but they ask you to do proofs too. They cover logic and foundations a bit too. So I think this is the ideal book for you. Another good book to look at is Gelfand's algebra. This has very easy material, but the problems are very good and nontrivial. It also develops math from a very mature perspective: not just "memorize this", but "this is why this is defined like this, etc. " I suggest you get both Gelfand and Lang and work through them both. Gelfand has more books like a book on trigonometry and coordinates. As it happens, both Gelfand and Lang are top mathematicians, unlike many authors of high school books. So they really know what they're talking about on a very high level. Sometimes that is not good, but often it leads to valuable insights.

 

[brunopinto90]

Yes, basic mathematics is a book that covers high school mathematics

Plug and chug exercises are like drills in sports, you practice, practice and practice to eventually become second nature. I had enough, because i just finnished a exam, that's why. I am more in a problem solving mood. But of course i will do most of the Lang´s book exericses.

Thanks for the tip, i will take a look on Gelfand´s book.

 

[Hellmut1956]

Good afternoon. I am planning on studying computer science or a math major, haven´t decided yet. I am passionate about programming, mathematics, pysichs and logic. I struggled at mathematics (3s and 4s out of 20, yes that bad!) because i didn´t see the beauty of it and now after becoming passionate, i am quite satisfied with my skills (got 16 out 20 in the national high school exam), but i could do much better. By the way i didn´t made any Math subject, so my exam performance was my final grade. I learned all the math by my self using Khan Academy, Explicamat (Portuguese website).

I am passionate about math, i took the liberty to dig deep and create insights, which most schools don´t do, the main reason, students fail miserably in the national exam, which tests students logical and analytical skills. I did so much better, despite self-learning, because i understood the concepts, didn´t just memorize formulas.

Since i am taking an engineer course quite similar to computer science or even a math major, i will be taking integral and differential calculus, complex analysis, discrete mathematics, linear algebra and calculus-based pysichs, i really need a deep understanding of the material covered in high school. I feel like i can to much better, so i am devising a plan to cover high school math material with more rigour, proofs included, so to speak, increasing my math maturity.

Why i am doing this? I don´t want to faill those math classes in the first year already. I want to be the best, i am willing to work to achieve such massive goal and for that i need the basics well developed just like a building a house.

I was thining of reading Basic Mathematics by Serge Lang. I don´t want some silly plug and chug exercises ( i had enough), i am looking for problem-solving exercises, word problems, proofs, logic, foundations, etc.. Will that book provide me such needs?

Short story: I want to develop a mathematics mind set and the foundations necessary to study harder subjects. What do you recommend me?

Thanks in advance.

Hi Bruno

Due to other reasons to do with my hobby I have to acquire the knowledge as given in a math bachelor, as well as bachelor physics. So first issue was to teach myself mathematical thinking and so I found an offer from the university of Heidelberg were for free the lecture were offered as videos. Talking to the professor he told me that he bases his course on the 2 books about Analysis from Terence Tao and his course with honours. The books I found legal and free as pdfs at the homepage of Terence Tao, Analysis I and II. What I liked about his approach was that he spends comparatively a lot of time to teach mathematical thinking and prove thinking by using the natural numbers and moving from there. So the kind of statement, "as it obvious..." becomes none existing. I can highly recommend this book in english as the teaching at the german university is in german!

As nearly 4 decades have passed since I studied mathematics at high school and at my study for mechanical engineering, I soon found out that I had to refresh those topics teached at high school. So i found the courses of Calculus from MIT, OpenCourseware, 18.01 and 18.02, Single and Multiple variable calculus using the also free pdf book from professor Strang very useful.

 

[Franco_Carr14]

Whenever I study Mathematics, I always find myself highly irritated, I feel like I always have to remind myself of what I have already learned to be put in the right mind set, I can't just read a book without thinking about this stuff because I feel like I maybe losing knowledge. I'm always looking for a mindset before I read, but I find it a very arduous task.

 

[Hellmut1956]

Franco, if I understood you right, you are aware of the fact that mathematics requires to be tuned to mathematical thinking. About the course of Analysis 1 from Terence Tao of the UCLA he himself comments that by following his scheme in his course with honours his students start with less abstract and new concepts like those dealing with natural numbers i.e. to learn the mathematical thinking and its application to solve the mathematical proves. So his students the first couple of weeks advance less fast than those students of the "normal Analysis 1" course but later they catch up and pass those students due to have had the learning of mathematical thinking and its application to tasks!

 

[Franco_Carr14]

Franco, if I understood you right, you are aware of the fact that mathematics requires to be tuned to mathematical thinking. About the course of Analysis 1 from Terence Tao of the UCLA he himself comments that by following his scheme in his course with honours his students start with less abstract and new concepts like those dealing with natural numbers i.e. to learn the mathematical thinking and its application to solve the mathematical proves. So his students the first couple of weeks advance less fast than those students of the "normal Analysis 1" course but later they catch up and pass those students due to have had the learning of mathematical thinking and its application to tasks!

If I understand correctly, I should first develop the basics? That's great advice! I think I feel irritated because of my difficulty to concentrate while reading, but that's more of a personal problem, not unless you are willing to spare some advice for reading.

 

[Hellmut1956]

Well, I would say it is an iterative process of reading, then applying the reading to some problem hopefully available in the book you read and verify if what you think you have understood fits to solve the problem. An example of a good learning book is the one about calculus 1 from Gilbert Strang that is made available for free in the material accompanying the course about Calculus single variable from the MIT in teir free offering within OpenCourseware available in the internet. Here the link to the course supported not just by videos of the lectures given at MIT, but also uses the book from Gilbert Strang. You might see that as part of this course even the Assignment lectures are recorded as videos.
But in general it is to say that between believing to have understood something while reading it and getting the ability to apply it is a way to go. That why iterations in which the "already understood" text of a book should be reread. Happens to be that you catch new facets of the topic read a couple of times with exercises and a couple of days between each run!

 

[faiziqb12]

Yes, Rudin is a difficult book. It's not really suitable for self-study

difficulty is due to low IQ and matametical intution
dont forget that the book is meant for you and not the professors

 

[bacte2013]

Franco, if I understood you right, you are aware of the fact that mathematics requires to be tuned to mathematical thinking. About the course of Analysis 1 from Terence Tao of the UCLA he himself comments that by following his scheme in his course with honours his students start with less abstract and new concepts like those dealing with natural numbers i.e. to learn the mathematical thinking and its application to solve the mathematical proves. So his students the first couple of weeks advance less fast than those students of the "normal Analysis 1" course but later they catch up and pass those students due to have had the learning of mathematical thinking and its application to tasks!

The books like "Analysis I" by Terrence Tao and "Numbers and Functions" by R. Burn focus on teaching the construction of real number system and developing how to apply the real number system to the real analysis. Both books are incredibly strong books, but I think first few chapters from both books are enough to devel the mathematical thinking and the understanding of real number system. Another good book, but one I do not like that much, is "The Real Numbers and Real Analysis" by Ethan Bloch. He has a same philosophy as Tao and Burn, but Bloch's treatment already assumes the mathematical maturity from prospective readers, and he also does everything quite rigorously.

 

[Hellmut1956]

Well, I believe and it is my personal opinion that most of us probably always will always have room to improve mathematical thinking. But Analysis and Linear Algebra are fundamental basics. So far I have reached the opinion that all of the mathematics you learn as part of a bachelor study besides learning mathematical thinking are just learning a toolbox so to be able to really deal with mathematics. This even applies for at least part of the master study courses. Once you are through your mathematical toolbox will help you to know which tool in the bos of techniques you will have learned is applicable to a specific question you might deal with!

 

[Primrose]

Hello, I am self-studying mathematics in English. Just finished high school and Rudin is so much painful. However, at Uni my courses are in French. Do you think I should look for French textbooks?

 

[micromass]

Hello, I am self-studying mathematics in English. Just finished high school and Rudin is so much painful. However, at Uni my courses are in French. Do you think I should look for French textbooks?

Reading math in English is a skill you're going to have to master eventually. Most advanced books and advanced papers nowadays are English. When you write a paper to publish it, you will have to do it in English. When you have to give an international talk, it will have to happen in English. So you're going to have to get good in communicating math in English anyway.
So if you really feel uncomfortable with English language books, then sure, go search for good French books. But know that there is a huge variety of good English analysis books out there, while there are not so many French books.

 

[Primrose]

Thank you so much Micromass. I will do my best to master both.

 

[emma watson]

Are you self-studying mathematics? Do you have any questions on how to handle it? Anything you want to share? Do so here!

Mathematics is a vast and constantly expanding discipline, with numerous major subject divisions such as algebra, geometry, analysis, topology and hundreds of subdivisions. Just as with languages, different branches of mathematics may have different degrees of usefulness to you, or different aesthetic qualities in terms of the beauty of their central ideas.

So which should you select? To sharpen your focus on just those areas that might be of interested and relevance to you.

 

[Ahmad Kishki]

Certainly, but don't like post 10 questions at once. Only post like 3 questions at once and more questions if they get resolved.

In my opinion, proofs can be learned best by letting somebody critique your proof. So ask somebody to rip apart your proof completely. It is really the only way to learn. Watching somebody else's proof doesn't teach you much. Computational problems are very different though.

please rip apart this proof for me: https://www.physicsforums.com/threads/closed-set-proof.830944/

i am self studying real analysis fro Understanding Analysis by Stephen Abbott, and i must say, i am having the time of my life!

 

[Ahmad Kishki]

I am currently self-studying 6 subjects at a time. But I'm a bit extreme. I think 3 should be a decent number.

wow! i can barely manage 1 subject! but i get so consumed mentally in the subject, i just can't think of anything else. How do u manage 6 subjects?

 

[ohwilleke]

I have much advise, but I don't really know what you're looking for. But here's some things I would have liked to hear:

1) Get in touch with the profs. Many profs are more approachable than you think (while some are absolutely not!). Get to know them, go to office hours, ask questions, etc. And I don't (only) mean to talk about the class, but talk about other things in physics/math too.

2) Don't be discouraged by your class mates. While in undergrad, and while teaching undergrad I have seen many classes with bright students, but with an atmosphere that is very bad. Many would care about the grades only, and others openly disliked the courses. This reflected on the entire class. Don't let yourself be discouraged by them.

3) Don't care about your grades (only). I have mentioned this before in point (1), but there is more than grades. Grades =/= understanding (although there is a correlation). Focus on understanding the topic, not only on getting good grades.

4) Think before you ask a question. Don't just go to a prof and start asking a lot of questions without first thinking about it for a long time. Of course, if you REALLY don't know, then ask the prof and don't be afraid to do so. But it is worthwhile to think things through first.

5) Be sure to have fun too. Life isn't only about learning.

This is really good advice, especially (1), (4) and (5). Even if you have no formal affiliation with a university, most profs truly enjoy spending a little time with an earnest young math scholar who is talented and is asking thoughtful questions. Most of the biggest names in mathematics and physics (e.g. Nambu, Einstein) had a few people that they had this kind of relationship with, and it does wonders for the isolation you can feel working away for whole courses with little human interaction as well. And, many famous people in these field (e.g. Emily Noether, Oliver Heaviside, Leonhard Euler, and Srinivasa Ramanujan) were on the student side of these kinds of relationships at some point in their lives. These kinds of people can also make excellent references for college or graduate school.

 

[ohwilleke]

Emma Watson's observation flows from something more fundamental. Math is a very mature discipline. There is almost nothing in the mathematics curriculum even up to the 500 level graduate curriculum (with a handful of isolated exceptions such as fractals and certain kinds of optimization problems in linear algebra) that wouldn't have been familiar to someone like Euler, hundreds of years ago.

Physics isn't quite as mature, but it is close. Classical electromagnetism is about 125 years old, and classical mechanics, Newtonian gravity and first year calculus are about 350 years old. Even pure General Relativity hasn't changed much in the last hundred years, although there have been some advances in cosmology and our understanding of black holes based upon it. Obviously, there have been some new discoveries made in physics more recently, mostly in high energy/quantum physics, optics and condensed matter physics. But even there, the Standard Model is more than 40 years old, except for the fact that neutrinos have mass and the precision with which some of the constants have been measured.

Unless you are studying a field that is very new (e.g. string theory), it isn't important to get hot off the presses texts. Pedagogy most certainly hasn't made any great strides in the last four or five decades (although it does feel a bit lame and depressing to read a book that boldly wonders if man will ever make it to the Moon, or still thinks its trendy to call black holes "frozen stars").

 

[bacte2013]

I have been reading following two books, and I would like to take this chance to recommend them to others.

"Foundations of Analysis" by E. Landau
"A Concrete Approach to Classical Analysis" by M. Muresan.

Landau's book is great to learn the number systems and their construction. He basically give clear proofs to even trivial properties of the numbers. This book is great read before jumping into the analysis texts. I found Muresan a good complement to Rudin as he provide different approach to the proofs and thought-process behind many proofs and definitions. Professor Micromass, I would like to hear your opinion about them if you read them before.

 

[micromass]

Landau is a very good book. It is a classic for good reasons. The book "Real numbers and real analysis" by Bloch is somewhat similar in approach to Landau, but covers more.
I don't know the text by Muresan, but it seems to have some cool and nontraditional topics.

 

[bacte2013]

I really like Landau too. I read portions of the Bloch but I did not like it as much as Landau since Bloch is not concise and clear as Landau (personal opinion and taste). I really regret not reading Landau earlier since I had been facing difficulty with the number systems and their rigorous construction when studying the Rudin and Apostol. Now I finished reading Landau, I have better ideas about how to construct the number systems and implement them to the proofs.

Do you have any recommendation for the introductory books about mathematical logic? I would like to investigate this topic, but I am not sure which will be a good place to start.

 

[Ssnow]

For mathematical Logic I suggest '' Mathematical Logic '' Joseph R. Shoenfield , I think is the best.