Share self-studying mathematics tips

[micromass]

What are your thoughts on sitting in on a class even if you have taken the class already? I have time available to me during the summer where I can sit in on two math classes I have taken before and did well but would like a review them and the instructor is known to be very tough so I was curious to see his teaching style. I could just review it myself with a book but the summer class goes at a much faster pace and its only a few week commitment and I feel it would force me to review.

If you can take the absolute boringness of seeing the material again, go for it! It could be a useful exercise. But I kind of think that self-studying (so you can choose what to go into more) is more worth it.

 

[micromass]

How long does it take you folks to make it through a whole book solving selected problems?

Depends on the book, really. But it can take me several months.

 

[ohwilleke]

How long does it take you folks to make it through a whole book solving selected problems?

I would typically finish a 16 week college course amount of material in 10-14 weeks. In part this is because taking a lengthy vacation from it is much more perilous than doing the same in a regular course. Get off track for too long and you may never get back. This said, the pacing of self-study for me is much more erratic in terms of material covered than a classroom course. I might spend three weeks on something that only takes a week in a classroom, because I'm stumped, and then breeze through two or three weeks of classroom material in a week because it all made sense to me.

 

[Hellmut1956]

I have gone through the description of the courses and first think I met was that you have to decide upfront if you want to learn pure mathematics or related to physics. I believe this is something you should seriously think about and make an as far as possible educated decision about it. The next think the document correctly states is, that the studying of the courses you have to take is very heavy!

This leads me to my suggestion. This suggestion reflects that due to health problems I am not able to sustain concentrated study efforts as required to follow the courses offered in a bachelor study of mathematics. but I have studied the course book, equivalent to the document yo link to, from the technical university of Munich, the institute of mathematics. Analysis and Linear Algebra are fundamental mathematical tools to grasp what ever is presented in courses later. So I found the video lectures from a german professor about Analysis to be the best fit to my way of learning. He bases his lectures of the 2 books from Terence Tao, UCLA, available for free legal download from the home page of professor Terence Tao. He starts rigidly to have its students learn to think as a mathematician by learning to apply the techniques of proofing starting with the natural numbers.Terence Taos course of analysis is course with honours, but thanks to his rigid methodology you do not phase the obstacle it represents at least for me, that mathematicians tend to see often issues as trivially obvious!

As to Linear Algebra I can only confirm, what has been written earlier in this thread, the courses from Gilbert Strang offered as part of MIT's OpenCourseware course offering. Different from what the one writing about his experience with this course and the "B" rating he received applying the knowledge from Gilbert Strang's teaching to his examin, I would only take this OpenCourseware offerings as a mean for preparing yourself for the courses at the university you will be assisting in person. Most of the other courses listed in the document you offered the link to are also in the course offering from MIT! It is my solid believe, that never ever you should learn for passing exams at school or university, but to learn and study to understand the topics for yourself. So preparing for the visit at the university studying with great effort and dedication the courses offered for example from MIT will take the pressure from you to follow all the courses you have to assist you at the university, as you would also already have a solid understanding, but it enables you to ask the questions left not fully understood as part of the self study and to grasp the concepts teached more extensively. A side effect will be that to follow the courses will not be as stressful as it would otherwise be and that your results in the exams I am sure will be excellent!

I found out myself, that in the 4 decades since I learned the mathematics a lot of my knowledge has eroded. So I did a step back and took the Calculus Single Variable and the Calculus Multivariable courses offered by MIT, if I remember properly those are the courses 18.01SC and 18.02SC. The courses follow the books from Gilbert Strang available for free and legal as PDF archives in the Internet. I even took this opportunity to learn to solve the numerous example problems in Gilbert Strang's book on Calculus with the software tool Mathematica in parallel to solving those assignments in writing manually to learn the contents and to learn to apply Mathematica to them!

 

[Loststudent22]

Is it reasonable to work through Calculus of Several Variables by Lang even though I was taught out of a easier book(anton) for single variable? I could pick up a copy of the first book from the library and probably go through it pretty quick since I know the topic.

 

[Hellmut1956]

Here the link where you can down load the complete book as a pdf archive for free and legal!

 

[Cruz Martinez]

Something I've been having some problems with, is how to avoid forgetting the proofs of particularly tricky theorems. Do you have any advise?

 

[tommyxu3]

When I tried to learn something before, what made me worried most was that there didn't exist many ones around me interesting in what I did. Thanks to the forums like this PF and others such as Art of Problem Solving, I have more opportunities to discuss my opinions with others! That's such a blessing for me!

 

[JorisL]

Something I've been having some problems with, is how to avoid forgetting the proofs of particularly tricky theorems. Do you have any advise?

The cliché; Try to understand what the proof does.

For me this often means making sketches for example visualizing the domains of functions when I have to look at compositions.
If it is in any way related to some geometric problem make a simple picture which shows potential problems.
This is not always possible or easy (a simple example is showing that the biggest fraction of the volume of a sphere is concentrated around the equator in high dimensions)

Next up is that you have to revisit theorems and such you are using. Here the pictures can speed things up again.
As they say a picture says more than a thousand words.

Finally think about the limits, if you need for example that a function is $C^\infty$, what goes wrong if this condition is not satisfied.
This is equally important as the picture for me. (and usually reasonably fun to do)

Note that this is my perception and method. For you another method could be better.

 

[Randy Johnson]

I am 17, and have finished all the mathematics courses my high school provides. I am staying back a year to play sports and work, but I am concerned I will forget or become "rusty" on the calculus, vectors, and functions, I have done so far. Is there any good textbooks you can recommend? I am very good at math and physics, and pick up on things quickly.

 

[micromass]

I am 17, and have finished all the mathematics courses my high school provides. I am staying back a year to play sports and work, but I am concerned I will forget or become "rusty" on the calculus, vectors, and functions, I have done so far. Is there any good textbooks you can recommend? I am very good at math and physics, and pick up on things quickly.

Is there anything specific you want to learn? What is your current knowledge? What do you intend to do with the math later in life?

 

[Randy Johnson]

Is there anything specific you want to learn? What is your current knowledge? What do you intend to do with the math later in life?

Very general, more to get a head start on university than anything else. My current knowledge for vectors is lines and planes intersections and relations. For calculus is basic derivatives (exponential, trigonometric, and polynomial up to 3 prime) , limits, graph sketching. I'm ok with proofs.
In physics we've done basic projectile, forces, electricity, waves and magnetism.
I initially planned to take some form of engineering (such as mechanical, physics, or electrical), in university, but recently I have been considering taking a general physics or math.

 

[micromass]

OK cool. Then I can recommend either studying a bit of calculus, or linear algebra (or both!).

You know calculus of course, but you can go deeper in it, for example, you can do integrals (which are among the most beautiful mathematical objects ever created!). I recommend the free book by Keisler: https://www.math.wisc.edu/~keisler/calc.html It will cover a lot of what you know, but I do recommend going over that stuff since Keisler has a truly original approach to calculus. Namely, he works with infinitesimals, which were the historic method of calculus, and which still remain very very useful in physics. If you truly understand this book, then you will have a very good intuition for calculus.

Linear algebra is a kind of generalization of geometry, but by using algebra. You know lines and planes in usual 3D-space, linear algebra generalizes it to higher dimensions. This is useful not only because it is cool, but also because many real-life phenomena have higher dimensions (you should see a dimension as a parameter or a degree of freedom). For this I recommend Lang's introduction to linear algebra (do not get his "linear algebra" which is more advanced). This will teach you vectors (which you know), matrices, vector spaces, etc. If you're comfortable with vectors and matrices AND proofs, and you don't mind a challenge, then you can't beat Treil's linear algebra done wrong (freely available again): http://www.math.brown.edu/~treil/papers/LADW/LADW.html

 

[Randy Johnson]

OK cool. Then I can recommend either studying a bit of calculus, or linear algebra (or both!).

You know calculus of course, but you can go deeper in it, for example, you can do integrals (which are among the most beautiful mathematical objects ever created!). I recommend the free book by Keisler: https://www.math.wisc.edu/~keisler/calc.html It will cover a lot of what you know, but I do recommend going over that stuff since Keisler has a truly original approach to calculus. Namely, he works with infinitesimals, which were the historic method of calculus, and which still remain very very useful in physics. If you truly understand this book, then you will have a very good intuition for calculus.

Linear algebra is a kind of generalization of geometry, but by using algebra. You know lines and planes in usual 3D-space, linear algebra generalizes it to higher dimensions. This is useful not only because it is cool, but also because many real-life phenomena have higher dimensions (you should see a dimension as a parameter or a degree of freedom). For this I recommend Lang's introduction to linear algebra (do not get his "linear algebra" which is more advanced). This will teach you vectors (which you know), matrices, vector spaces, etc. If you're comfortable with vectors and matrices AND proofs, and you don't mind a challenge, then you can't beat Treil's linear algebra done wrong (freely available again): http://www.math.brown.edu/~treil/papers/LADW/LADW.html

Thank you so much, I appreciate your time and advice. This truly means the world to me. Also, do you have any advice, suggestions, or anything else you'd like to share about future studies in university or future career paths?

 

[atyy]

You know calculus of course, but you can go deeper in it, for example, you can do integrals (which are among the most beautiful mathematical objects ever created!). I recommend the free book by Keisler: https://www.math.wisc.edu/~keisler/calc.html It will cover a lot of what you know, but I do recommend going over that stuff since Keisler has a truly original approach to calculus. Namely, he works with infinitesimals, which were the historic method of calculus, and which still remain very very useful in physics. If you truly understand this book, then you will have a very good intuition for calculus.

It's absolutely correct, but shouldn't an introduction start off with the more standard approach? Do Robinson's infinitesimals really correspond to physicists' infinitesimals?

 

[micromass]

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.

 

[micromass]

It's absolutely correct, but shouldn't an introduction start off with the more standard approach?

That is a matter of taste, I guess. If you do it right then there is very little difference between the two approaches. Both approaches have limits, derivatives, integrals, etc. The only essential difference is how limits are defined. There are other differences too, but it shouldn't be too difficult to translate between the two approaches.

A first course in calculus should not focus much on epsilon-delta definitions. It should be brought up, but it is too difficult for the students at that stage. Infinitesimals on the other hand are much more intuitive and do provide a solid basis for calculus.

I do think it would be a mistake to focus only on one type of approach. Both approaches have their benefits. The standard approach is beneficial because it doesn't need mysterious infinitesimal numbers, and because everybody works with this. The nonstandard approach is more intuitive, more historical, and offers useful ways of thinking. Keisler has both approaches.

Do Robinson's infinitesimals really correspond to physicists' infinitesimals?

Not exactly, but they come closer to physics' calculus than the standard approach. Face it, who uses epsilon-delta definitions in physics? A lot of the techniques in nonstandard calculus come up in physics, for example integrals as sums, infinitesimals, geometry with infinitesimals. So I feel that students might feel more comfortable if they already saw infinitesimals in calculus, even though they're not exactly the same thing. If you want a better correspondence with physics, then you'll have to look at constructive mathematics, and especially at synthetic differential geometry. But that would be completely unsuitable to teach newcomer to calculus. http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/comment-page-1/

 

[one]

Good evening everyone,
Over the summer I have been trying to study some extra mathematics to prepare for my next school year, but laziness has impeded my progress :)
I would like to know if there is some place on PhysicsForums for "study groups"; say, if several people wanted to study a specific subject, they could come up with a schedule of what to read and then they could have a thread for discussion, questions and such. Is there anything like this set up already?

 

[ohwilleke]

Something I've been having some problems with, is how to avoid forgetting the proofs of particularly tricky theorems. Do you have any advise?

Unless you are going to be tested on them, there is no good reason to memorize proofs of particularly tricky theorems. The whole point is a theorem is to avoid having to reinvent it from scratch each time you encounter a situation where it applies. Doing it once gives you confidence in the method that produced the theorem and a sense of how that kind of proof is done, but isn't something that you need to use on a regular basis. If there is some reason to avoid forgetting the proofs, however, recopying them into a journal from your local book store (I prefer the ones with pink unicorns myself), is an easy way to refresh your memory when you need to.

 

[arturodonjuan]

One thing I'd like to share from my experiences is that far too many of my peers get discouraged from self-studying mathematics because they "get lost". A mathematics textbook is (typically) not a story book, and so it shouldn't be read like one. I've learned that I personally need to go very slowly through every single statement/paragraph, theorem, lemma, and corollary very carefully, making sure I understand what is being presented before I move on. I know this may seem completely obvious, but I think it's worth saying given the ridiculous number of people that get discouraged by self-studying mathematics for this reason.

 

[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!