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Demeter Exekutor

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Aug 3, 2024, 1:52:56 AM8/3/24

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I am very interested in mathematics, particularly mathematical physics, but I currently do not have access to an institution or any professors. As a result I have mainly been self-teaching myself through textbooks. My usual routine is to very carefully go through a chapter in a book, making sure I understand every step in every proof, and then move onto the exercises. Going through the proofs this way helps me measure which theorems/results are more important, and also helps me learn common methods of proof. This part of my studying is going fine and I'm able to finish the chapter and understand most of the material, sometimes with the help of other books or online resources such as math stackexchange.

The problem I have been having is with the exercises. I use these as a way to reinforce the material and test my understanding, but also as a way to develop my problem solving skills. I very rarely am able to solve a problem in any graduate math textbook I have read (which I have already finished 2-3), and have trouble even getting started with them. Eventually if I make no progress I open a question on stackexchange, or try to find a solution online which I usually am able to follow but I would not have thought of myself. I want to get to a point where I can actually do these exercises and not just be able to understand the solutions. I am beginning to question if I'm just not cut out for it or if my approach is wrong.

In summary, I am able to read about these topics and understand proofs written by others, but I struggle on actively doing the math myself. I am looking for some advice on how to develop these problem solving skills and do better on the exercises. I have taken a look at similar questions on this site such as:

The advice given in the intersection of all these posts seems to be more practice and to get help/hints from your professor. As someone who is self-studying and often gets stuck on where to even begin, I am not sure how to apply this advice and I would love to hear what working mathematicians would advise. For example, when I don't know how to start or I am stuck what should I do? I often think about the problem for some time but get nowhere, only to find a clever trick or small lemma should have been applied.

I expect that what you lack is appropriate mathematical experience and maturity. Graduate-level textbooks often assume a level of mathematical maturity that is different than the "logical prerequisites". You're also hampered by the lack of mentors/classmates to discuss things with.

My suggestion is to begin by going through easier (e.g., undergraduate) books in your area of interest, where you can do many of the problems on your own. That will help you naturally improve your problem solving abilities, and make it easier for you to understand more challenging books later.

You can also look on Math StackExchange (e.g., questions with the [book-recommendation] tag) for recommendations about books for either your particular interests, problem solving, proof writing, etc. And if you don't find exactly the the kinds of suggestions you want, ask a new question there.

The ruling class doesn't want critical thinkers.
It wants trained workers.
Apparently you've outstripped your school's ability to educate you. So move on from the Dummy Down Dunce Dance school you're currently attending. Learn math on your own.
Choose the topics that most interest you and study them.That is where you'll excel. Skip that I've got to curriculum, it's just baggage. If you bump into something that you like but lack the background, then you'll be motivated to study the background material.
Skip that high school stuff and look for a community college that'll accept you. You'll likely be happier there. Visit a community college, meet with a math teacher, and find what opportunities be there for you. I have a friend who got so bored with high school, she attended a community college instead where she was happier. I also know a self tutored math student who was accepted into a liberal arts college based upon his success in a college math exam.
PS. Skip the most modern text books and look for books written for mathematicians where the cook book method is ignored in preference to concepts, theorems and proofs.
Good luck with your struggle living in a country where being of above average intelligence is a handicap and with your new adventure beyond high school into adult education.
PSS. If you want a tutor, I'll give you a throwaway email address by which you can contact me. I have successful experience tutoring both elbow to elbow and by phone. Being retired, I've no need of payment.
I also suggest you use this web cite for questions and when you get stuck understanding a concept or a formula.

I wrote a book that may be helpful, as it covers the mental process of problem solving, something rarely taught in school. It has a series of engineering problems solved with the top-down approach, as a boot camp for the reader to get the hang of it.It's titled "The Top-Down Approach to Problem Solving", ISBN 979-8464073296. You can find it on amazon as paperback and ebook.I hope you like it.

I am OK at solving novel problems. I've been through general University calculus, and have always gotten mainly A's and B's in my maths courses. But I've mainly achieved this through picking up patterns, etc. and I always skip steps in my head which leads to problems when I get to tougher problems down the line. I find myself weak on the fundamentals, so I cannot solve higher level problems. I am taking discrete maths right now as a part of my Computer Science degree, but I find that the kids coming from math backgrounds are running circles around me.

As a result of all this, I'm not as good at solving novel problems that have new situations and contexts that I have no seen before as I would like to be. I don't intend on trying to become a mathematician by any means, but I believe that learning how to properly approach problems in maths will help me learn to approach other problems, both in Computer Science and life in general.

I am really trying to take a more math centric approach to solving the problems I come across in Computer Science and, right now, discrete maths. But I would like to maybe go backwards a little bit in my free time and work on bettering my fundamentals and learn how to better problem solve with topics that are more easily tractable.

I second Kolmin's recommendations especially "The Art and Craft of Problem Solving" by Paul Zeitz. It is a fantastic book. Another book on problem solving, focusing more on ways to think, for solving tough problems, is written by the greatest of them all, namely George Polya.

I highly recommend this book. Information in the II volume is priceless and there are hardly any other books which have given information as useful as in this book. This information may not transform anyone instantaneously into a problem solving genius but atleast, it mentions the kind of though process that is necessary for solving tough problems. This, I think, in turn helps to gain mathematical maturity at a much higher rate than otherwise. I can attest to the fact that this book made a tremendous difference in the way I thought about problem solving.

For the most part, this textbook is a great guide for understanding group dynamics and managing relationships in group settings. I appreciate the learning activities (e.g. quizzes, reflection & discussion questions) throughout several chapters. A few chapters seem out of place without relevant context or additional adaptation to demonstrate relevance.

The book seems accurate enough. I do question the accuracy of Wikipedia articles, however, which the text relies upon several times. I would supplement that material with other OER and/or journal articles.

Several sections are presented in a logical, clear fashion. I would appreciate covering group theories earlier in the text (this text places group theory last). The two writing chapters, for example, seem better suited for a writing course. The chapter on "Intercultural and Plane Crashes" offers necessary insight but the audience is clearly meant for those in aviation. With a few chapters seemingly out-of-place, I found the difference in content to be confusing. Some additional context and/or learning objectives at the beginning of those chapters would be helpful.

The book is written in accessible language, with practical learning activities and related resources interspersed. It was helpful to see the sample course syllabus and schedule, because it allowed me to consider similarities and differences with existing courses that may benefit from adopting this textbook. Although the conceptual frameworks provided in the textbook are relevant for graduate students, they are sometimes presented in a way that seems more appropriate for undergraduate students. If the book was used for a graduate-level course, I believe it may need to be supplemented with scholarly publications that highlight the related research.

The concepts are relevant for present-day application, including descriptions of many classical psychological experiments. However, in several chapters, I was somewhat disappointed the references were not more current and reflective of recent research.

The text is written in accessible prose, and many of the chapters contain appropriate attention to terminology. While many images and textboxes are visually appealing, some of the figures are not as crisp as I would like them to be.