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Many, perhaps most, math textbooks are very hard for most people to dive into. A few textbooks are delightful, and encourage people to learn math by their engaging tone. What textbooks have helped draw you into the subject, and what was it about them that did that? What recommendations would you give to an author based on your experience reading good and bad textbooks?

[I am asking this question because the closed question "Why are most math textbooks so uninspiring; unless you already like Math?" seemed like it raised an important point - which needed reframing.]

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    $\begingroup$ The answer will depend quite a bit on the student's level and background. And a textbook useful for classroom use isn't necessarily ideal for self-study... That said, some of the books I found inspiring and fun were deemed abstruse by classmates. $\endgroup$ – vonbrand Jul 24 '15 at 11:15
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    $\begingroup$ Motivate before using an "abstract" definition. My first year linear algebra course was so bad that it took me another 2 years to understand the tremendous applications of vector spaces outside $\Bbb{R}^n$ or $\Bbb{C}^n$. $\endgroup$ – Paracosmiste Jul 24 '15 at 13:25
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    $\begingroup$ I think examples are fairly important. Don't define a "Sheffer sequence" and leave it at that — give me a few to play with. (Also, nonexamples. If you define "delta operator" and give me a few, that's good, but it's also nice if you could give me a few things that aren't delta operators and explain why they aren't delta operators.) $\endgroup$ – Akiva Weinberger Jul 25 '15 at 0:19
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    $\begingroup$ Coming at this late, but I thought the word "accessible" in the title referred to disabilities, etc. Is there another word that captures what you are looking for? $\endgroup$ – mweiss Apr 5 '18 at 22:20
  • $\begingroup$ Related mathoverflow.net/q/13089/36173 $\endgroup$ – Paracosmiste Apr 18 at 14:39
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Among many other distinctions: textbooks that are stern, hostile-skeptical, unconfiding, are obviously-reasonable construable as "unfriendly". That is, if the author makes clear that you, the reader, are unworthy, and probably unable to understand the point anyway, it's off-putting. Similarly, as we know too well, it is possible to take a similar adversarial/hostile approach in "teaching", and students often default into an assumption that this is the case... which combines in a bad way with such a textbook.

The hostility can be veiled, as in text that is primarily definitions and theorem-proving with few examples, and "exercises" that are exclusively the working-out of the standard examples and counter-examples, but without any models in the text proper. Thus, a bait-and-switch on the reader, sometimes rationalized by the mythology that "problem-solving" is something that students can be coerced into, rather than shown by example, etc. Also rationalized by the unfortunate mythology that upper-division math starts to become a "test of talent", offering believers in that an excuse in case they're inadequate teachers, etc.

I see the same issues of "skepticism", let's call it, and "(passive-aggressive) unhelpfulness" appearing in texts and "teaching" throughout undergrad and graduate-level teaching at my R1 university. I've heard rationalizations that "if there are no losers, winning is meaningless"... and codeword "challenging" for unhelpful texts.

There're also the misleadingly-thin texts whose text is definitions, with most of the major theorems left to the exercises. Ack. Really nothing attractive in such a thing.

Sources (written or live) that show fun things, live(ly) examples to pique curiosity prior to "definitions", rather than throwing the important examples into an ocean of exercises so they'll be lost among drill-and-kill stuff.

Many people who're "successful" in mathematics have developed a resistance to that toxicity, although it seems to me a sad waste of energy to have to defend oneself in such fashion.

Somehow the style of mathematics textbooks has mostly drifted into fairly extreme stodginess and fussiness, although some of this seems to be merely meeting societal expectations (in the U.S.). It really cannot be that every detail is as important as every other, yet this seems to be a common presentation... seemingly compelling the reader to have to suspend their own judgement for the whole business to make sense, such as it may.

Worst-case: "math as rules handed down by authority" (The State, possibly?) No one should respond positively to that. :)

But the "standard curriculum(s)" I see in U.S. high schools and undergrad mostly seem to be the most inertia-laden of any part of the larger curriculum, and even many parents and students have a powerful expectation that math will not be interesting or fun, and will actually react badly if it is made into something else. I've had people (both students and parents) privately tell me that the best features of "math" do involve it being "a test of character" for them or for their kids. Yikes! :)

This creates a bizarre environment for successful (=adapted) text-book writing...

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    $\begingroup$ OK. Uh, are there any textbooks or even online references that you found accessible and engaging? Gerhard "Would Like A Positive Answer" Paseman, 2015.07.24 $\endgroup$ – Gerhard Paseman Jul 25 '15 at 5:52
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    $\begingroup$ @GerhardPaseman, as you can infer, I am put off by most textbooks. :) I do like Alain Robert's book on non-standard analysis, R. Strichartz' book on distributions, Folland's Tata lectures on PDE, Varadarajan's "Intro to Harmonic Analysis on Semi-Simple Lie groups", Edwards' book on Riemann's zeta, Gelfand-Graev-etal six volumes on generalized functions (including the 6th volume, on automorphic forms), Coutinho's book on D-modules, ... I've tried to write notes/books myself that were engaging, etc. The very concept of painstaking, drawn-out version of "undergrad" math is itself the problem. $\endgroup$ – paul garrett Jul 25 '15 at 14:44
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    $\begingroup$ I'm skeptical your approach works any better. I once spent the last week of an honors linear algebra class deriving the formula for the cubic discriminant using linear algebra on the space of symmetric polynomials. The class was bright enough to understand at least the basic idea. I got zero interest on the question. In fact, I know this persuaded one student to NOT major in math because it helped her realize that much of mathematics is about solving purely internal problems - problems that are about how to solve other mathematical problems (like telling when a cubic has a double root). $\endgroup$ – Alexander Woo Jul 28 '15 at 23:02
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    $\begingroup$ @AlexanderWoo, while I'd certainly not argue that all "internally motivated" problems are interesting to everyone, or even could be presented (honestly) as such, at some point a person's reaction to such things is sort of a diagnostic/litmus test for their natural inclinations. That is, after bluster and adversarial-ness is removed, it still doesn't follow that everyone finds math fun. And I don't want to hype anyone into thinking so ... $\endgroup$ – paul garrett Jul 29 '15 at 0:13
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I have no doubt there are more modern textbooks, but I found Fraleigh a fine introduction for unsophisticated undergraduates:

Fraleigh, John B. A First Course in Abstract Algebra. Pearson, 2003. 7th Edition.

The many short chapters, with both self-testing T/F questions, and achievable exercises, makes it easy on both the teacher and the student. He includes historical remarks, copious examples, and distinguishes in exercises between: computation, concepts, theory, and "proof synopses."


          Fraleigh
          (Fraleigh, pp.66-7, Sec. 6, Groups & Subgroups.)


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    $\begingroup$ Ok, I up-voted, and maybe the things you mention are what ("typical"?) students need, but I find it oppressive to do that sort of thing. It too-strongly reminds me that this business is so commodified that we "need" to do such stuff. I object mostly on the grounds that such diagnostics wildly delay getting-to-real-things. Typical undergrads, even pretty-good undergrads, are shockingly unacquainted with phenomena familiar to mathematicians in the early 1800s, ... due to the pedantic tendencies of "contemporary" textbooks. As though "defining a group" were a cottage industry... Problem $\endgroup$ – paul garrett Jul 25 '15 at 23:19
  • $\begingroup$ "What recommendations would you give to an author..." I think, with the books you (JO'R) have written, you are especially well-positioned to comment on some of your own thinking as you put together a text. (Maybe you have already written out these thoughts elsewhere...?) $\endgroup$ – Benjamin Dickman Jul 30 '15 at 3:49
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    $\begingroup$ @BenjaminDickman: I hesitate to reply, for two reasons. (1) Any recommendations not followed implicitly criticize authors, who have their own ideals. (2) It would be evident that I don't live up to my own recommendations. :-) $\endgroup$ – Joseph O'Rourke Jul 30 '15 at 12:14
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Here are some examples of really accessible textbooks:

In my opinion, the best textbooks take an approach of:

  • Limiting the scope to things that a non-expert reader is likely to find interesting.
  • Consciously treating the field as containing mysteries:

    • Asking "How can we figure something like this out?"
    • Showing that even experts make mistakes, and that ordinary people can figure things out.
    • Providing examples of the limits of what is known at the time.
  • Illustrate the ideas with pictures and/or examples.
  • Show a clear, practical, step-by-step way to solve each solvable problem.
  • Show a clear, practical way to check how accurate (or useful) each answer is.

These books tend to be well-organized, with big fonts and clear summaries. They emphasize "you can do it yourself" sanity checks rather than proofs.

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I wish I could remember more about why I found these books engaging. Although some of it is presented in a near-conversational style, I think a selection and arrangement of topics is also important. Two examples from my undergraduate education are Martin Braun's Springer undergraduate text on differential equations and applications (I fondly recall the section on the Tacoma Narrows Bridge collapse) and Munkres's undergraduate Topology text. A further example is Concrete Mathematics by Graham, Knuth and Patashnik. Each of these gave me the impression they were written by humans, or at least mathematicians who were willing to show a human side. However, I found none of these as engaging as books by Martin Gardner or Isaac Asimov.

I wouldn't mind having a Gardner or Asimov-like guide to read first. After that, I might be willing to tackle less friendly texts once I have had someone reassure me as to what was going on, and why some concepts were key to understanding the rest.

Gerhard "Should Read More Modern Authors" Paseman, 2015.07.24

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    $\begingroup$ "what was going on, and why some concepts were key": This captures it perfectly. This is all too rare in textbooks (and, not coincidentally, difficult for authors to provide). $\endgroup$ – Joseph O'Rourke Jul 25 '15 at 13:48
  • $\begingroup$ I have certainly seen Gardner and Asimov given descriptors such as engaging; but never have I heard as much applied to Munkres' "Topology"... $\endgroup$ – Benjamin Dickman Jul 30 '15 at 3:46
  • $\begingroup$ These things are relative. Of the books I have on my shelf, the ones I would pick up just to read (without a specific agenda nor to help for research) are those three. I might browse the other books on my shelf, but I have to motivate my self more to engage with those books. Then again, there are books ahead of those three that I would reach for. Gerhard "Engage Relative To The Reader" Paseman, 2015.07.31 $\endgroup$ – Gerhard Paseman Aug 1 '15 at 4:18
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What makes a textbook engaging:

  1. The absence of "it can be shown that" or "it is obvious".

  2. Clear writing. (Just general English quality of any writing.)

  3. Friendly "person to person" style of discussion. The actual definitions can be precise in terms. But the explication should be more the way you would talk when tutoring.

  4. Absence of typos or other errors like that (especially in formulas, derivations, problems.

  5. Answers to the exercises

  6. Programmed instruction.

  7. A preface aimed at the reader (student), not the teacher/specifier.

  8. Mentioning common errors ("watch out"). Note to convey this requires the writer know STUDENTS as well as he knows the content.

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I read Henri Roudier's Algèbre linéaire (in french) after reading Serge Lang's book. They are polar opposite in term of exposition, clarity, engagement and efficiency. I was forcing my self to read Lang's book back then to learn linear algebra but it doesn't seem Lang put any effort in this book; maybe he finished it in a year or two while copying the first chapter from his other books. It feels more like a handbook of theorems than a textbook.

On the other hand I read Roudier's book without much difficulty. While reading the book you feel like the author is human, like he's your professor in class or your more experienced colleague. He explains the concepts and motivate them, he proves the theorems and gives many remarks about the utility of these results, he warns you about common mistakes or abuse of notation, etc. The book is better than most of the famous linear algebra books.

For abstract algebra there's Godement's book. Not as engaging as Roudier's but still very clear.

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