It's pretty useless if you want to learn anything
- Senior professors who actually suffer from the curse of knowledge and really forgot how it is not to know certain things, so they make tons of assumptions that are obvious to them.
- Junior profs who could actually explain the topics in an accessible way but do not feel secure enough and engage in a strange game of showing off. I know the same people could do a good job in the classroom, but once they get down to writing, they start to be afraid of being judged by their Senior colleagues so they follow the trodden path.
I gather the books that don't fail into these two categories for my daughter so when she grows enough to be able to grasp these concepts, she won't have to do dig through tons of crap.
It's completely artificial padding and unnecessary: you can teach calculus in 100 pages, you don't need 1000 pages (cf. Silvanus Thompson book on calculus or my books). I think the padding is done to make the exorbitant price tags seem more reasonable, so this is why I'm optimistic about this book since it's coming from the Eastern Block (no padding).
Anyone who has actually tried to teach students knows this is false. Merely trying to be understood fails with high probability. The average math or physics grad student upon entering already knows more than they have any chance of explaining thoroughly; distinguishing between senior and junior professors puts that line much further away than it really is.
It's plausible that a sort of follow-the-template dynamic entrenches bad pedagogy, but I would think that the reason authors defensively stick to the old patterns is not because they worry about being judged, but rather the concern that they will do students a disservice if they are not "better than the Beatles":
https://pubmed.ncbi.nlm.nih.gov/22378269/
In other words, rather than risk being blamed for using a progression that works poorly, it's safer to follow the old patterns, so that tradition is blamed instead — nobody ever got fired for teaching IBM^W the geometry sandwich.
- Explain the reason first instead of jumping into the definition straight away. I'm not taking about applications in physics etc., just a simple sentence like, "We have to learn series first in order to understand limits, and limits are necessary for understanding differentiation." Just one short sentence is enough to create a map in my mind and actually give me a decent reason to learn the topic. Seems obvious? Most math books chapters start with a definition.
- Give examples. Really. How am I going to even remember the topic if you have failed to give even one example?
- Give exercises for self-study. This is where the actual learning happens: at this point I can text whether I understood the theory or not. Moreover, it is through exercising that retention happens. Without exercises I can force myself to learn 50 pages and have only a vague memory of it the next day.
- Provide the solutions to the exercises. I get it, if it's a textbook, you want to separate them - that's fine. But not providing them at all means the books is only half-useful for self-study.
If a book has all these, I already consider it decent enough. Additional points for explaining particularly difficult points in more detail (good profs know well where their students are lost most often). If it makes sense, providing examples of practical application in sciences is always useful as it gives me some mental anchors connecting ideas and helping them to stick.
- Warner, Pure Mathematcis for Beginners
- Devlin, Introduction to Mathematical Thinking
- Stewart, Concepts of Modern Mathematics
- Herrmann, Sally, Number, Shape, and Symmetry
- Baylis, What is Mathematical Analysis?
- Feil, Krone, Essential Discrete Math for Computer Science
- Rotman, A First Course in Abstract Algebra with Applications
- Banjamin, Chartrand, Zhang, The Fascinating World of Graph Theory
- Zou, Mult-Variable Calculus: A First Step
- Hubbard, The World According to Wavelets
- Sayama, Introduction to the Modeling and Analysis of Complex Systems
- Darst, Introduction to Linear Programming: Applications and Extensions
- Sourin, Making Images with Mathematics
- Gallian, Contemporary Abstract Algebra
And many others. Of course, all such lists are completely arbitrary. Once I get familiar with a certain topic, elaborate explanations seem redundant and I feel like shouting, "Get to the point already!" - whereas the same explanations can be extremely helpful for a beginner.
That is an amazing book. I will also recommend "A walk through combinatorics" by Miklos Bona for simple explanations and well made exercises with solutions present in the book itself.
Not completely, because they'd still all read like they were written by a math grad student who's trying to impress their peers rather than communicate clearly. But it'd help a lot.
But that’s provided your general mathematics education is something like one or two semester-long courses away from the thing you want to learn. Otherwise, they’ll frequently be useless, and you’re better off turning to gentler introductions.
Wikipedia is not unique in this; many other technical reference books are the same, including the Springer Encyclopedia of Mathematics (a rebranded and somewhat expanded version of the Russian-language Matematičeskaja ènciklopedija), probably the best general reference on university-level mathematics ever (unsurprisingly, as a lot of it has been written by then-current or -future stars of Soviet mathematics, that being one of the few legal ways to earn additional money while holding a job in academia). Few references are good introductions. You don’t learn C from the ISO standard—or even Scheme from RnRS, as wonderfully written as the latter is.
I am quite literally furious over an accusation of a cleverness contest in a source the quality of mathematics Wikipedia ... But to direct this fury at you would be both wrongheaded and useless. Only, any environment where this kind of behaviour exists at all, in any way, is best exited as soon as possible and forgotten about. It’s just that if you happened to suffer such an environment previously (possibly unwillingly, such as in school), you may see signs of this even where there are none. The best way to avoid this false impression is probably to look not at whether some people (appear to) flaunt their knowledge, but whether others are scorned for not having such (to be distinguished from scorn for being unwilling to learn).
Pure mathematics departments are generally among the friendliest places I’ve been to, if you just show up with a question (and display signs of having tried to find an answer by yourself, even if the result is a completely arse-backwards, mangled parody of the subject). Applied mathematics departments too, by and large, but there’s a small minority of them where people are jaded by having to teach unwilling students and justify their existence to narrow-minded bureaucrats, so unfortunately I can’t just recommend them unreservedly.
> These definitions, which are not at all simple for the > beginner, came to be used in the wrong context. > Textbooks presented them before any explanation was > given of the theory and its applications, there > by complicating an understanding of things that > were intuitively clear.
hear hear!
This problem is like most of the Wikipedia pages that I come across on 'complex' topics of not just maths but anything remotely intersting technically. I just give up in the end. Life is too short to be bothered with some editor's 'dick measuring' competition about how clever they.