College algebra

What Hacker News says

1. downbootsJan 2023
   Algebra Baldor !

   College Algebra Heineman

   Discrete Math Rosen !

   Linear algebra D lay

   Calculus Stewart

   Nonlinear Dynamics Strogatz +

   Combinatorics Mazur +

   Statistics *

   ESLR +

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2. jeremyisDec 2014
   I went from being a below average math student in high school to a really good one and it started by reading this text book and doing the problems in the back: http://www.amazon.com/Stewart-Redlin-Watsons-College-Algebra...
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3. rnewmeNov 2024
   List of good books, sorted by difficulty:

   - 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)

   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.

   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).

   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.

   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.

   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.

   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.

   Pay graduate students at your local university to tutor you.

   Take walks, they're essential for learning math.

   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.

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4. jcrOct 2012
   1.) My first compound question would be (as it has been in the past), "What are some good texts? And for what levels? (beginner, intermediate, advanced?)"

   For example, I've been eyeing the following two books since they're commonly used, but of course, commonly used is not a guarantee they're any good.

   James Stewart 2011 Algebra and Trigonometry 3rd Ed ISBN:0840068131

   James Stewart 2011 College Algebra Concepts and Contexts ISBN:0495387894

   As usual with university texts, we're dealing with the variance in "required material" and preferred teaching style for particular courses along with the textbook mafia exploiting the differences in opinions with multiple overlapping titles. I wouldn't quite call the multiple versions of the same material "cut-and-paste textbooks," but they're still really close.

   Sure, I've got the luxury of revisiting fundamentals for fun even though I don't necessarily need to revisit them again. As expected, when done, I'll probably have my own uneducated opinion of how good those texts are at teaching the basics, but that's beside the point -- You have the clear advantage of having far more experience with math, and hence, you'd have a much more informed opinion of good texts.

   1.a) "Other references or resources besides textbooks?"

   2.) My second question would be, "Why limit it to just `Linear Algebra`?" This could be my own academic ignorance showing through or possibly a language barrier of some sort, but there are so many different and useful applications of "Algebra" that limiting it to a specific area seems, well, overly confining and constrained.

   If you're just trying to be humble and only stick to stuff you know really well, I can empathize. There are a lot of smart math folks on HN, and some of them know particular fields extremely well, so venturing outside of your field of expertise can be a risky proposition. On the other hand, I know you're damn good at a lot of different fields of mathematics, so even if you aren't the resident expert of a particular field, your insights can still be very useful and beneficial --A good example is me asking you about textbooks even though I know you don't do the whole regular teaching thing, let alone spend your time evaluating each new edition and variation of books for the sake of potentially using them in a class. If someone wanted to know more about the application of Algebraic Theory to Automata Networks, your opinion is still a hell of a lot more informed than mine. (And no, I don't really understand it... yet!)

   3.) If you want to stick to the topic of just Linear Algebra, "Where is Linear Algebra useful?" The pure math folks are always amazing, but the engineering types want to know how to apply the knowledge to do useful work. The first step is simply knowing where things could be useful.

   3.a) "Why is Linear Algebra useful in particular areas?"

   The things you consider to be "completely elementary" or even blatantly obvious, probably aren't. I'd bet the reason why this submission hasn't resulted in a whole lot of questions is due to people being afraid of looking dumb, publicly. Some of the knowledge you take for granted is actually intimidating to others, or worse, they actually dreaded their mathematics classes and try to avoid the topic out of fear.

   http://24.media.tumblr.com/tumblr_l0z6j5sHtL1qz7s3bo1_500.jp...

   > So, exactly what is it that you think you want to know about Linear Algebra?

   If people don't know where Linear Algebra is applicable to do useful stuff, they won't think of things that they want to know.

   Without using a long string of expletives, I'm unable to express how much more qualified you are than I am for giving answers to 3/3a.

   4.) Lastly (and rhetorically), "Why do people tend to discount the things they know?" --Even if learning math was easy for you, you still put in a lot of effort, so you ought to give yourself more credit. :)

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