How do you know when a textbook is too difficult for you?

Question

Not sure if this is more appropriate for here or for Math.SE, but here goes:

How does one who is self-studying mathematics determine if a textbook is too hard for you?

Math is hard in general, but when does a textbook cross that line from being challenging to being nearly intractable?

Sometimes I can't tell if I'm just being challenged when I have to re-read one paragraph ten times to understand what the author is saying (even if I understand all the components of their statement individually), or if the book is simply not at the right level for my current background.

Answer 1

I don't know how to decide, in general, whether a book is the right one or not. In particular cases, it might make sense to ask here, with the details.

But I have a great answer for the related question, "How do I approach reading a math book?"

It's very different than reading other books, and takes practice. (If a class I'm teaching uses a good textbook, I require students to take notes on each section we cover. That at least gets them started on actually reading a math book.)

Here is a great article about how to read a math book that I've modified for my students. (I mostly took out stuff that didn't seem vital, so my students would read the article itself!) One of the authors, Shai Simonson, has written a great book about learning math, Rediscovering Mathematics, which you might also find useful. (I avoid amazon, so here's where I'd look to buy it.)

I tell my students to expect reading the textbook to go slowly, and to expect to read it 3 times. Skim the first time to get a feel for the big picture, read carefully the 2nd time and mark anything that doesn't make sense, and work ahead of the author (in examples) in your third read. I think you'll find all that in this article, and lots more.

Answer 2

When you are self-studying, it is a much harder environment because (a) you lack external "whip" of the grade. And (b) you don't have a teacher, lectures, etc. With that in mind, you need to pick the books that are most beginner friendly. Especially useful are "programmed instruction" books (e.g. those by Stroud). Please ignore all the people telling you to learn calculus from Spivak or RA from Rudin. That's just Internet jocks, pecker-flexing. You can always go do the Spivak or Rudin LATER. But pick something easy to start with, that you won't give up on. Often Amazon reviews are helpful to find such books. (And also ones that have the answers in the back...very important for self studiers.)

Answer 3

Lovely question; I await better answers than mine; meanwhile here's my provisional one(s)...

tl;dr

(a) tempo (b) flow (c) practice-to-success

Note 1: These are related but different enough to merit separate discussions.
Note 2: The commonality and differenc s are clearer in music/arts than in math so I'll use that as a running example.
[Yeah when I was in school I dangled for quite a while choosing between STEM and music]

Longer Version

Tempo

Think of a normal person walking, an old disabled person, an infant just beginning to try and an olympic champion at their peak. Each has vastly different details of movement, but that's too reductionistic. Each has a very different tempo. You need to have a sense of the perimeter of your tempo and not fall wildly outside.

Note at the start this almost always will require some downscaling. Even the math prodigy will not be able to read math with the speed of novels, newspapers etc. So at the start you set the limits and then you work within that pushing against (ultimately yourself) a little at a time. Little is important as I show below!!

Flow

Compare Dinnerstein with Sokolov playing the same Bach. Do they sound same??

I'm not going to say which I prefer, still less which you should prefer. Reason I'm giving two very different renderings is to show that hi-tempo can be be enjoyable but is not necessarily better. One can tarry along enjoying -- so to say -- the flowers by the wayside.

When you're unable to digest a book/author X and resonate with Y, it can mean X is hard and Y is easy. But it can also mean quite simply that Y suits you X doesn't. Respect your own innate preferences.

One of the biggest figures of math-in-Computer-Science, Dijkstra, towards the latter part of his career was given a festschrift that summed up his work. The title is Beauty is our Business.

So that's flow, which is kin to tempo but different.

But this all applies to 'the great'. What about all of us who are far from there? In music as in math... it's practice.

However there's helpful, not so helpful, useless and outright damaging practice. The physical aspects which apply in music don't carry over much beyond specific instruments and genres. The psychological aspects are common to both music and math. I'll try and transmit a very crucial lesson I received from a music professor. And his lesson was:

Practice to Success not to Failure

[Firstly note how he adroitly changes the cliche Practice to perfection]
He said to us: When you walk the walls of a music school you'll often hear this: Then he sat at the piano and demonstrated:

He started playing the well known Mozart.
At first it was fine.
Then he started making mistakes.
Then he started showing more and more frustration.
And getting worse and worse.
Finally he slammed the keyboard and got up.

^{Of course a description does not work anything like a real demo; the closest I can think of are the pantomimes of Victor Borge but this was more literal.}

What he said after this was memorable and (I believe) applies to music math or any field where there is significant difficulty to be overcome. He said:

I believe this actually damages the nervous system. When you practice, stop when you're doing well not when you are doing badly. Practice to success not to failure.

I believe this applies very much to math. A good dose of math — whether reading, proving, problem solving, whatever — should have a resultant residue of a delicious coffee after a good sleep. It should refresh and invigorate. If its leaving a residue of frustration you're doing it wrong.