Geometry

What Hacker News says

1. TorstenSchaal64May 2022
   Not sure if they can replace a good math teacher but some top picks for math books include "Geometry" by Euclid, "Algebra" by Harold Jacobs, "Trigonometry" by Charles P. McKeague, and "Calculus" by Michael Spivak.
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2. less_penguinyMay 2019
   Geometry: Seeing, Doing, Understanding by Harold R. Jacobs

   Preview: https://books.google.de/books/about/Geometry.html?id=u5o6DwA...

   You can purchase a complete solution manual separately, which is very handy.

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3. mathsciencekidSep 2016
   I grew up in a single-parent household back in the 70's before computers and startups were the thing they are now and although my mom was an uneducated immigrant, I was fortunate to grow up in the bay area and get an education at a good public school where my teachers introduced me to

     Harold Jacobs's "Geometry"
     Nietziche's "Thus Spoke Zarathustra"
     Freeman Dyson's "Disturbing The Universe"
     Nigel Calder's "The Key to the Universe"
     Jacob Bronowski's "The Ascent of Man"
     Douglas Hofstadter's "Gödel, Escher, Bach"
   

   I especially loved spending time in the library, reading OMNI magazine, Scientific American magazine (especially Martin Gardner's column) and the stories of Issac Asimov and Larry Niven.

   Those authors made me want to go to college and learn from people like them - people capable of thinking big ideas about science and civilization - and while none of those books ever helped me raise money or start a business, they did help me help me overcome my personal feelings of inadequacy from having come from a poor family and eventually become accepted into a technical community which appreciated an ability to think and ask questions.

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4. ahoyhereMay 2009
   In many countries, math has long been recognized as a tough subject that can be mastered through hard work, said Tom Loveless, a scholar at the Brookings Institution who analyzes international math test results and cultural differences. Efforts to make math more fun or dress up textbooks are not the answer, he said.

   OK, yes, but the state of American textbooks is TERRIBLE. They don't need shiny graphics or examples using social cliques. They don't need personality quizzies. (Ex-Winnie Cooper, I am pissed at you.)

   In the lower grades, some schools are subsidizing textbooks by purchasing ones from companies that use brand names in the problems. E.g. "If Tommy has 3 Oreos..." That is not the kind of dressing up they need, no.

   But if you compare your basic Silver Burdett piece of shit to a book like "Mathematics: A Human Endeavor" or Geometry by the same author (Harold Jacobs), you'll see the difference.

   I have the Geometry book. It uses Escher, Alice in Wonderland, and other great and useful devices for explaining math. (Hofstadter uses the same devices - not a coincidence, I am sure.) It uncovers math in everyday life and unexpected places. It reveals. It pulls back the curtain. That is the magic of learning.

   It does not pander, which is what all the other textbooks do. And, in fact, the entire school system does.

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5. 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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