# Learning math - Maths: A Student's Survival Guide (ISBN-13 978-0521017077) - Review Text in Preliminary Mathematics - Dressler (ISBN-13 978-0877202035) - Fearon's Pre-Algebra (ISBN-13 978-0835934534) - Introductory Algebra for College Students - Blitzer (ISBN-13 978-0134178059) - Geometry - Jacobs ( 2nd ed, ISBN-13 978-0716717454) - Intermediate Algebra for College Students - Blitzer (ISBN-13 978-0134178943 ) - College Algebra - Blitzer (ISBN-13 978-0321782281) - Precalculus - Blitzer (ISBN-13 978-0321559845) - Precalculus - Stewart (ISBN-13 978-1305071759) - Thomas' Calculus: Early Transcendentals (ISBN-13 978-0134439020) - Calculus - Stewart (ISBN-13 978-1285740621) ## Learning Tips Beware, if you have a [toxic memory](https://supermemo.guru/wiki/Toxic_memory) you have to get rid of it first. Learning Mathematics is a tedious task that requires long periods of conscious effort and patience. I offer some tips which I consider to be of great importance when learning any subject within Mathematics (which could be applied elsewhere). * The main goal of learning is to understand the ideas and concepts at hand as "deeply" as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By "deeply" we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no "perfect" state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: "Young man, in mathematics you don't understand things. You just get used to them", which I think really means that getting "used to" some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination. * Motivation for any new concept is a must. This includes historical development of the subject which is sometimes crucial to understanding, analogies, drawings, and many other methods. Thought is induced by problems, questions, and misconceptions; thus, knowing what questions were asked in the mind of the mathematician who developed the subject and the problems he confronted really helps guiding thought in the right direction of understanding. * Always question the way the subject is presented. This includes questioning everything from the way terms are defined, to the way theorems are proved, even questioning whether the subject deserves the time and effort mathematicians put to it. Sometimes, there could be many different ways to define something; however, a particular definition is chosen among others for some conveniences and goals, so learning about these conveniences and goals would motivate the use of that definition. Some other times, more than one definition are studied independently so one can easily see the consequences of different definitions. We could use a good quote here from the mathematician Paul Halmos: "Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?". * Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap). * Be **metacognitive** (from **Metacognition** which literally means "beyond cognition", i.e., "beyond knowledge"), that is, be aware of your own knowledge and thoughts and consciously think about how you think and acquire knowledge. Thought is not passive, but an active process that could reflect on itself. Metacognition and consciousness help us monitor and regulate our thought processes to increase our potential to learn. This gives us the ability to evaluate our own performance by utilizing past thought experience. * Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding. ## Reading Tips How should one approach books? Should the reader go through every word from the first page to the last page? Should you solve every single problem? These questions are typical regarding book reading, and answering them is not a straightforward task. I will provide general guidelines, and accordingly the reader should find suitable answers for these questions. * What is the most fundamental purpose of reading? To learn, of course. So determining what you want to learn, determines what you should read. Not only what books to read, but also what chapters within a book to read. Sometimes it suffices to read the first chapter of the book, and in other times you have to go through all the chapters. However, one isn't always sure what to read and what to leave, and in that case only read the part you are sure you will need, then after going through other books you will eventually know whether you need to return to the book to read more. Moreover, reading books is not always a "linear" process, that is, sometimes going back and forth between multiple books is necessary. The reader should be critical to himself, and he has to assess precisely what he knows and understands and what he doesn't. * Sometimes, skimming (pass quickly through the text to note only the important points by looking for certain keywords) is possible; however, in some cases, you might arrive at a paragraph that you will need to read word by word. That is left to the assessment of your understanding. Patience is the key when dealing with books, so don't expect to go through a 100+ page book in one day and understand everything completely unless you have reasonable prior knowledge of the subject. * When reading about a new concept, try to predict what the writer will say before you read it. What (important) questions would you ask about this concept? how would you answer them? and what would you deduce from these answers? Before you read a proof of a theorem, try to prove it yourself first. If you could carry out the proof entirely on your own, then you will become more confident of your knowledge. If you become stuck, then when you read the proof you would embrace what you didn't know and you would hardly forget the proof afterwards. * It is possible to find repetitive exercises, in other words, you could go through several exercises that have the same idea which could be solved by the same method. In this case, solving one of them could suffice. Don't always count on your intuition, since one can think he has solved the exercise by just looking at it and at the end he finds out otherwise. Going through all the exercises of a chapter/section is up to you and your assessment of how good you did with the exercises you solved (and again, depending on the assessment of your understanding of the subject). * Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory. * Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note. ## Other Notes * If you want to "bring a nuke to a knife fight", you could work your way through all of the Art of Problem Solving books. They go much deeper than a standard curriculum so your foundation would be extremely strong (especially if you use Anki to schedule your review of problems/definitions you've understood and solved). Completing it would mean there's unlikely to be any math book that's outside of your reach. This is complete overkill, though. * Pay graduate students at your local university to tutor you. * Don't do exercises unless you want to. Completionism is a trap. * Take notes. Rewrite things in your own words. Imagine you're writing a guide for your past self. * Ask questions. Anytime you write something down, pause and ask yourself. Why is this true? How can we be sure? What does it imply? How could this idea be useful? * Cross-reference. Don't read linearly. Instead, have multiple textbooks, and "dig deep" into concepts. If you learn about something new (say, linear combinations) -- look them up in two textbooks. Watch a video about them. Read the Wikipedia page. _Then_ write down in your notes what a linear combination is. * Consider taking notes in LaTeX, but solve problems in paper textbooks. * The most key piece of advice might be to take walks. Walking is essentials for mathematics. * Depending on what you want to get a degree in, you can focus on particular areas and gloss over others. If CS, then discrete mathematics and logic are the most important (plus stats/probability/linear algebra for machine learning and AI). If engineering, then trigonometry/calculus/physics is more important. * Learning is a social activity, so maybe enroll in a community college course or find a local study group. I find it's especially important to have someone to discuss things with when learning math. I also recommend finding good public spaces to work in—libraries and coffee shops are timeless math spaces. Along those lines—as you're learning/reading/practicing new concepts, imagine explaining them to someone else (Feynman method). * Do lots of problems by hand on pen and paper, there is research and eons of practical experience that shows that doing math is a kinesthetic experience (that is, there's literally “muscle memory” for math). Draw pictures and graphs on paper. Keep all this scratchwork and doodles and stuff in a notebook. * Learn to process the “I have no idea what's going on” thoughts and feelings you get when you're faced with something new and challenging, or seem to continually forget things you've just (re)learned. That's par for the course, you just have to “feel the burn” and keep going. * Use tools like WolframAlpha to check your work and explore problems in more depth (like visualizing functions, seeing alternate forms). Get the app and use the solution explainer thing—essential for e.g. solving integrals. * It might be worthwhile to learn to solve and verify problems using some sort of (mathematical) programming language or a CAS. Could be anything, but something like SageMath comes to mind. Honestly, even Excel is pretty good for this. Being able to do simple things plot functions, verify work by plugging in values, simulate random numbers, etc. goes a long way. Developing this skills becomes even more useful (essential) when you're at the college level and beyond. * Khan Academy is not enough. It has broad enough coverage, I think, but not enough diversity of exercises. College Algebra basically is a combination of Algebra 1, Algebra 2, relevant Geometry, and a touch of Pre-Calculus. College Algebra, however, is more difficult than High School Algebra 1 and 2. I would tend to agree that you should start with either Introductory Algebra for College Students by Blitzer or, if your foundations are solid enough (meaning something like at or above High School Algebra 2 level), Intermediate Algebra by Blitzer. Basically, Introductory Algebra by Blitzer is like Pre-Algebra, Algebra 1, and Algebra 2 all rolled into one. It's meant for people that don't have a good foundation from High School. I would just add, if it is still too hard (which I doubt it will be for you, based on your comment), then I would go back and do Fearon's Pre-Algebra (maybe the best non-rigorous Math textbook I've ever seen). Intermediate Algebra is like College Algebra but more simple. College Algebra is basically like High School Algebra 1 and 2 on steroids plus some Pre-Calculus. The things that are really special about Blitzer is that he keeps math fun, he writes in a more engaging way than most, he gives super clear--and numerous--examples, his books have tons of exercises, and there are answers to tons of the exercises in the back of the book (I can't remember if it's all the odds, or what). By the time you go through Introductory, Intermediate, and College Algebra, you will have a more solid foundation in Algebra than many, if not most, students. If you plan to move on to Calculus, you'll need it. There's a saying that Calculus class is where students go to fail Algebra, because it's easy to pass Algebra classes without a solid foundation in it, but that foundation is necessary for Calculus. Blitzer has a Pre-Calculus book, too, if you want to proceed to Calculus. It's basically like College Algebra on steroids with relevant Trigonometry. Don't get the ones that say "Essentials", though. Those are basically the same as the standard version but with the more advanced stuff cut out. ## Resources - pauls online notes - https://tutorial.math.lamar.edu/ - https://www.youtube.com/@patrickjmt/videos / http://patrickjmt.com/ - https://math.hmc.edu/calculus/ - https://virtualnerd.com/ - https://www.youtube.com/user/professorleonard57 - https://ocw.mit.edu/courses/find-by-topic/#cat=mathematics - https://4chan-science.fandom.com/wiki/Mathematics - https://math.mit.edu/academics/undergrad/roadmaps.html - https://github.com/TalalAlrawajfeh/mathematics-roadmap - https://artofproblemsolving.com/store/recommendations.php Review again later - https://learnaifromscratch.github.io/math.html - https://news.ycombinator.com/item?id=22400375 - https://news.ycombinator.com/item?id=19811715 - https://news.ycombinator.com/item?id=20446796 - https://www.neilwithdata.com/mathematics-self-learner Stats: - https://projects.iq.harvard.edu/stat110/home (see book tab; Introduction to probability, blitzstein) - https://probability4datascience.com/ Old/unsorted notes: ``` https://terrytao.wordpress.com/career-advice/ Building mathematical maturity starting from nothing 1. Terrence Tao's Analysis 1, 2. "An Introduction to Mathematical Reasoning" 3. "How to Prove it" 4. Hammock's "Book of Proof" From primary to graduate: Use Khan Academy. Start at a level that feels too easy, even if it's elementary school math. The key to learning anything is to start at a level that feels too easy and gradually increase difficulty. As you finish a subject, see if there's a corresponding book in the Art of Problem Solving store [0]; you can revisit the subject at a deeper level that will strengthen your foundation. The AoPS books will also expose you to areas useful in programming like discrete mathematics. Before any of the above, take Coursera's Learning How to Learn course. You'll learn lots of effective strategies to get the most out of your efforts. For example, you can use Anki [1] to remember definitions and concepts you've managed to understand and to schedule review of problems you've already solved. [0] https://artofproblemsolving.com/store/recommendations.php [1] http://augmentingcognition.com/ltm.html core math subjects for engineering (khan academy) 1. Single Variable Calculus, 2. Linear Algebra, 3. Multivariable Calculus 4. Differential Equations university basic and addvanced https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-1/ Parts IA, IB and II of the Mathematical Tripos (Bacc) https://en.wikipedia.org/wiki/Mathematical_Tripos#The_modern_tripos https://www.maths.cam.ac.uk/undergrad/files/schedules.pdf the euclid https://personal.math.ubc.ca/~cass/euclid/euclid.html https://personal.math.ubc.ca/~cass/euclid/byrne.html ```