How to choose a textbook that is pedagogically optimal for oneself?

I recently learned about mathematical maturity: https://en.wikipedia.org/wiki/Mathematical_maturity

Previously, I thought certain math topics were "hard" (e.g. category theory) while others were supposed to be "easy" (e.g. Calc I). I beat myself up for struggling with the "easy" topics and believe this precluded me from ever tackling "hard" topics.

I was thirty-something years old when I finally realized math has a well-documented maturity model, just like emotional maturity or financial maturity. This realization inspired me to go back and take a few math classes that I had previously labeled as "too hard," with the mindset that I was progressing my math maturity.

My point is that choosing an "age-appropriate" (in terms of math maturity, not actual calendar age) textbook is important. I also find it extremely helpful to chat with people who are more mathematically mature than I am, in the same way it's helpful to seek advice from an older sibling.

If you look for optimal you are going to spend more time looking for that textbook than learning.

The optimal solution is to find a good enough textbook and start as soon as possible to learn and tonstop procrastinating.

This echoes my own experience. I made an attempt at andrew Ng's revised machine learning course last year. I could understand well enough at a high level how gradient descent worked but its been 20 years since I did any calculus and the level to which my ability to even do algebra had atrophied greatly. To their credit, they derive the equations for you but I did't see how I was going to be able to really apply any of what I learned when the caclulus "handwaiving" is probably the most crucial step.

I'm a fulltime CTO so finding textbooks that can fill in the gaps and finding endless problem sets to solve was just not going to work. Luckiy, A good friend of mind from hack reactor clued me in to mathacademy. I would argue thats its probably one of the biggest underated resources for getting back in mathematical shape. I've been setting aside an hour a day to just grind through the lessons and problem sets that it throws at me. it uses spaced repetition along with an inital placement test to figure out what you're weak at and just hits you with those problems as you improve.

echoing the sentiment in the article, you'll get better just grinding though different problem sets consistently each day with the occasional metaphorical boss battle. Once you realize that, actually getting better at math is more of a logistical challenge (having to track down skill appropriate problems to cut your teeth on) Mathacademy basically automates that completely for you. I've gone from giving up on ever getting into this machine learnign stuff to looking forward to spending next year taking on deep learning.

PS: not paid by mathacademy.com. just an incredibly pleased custoner

Also PS: didn't realize you worked at math academy. any plans on expanding into physics problems? would LOVE these ideas to delve back into phsyics. (especially circuits.)

Don't!

Faithfulness to a single source is the biggest reason I see for failure In students. Be promiscuous. If a page, chapter, or even whole book bores you, scan ahead, put it on trial for a bit, and if it doesn't redeem itself quickly, replace it. The same goes (to the extent possible) for courses, teachers and even whole media. Only once you've tried the whole universe do you have reason to lower your standards and try something again from that universe that didn't meet your earlier ones. A book isn't a friend. There are no brownie points for completion.

Also most subjects are like that too. If you really want to know a natural language and hate the verb rules, focus on the rest of the language. If you soak up the verbs more slowly you'll still be hnderstandable, and you'll have fun, and most importantly you won't give up.

And programming languages are especially like this. Don't like class methods? Good! They suck anyway. Keep your functions pure. Don't like generics? Well that's a shame but it didn't stop the first many generations of Go programmers who couldn't use them if they wanted to. Etc.

There needs to be more resources for self-learning. Solutions need to be provided for problems, with clear explanations. It's a different mindset from a formal academic setting, where there's a strong focus on cheating prevention.

For a business I had to learn how to design parts for mass production injection molded plastic. It’s simple in concept but the devil is in the details, of which there are a great many.

I couldn’t find a general non-fiction book with the information I needed, so I found and ordered the best textbook I could find on the subject.

Teaching yourself from textbooks, I think you just have to be prepared for a serious grind, involving lot’s of looking up math and other terms that you either forgot or never knew, trips down the Wikipedia rabbit hole, etc.

Those books are, for the most part, designed as teaching tools to accompany classroom learning — sometimes the whole class is going to come and not have a clue what they’ve read, and it’ll be via class or office hours they figure out WTF is going on. These books are not designed for autodidacts.

I could be less charitable and talk about a lack of competitive pressure and perverse incentives for selection of academic books, but I’ll leave it at that.

Worked out for me and the manufacturer I was working with said we were the most professional part designers he’d worked with (we were helped tremendously by software I’d written), he wasn’t a bullshitter generally, so I’m inclined to believe it.

You can be successful but it’s going to take a lot more energy than it would with a nice trade book with an animal on the cover.

I have a PhD in math and read many textbooks. The tried and true approach: gather 20 books on the subject and read the first couple pages of each. It should jump out to you right away which one is the best for you.

My credentials for textbooks for self learning:

1. Must explain stuffs in a clear way.

2. Must give enough examples.

3. Must have many exercises AND a solution book for at least some of them.

Context: prepraing to study all undergraduate Math and Physics courses to get a holding of General Relativity. Since I graduated as a Math Master but forgot most of it, I have to start from Calculus and Linear Algebra. I count about 8-10 courses for the journey.

I download the most textbooks I can, for evaluating which appeal to me, and then use that one. If I don't like a chapter on it, I look for another one.

I thought it was just me but 2 mathematicians I look up to.

The guy behind Stat Quest & Harry Crane.

Both have explicitly said that there is simply no good book for their maths fields (statistics & probability).

This really needs to be fixed.

Since I alot of people think they are "not gifted" at maths when the real problem is that there is simply very bad study material.

My 2 cents on the topic is that for the most part I've had a lot of success choosing what to be interested in based on good book recommendations on the internet rather than looking for good book recommendations on the internet based on what im interested in

If you're learning for fun, probably every topic in the history of the universe can be interesting given the right approach

Some books that meet some of the described criteria:

* the books of RP Burn: https://www.amazon.com/stores/author/B001HP60DI/allbooks?ing...

* the abstract algebra text by Dos Reis and Dos Reis: https://www.amazon.com/gp/product/1539436071?psc=1

* the books by AOPS: https://artofproblemsolving.com/store

My experience with ex-USSR published books in mathematics and physics was very pleasant -and maybe the most pleasant ever. They rather provide simple examples and easy to solve exercises which thing always boosted my confidence psychologically to the point when I faced exams I was thinking: "This is just another set of exercises which I will succeed to solve, like I did multiple times".

I also had the same experience with the American published Schaum's books.

Apart from these, I almost got PTSDs with other publishers: they give you very hard to solve exercises to the point you would feel you are useless and incompetent to face even the easiest exams.

There is no optimal textbook. This is a constraint satisfaction problem, not an optimization problem.

ANY textbook you sit down and read, and solve its problem sets is infinitely better than ANY textbook you don't.

Stop bike shedding and start studying!!!

I found the text and workbooks for my mathematics courses at the Open University in the Netherlands absolutely fantastic. They are created / supervised by a famous (educational) mathematician in NL named Jan van de Craats.

The method was designed for self study, and the absolute best I had ever worked through. Perhaps material from other similar institutes are of similar quality?

https://nl.m.wikipedia.org/wiki/Jan_van_de_Craats

I also realized that a lot of specialized areas in Engineering are just preoccupied with a few specific concepts —- most are not incredible wizards.

I was weak in matrix/linear algebra. All the graduate students seemed obsessed with matrix decomposition, eigenvalues, and Hermetian forms.

After taking an optimization course (heavily matrix based) I realized they were just using the same small bag of tricks for everything.

I'm currently working through calculus. I picked up Spivak's and Apostol's books-- probably the most recommended calc books on the internet. Aaaand... they're ok. There are many parts that are confusing, not because calculus is "hard", but because the authors didn't do any user testing. If they actually reworked the books to minimize real students struggling, the books would have been much much easier to self-study from.

I eventually found David Galvin's calculus notes[1] from University of Notre Dame. He basically follows Spivak closely, but reorganized the material a bit in response to user testing. The notes aren't perfect, but much much easier to follow. Same experience with Terence Tao's linear algebra notes[2].

I think book authors, even very highly respect ones, often kind of suck because they optimize for writing a beautiful book, not for minimizing student confusion. Once you struggle through the confusing parts, yes, the book is beautiful. But it's supposed to be written for people to learn, not for experts to appreciate! Notes written by professors who teach smart kids, optimize for minimizing confusion, and do real user testing are often much better than the best books, in my experience.

[1] https://www3.nd.edu/~andyp/teaching/2020FallMath10850/Galvin...

[2] https://terrytao.wordpress.com/wp-content/uploads/2016/12/li...

Use the darknet. Get a list of possible texts, then download them. Figure out which one teaches the way you need to learn, then go buy that one. Used.

Exercises in textbooks usually focus on proofs, but mathematics isn't just about proving theorems. Mathematics is also about:

1. Understanding mathematical concepts (e.g. what is an "acyclic" relation? What is KL divergence?) and theories (several interrelated concepts, e.g. decision theory). This also includes knowing why those concepts are important in the first place, which is often neglected.

2. Knowing the meaning of mathematical notation and technical terms, e.g. to be able to read papers in some field. Papers are often full of mathematical and other jargon while otherwise not necessarily being difficult to follow.

3. Learning mathematical formulas (e.g. Bayes' rule) and algorithms (e.g. differentiation), in order to solve specific problems by calculation or computation (mostly in applied mathematics, more rarely in pure mathematics)

4. Proving conjectures (mostly in pure mathematics, less often in applied mathematics)

5. Learning how to formalize informal problems using mathematical concepts and theories (by applying conceptual understanding gained by 1) in order to understand the problem better, or to make it easier to solve, e.g. by employing calculation (2). (This is often done in engineering and science)

Problem sets in textbooks often focus on proofs (4) or some more difficult algorithms (3) but less on the other applications of mathematics.

They could also check conceptual understanding (1) by asking the reader to explain some concept in their own terms, or how two different concepts relate to each other, or which concepts various example cases have in common, or how the cases differ on a conceptual level. Though verifying the answers might require a human teacher.

5) could be taught by coming up with word problems from a scientific or engineering (or economics etc) example, where the solution is easy once the correct formalization is known.

Unfortunately it is hard to come up with such artificial word problems in which the correct formalization is unique, non-trivial, and doesn't require technical background knowledge from engineering/science etc.

Moreover, in the real world, the difficulty with formalization is often to recognize in the first place that there is some problem that could be formalized, which can't be replicated in an artificial word problem.

Overall, coming up with good exercises, especially for 5, but also partly for 1, might require the writer of the textbook to know a lot of possible practical applications. Writers of math textbooks are often mathematicians, so they probably don't know a lot about engineering, computer science, empirical science etc in order to come up with good word problems.

Ahhh, the good old gym analogy...

We use it

We love it

And it is our mainstay for understanding all things personally growth related

Where would we be without it?

We would be lost in darkness and ignorance

Just study all of them.

I avoid any book that have complicated equations on the first few pages. There is a place for that but in this case the author is just trying to show everyone how smart and complex they are rather than trying to teach people.

IMO textbooks are dead and they were never a great source of knowledge to begin with. The idea of reading some piece of text written by someone, then edited by someone else, and expressed in an "appropriate" style and language (think formal language which uses fancy jargon), and then having to robotically solve some end-of-chapter problems is just absurd. My experience tells me the "gems" of knowledge are often found when authors and experts just say whatever the fuck they want on a forum or in personal discussions. That obviously presumes those authors actually know what they're talking about. So many math, physics, engineering, etc. books are written by people who had no business talking about those topics.