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Comment by hdjjhhvvhga | Hacker News Reader

Having read numerous books on various aspects of mathematics written by the academia, my pet theory is that there are two kinds of people who write incomprehensible math books:

- 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.

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I recognize your two categories of bad math books and would like to add a third: commercial textbook publishers "padding" the pages with filler. It seems every book made for first-year undergraduate students is 10x thicker than it needs to be: there are these huge color images and lots of repetition, which makes students just tired and not want to read the book.

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).

I appreciate your No Bullshit guides a lot! Especially the fact that you provide the solutions to the exercises so that they're perfect for self-studying. I can imagine someone might like to have hundreds of color images - the assumption is they can help in absorbing certain concepts by linking imagery with abstract ideas.

My objection to this argument is that it seems to present "comprehensible" as the default outcome, and then derive confusion as a result of problematic thinking.

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.

OK, a fair point. Let me be specific. What I consider a good math book for undergraduate students should have the following:

- 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.

Would you mind sharing the names of the books that you’ve gathered for your daughter?

Sure. I feel that many contemporary undergraduate/college textbooks are actually fine in this regard (like Topics in Contemporary Math by Bello, Britton, and Kaul). As for the rest, some of my favorites:

- 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.

> - Gallian, Contemporary Abstract Algebra

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.

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