# How to learn math from textbooks in the right way?

I am a freshman studying math.

Everytime I learn Mathematics, I do it the slow-way. I tend to read the definitions until I make sense of what is written in the texts. I can spend a whole day trying to get one page of the textbook. I also do all the theorem proving by myself, even it takes me a lot of time to do that. I always get a good depth of understanding learning this way. I can also do a lot of hard problems given in the textbooks. (Not in a trivial way, it also takes a lot of time to get the jobs done.) The works tend to take a longer time than everybody else, but I do get the feeling I know the material better than other guys in my class.

The downside of it is that I don't have enough time to get all of my tasks done in a reasonable amount of time. I always wish to be able to do research in Mathematics, so I need a deep level of understanding. But as a slow runner, I think I have to change somehow.

So this question is made to ask for advice and tips in learning Mathematics. Also, please share with me how you work with textbooks. Thank you, Stack Exchange.

• I'd recommend asking this on Math.SE instead of here. That said, I suspect that most mathematicians (including me!) think this is an excellent way to learn mathematics. You might want to also adopt complementary study strategies, but please keep doing what you're doing!
– Anonymous
Dec 5, 2016 at 11:15
• @Anonymous Thank you for your answers. But I want to speed it up; do you think of any way to do it?
– Michael D Nguyen
Dec 5, 2016 at 11:33
• You'll get better with practice. If you need to complete a homework set in a hurry, you can skip reading the proofs and just try to apply the statements of the relevant theorems. But overall, learning mathematics is a marathon, not a sprint. If you are succeeding in learning the subject thoroughly, and you are enjoying the experience, and you are doing well in your classes, and you are whetting your appetite for further study and research, then I would suggest you keep doing what you are doing!!
– Anonymous
Dec 5, 2016 at 12:20
• @Anonymous This question is off-topic on math.SE.
– user1362
Dec 5, 2016 at 12:31
• I myself would like to see this migrated to either Math or Mathematics Educators. I am very interested in the answers!
– mhwombat
Dec 5, 2016 at 15:56

I tend not to answer the questions that I have no professional knowledge about or the questions that are too general to be answered here. But, your question reminded me of my own undergraduate years when I had the exact same strategy as you! I am writing this to confess that it was not a right strategy. Let me explain, using two examples.

Suppose you have just learned the definition of convergence of sequences. It is a hard definition and takes a good time to grasp. Then you have a couple of standard theorems: uniqueness of limit, the limit of the sum is the sum of the limits, and so on. If you are as you described you are, as soon as you see the proof of one of these theorems, you will be able to do the rest (yet, it would not be easy, but it is mainly algebra).

Now suppose you want to know whether you can count the real numbers or not and then prove your conjecture (just in case that you do not know the answer, I do not give more away). It was an act of genius to find and prove the answer to this question. Should you spend your time to solve this problem yourself? The answer that I wish someone gave me when I was younger is NO. You need to learn to respect what the others did, and more importantly, learn how they did it and move.

You need to find a balance between what you are able to do yourself and what you should learn from the others (lecturers, textbooks, friends,history, and so on). At the end, I really suggest that you look at some of the answers given to this question "Are there proofs that you feel you did not “understand” for a long time?" There you see examples of great and well-known mathematicians who were even memorizing some of the proofs when they where undergraduate students. There you see we all need time to understand some of the concepts, definitions, and proofs we read and that understanding only comes when we see those in the light of what we haven't learned yet!

• Very good. Indeed, many things are incomprehensible except "in hindsight", exactly as you said, in light of what we've not yet learned or seen. Many issues will simply evaporate in hindsight... so it would have been a mistake to spend too much time on them. Dec 5, 2016 at 23:08

Everytime I learn Mathematics, I do it the slow-way.

Slow is not good. But superficial is worse.

I tends to read the definitions until I make sense of what is written in the texts.

I wish every student did this. Fantastic, well done, don't stop doing this, ever.

Do, however, consider using alternate sources—multiple alternate sources—to look up key definitions. Think for yourself, and ensure that what you are reading makes sense to you.

The test of any “truth” is whether it is true for you. If, when one has gotten the body of data, cleared up any misunderstood words in it and looked over the scene, it still doesn’t seem true, then it isn’t true so far as you are concerned. Reject it. And, if you like, carry it further and conclude what the truth is for you. After all, you are the one who is going to have to use it or not use it, think with it or not think with it.

I can spend a whole day trying to get one page of the textbook.

To be frank—to me, this means your textbook is probably badly written. Sorry.

(Or, it means you're studying at the wrong level. A middle school student given a well-written college math text would spend a long time on each page, too.)

I also do all the theorem proving by myself, even it takes me a lot of time to do that.

Good. Keep doing this. There is no better way to learn math than by actually understanding and working through the concepts yourself.

You may want to start differentiating the important theorems from the unimportant. Start considering which theorems will expand your knowledge and understanding of math, or even just which theorems you are interested in. Which of the theorems have application or relevance to the understandings of math you already have?

It's not necessary that you perfectly understand every theorem you ever read...and even if you do want to attain that, you may attain it more rapidly with important theorems and proof techniques under your belt.

The works tend to take a longer time than everybody else, but I do get the feeling I know the material better than other guys in my class.

You're probably correct on both counts.

First, the article I've linked above contains some excellent tips on learning in general, not limited only to mathematics. Start with that. (You can turn off sound if you like, but the one-minute video is actually pretty cool.)

Then, from my own experience in studying mathematics, I will add some specific points:

• Ensure you get clear, accurate and precise definitions for every mathematical term used in your text. (And every other term, too, for that matter.)

If you have to fumble around mentally to remember what five of the words used in the sentence mean, you're not likely to catch the meaning of the sentence.

• The same goes for notation. Play with newly introduced notation until it is as clear to you as English, or maybe even clearer. The notation itself must not be a barrier to your understanding. Play with it, and practice, practice, practice.

You're learning a language. Learn it well.

• The ideas you are studying must become thoroughly familiar, common ground to you. You need to drill on the basics of the branch of math you are studying until you know them cold.

You haven't mentioned any specific branches of mathematics, so I will mention a couple examples myself from my experience as a high school math teacher:

Trigonometry students start flying after they are drilled and drilled and drilled and drilled on a simple right triangle diagram with a letter for each acute angle and a letter for each side.

"What's sine x?" "Uh, P over Q." "Good. What's arctan F over P?" "Uh, angle c." On and on and on (ensuring you wait for the correct answer each time no matter how long it takes, and then validate the correct answer when given) until the student gets every question correct without appreciable hesitation.

With several right triangles of different sizes drawn in random orientations, and with all different letters, this makes an excellent drill.

(Actually it's several drills, all with the same setup of several triangles. First, just do sines of angles. Then cosines of angles. Then tangents. Then randomly alternate between sin, cos, tan. Then cover cotangents. Then secants. Then cosecants. Then randomly between all six basic functions. Then start into the inverse functions: arcsine, arccos, etc. Always back it up if the student has trouble and make sure you allow the student to succeed; he should win at the drill.)

There is a similar set of drills for Calculus. (Gosh, I should probably write these up in a book....)

The point is that you need to get the basics of the branch of math perfectly understood without the slightest hesitation, so you can read fluently whatever is expressed about that branch of math.

One of the most thorough students I ever taught had this as his only shortcoming: He got everything 100%, but he didn't work toward fluency, so he remained slow. He didn't see $8!/6!$ and think instantly, "(seven times eight equals) 56" too fast to even think the words.

• The only other advice I have, is work everything out on paper or better yet, in your head. If you are using a calculator or computer to "avoid errors," you shouldn't be using a calculator or a computer yet.

Once you are 100% confident in your ability to compute the answers yourself, with pencil and paper, and are solely using a computer to save time—then, okay.

But, if you really know your stuff, the absence of a computer or a calculator won't bother you much. It hardly saves any time at all, if you get sufficiently fast and well-practiced.

• Downvoter, please comment. The above approach allowed me to pass through high school calculus by age 13. Why downvote it? Dec 9, 2016 at 1:01
• I could imagine that someone could object that "drills" are not-to-the-point. Although the "interrogation" model does unfortunately fit much school-math, this is not a good long-term mathematical world view. Sure, in the short term, exams are usually time-limited. But longer-term, there are no time limits, and the game (such as it is) is not about "lightning-fast reflexes". The "adversarial" aspects of school math are unfortunate and (in my opinion) not productive. Dec 13, 2016 at 0:45
• @paulgarrett, fair enough. I can see that perspective, although I was trying to make it clear in my answer (perhaps I failed) that drills are to acquire fluency with the basics of math. Nothing to do with adversarial aspects (which I do not recommend), nor with "lightning fast" anything...except for lightning fast understanding what you are reading about math. :) Dec 13, 2016 at 0:51
• I understand. A potential issue is the (unfortunately standard) parody of acquisition-of-fluency into speed tests. The genuineness of the issue resides (I think, at least) in the palpable fact that people tend to over-simplify, so nuances are lost. But, yes, I would whole-heartedly agree that engagement with ideas is critical. The appropriate form of "engagement" might vary from person to person, given the variation of mentation. E.g., I am not convinced that (to take an extreme) "doing all the exercises" is the best investment of time-and-energy. "Keeping things in one's mind" is best? Dec 13, 2016 at 0:58

Jump over the unnecessary parts. The point is that not all of the proofs from every theorem needs to be learned. Ones the theorem is proved, it can be taken for granted. Some proofs must be learned by heart, but those are the ones close to the subject one specializes to. There is simply too much to learn. Building intuition to the theorems is possible, without memorizing the proof. Learning to form the abstract overall picture is also important, because science is pushed forward by proving hypothesis, which in the end are just hunches about how the abstract overall picture could be.

Mathematicians tend to trivially see things, that's why there is unsolved problems in a form: proof if you can. This leads to mathematicians building bridges, math is not the same as building a house block by block. Seeing the solution is a great heuristic, even if the hunch is wrong.

The rule of thumb is to learn strategies and patterns which can be used for the proofs. Also if you are learning methodology, learn the main method. You only need to understand Principal component analysis (PCA) to be able to learn fast every PCA type of method.

EDIT: clarified the overall by merging with a comment.

• Sorry, but -1. OP said that h/she wants to become a serious mathematician. Taking the time to understand the proofs is extraordinarily valuable.
– Anonymous
Dec 5, 2016 at 12:15
• I edited my post to clarify. The point is that not all of the proofs from every theorem needs to be learned. Ones the theorem is proved, it can be taken for granted. Some proofs must be learned by heart, but those are the ones close to the subject one specializes to. There is simply too much to learn. Learning to form the abstract overall picture is also important, because science is pushed forward by proving hypothesis, which in the end are just hunches about how the abstract overall picture could be.
– user3644640
Dec 5, 2016 at 12:29
• Most proofs should neither be (permanently) memorized nor taken for granted. One should take away a good understanding of the overall strategy of the proof, and a confidence that one could either fill in the details or recall them quickly upon looking them up. Without that, you can't say that you know the theorem.
– user37208
Dec 5, 2016 at 15:53
• What is PCA? [and more characters] Dec 5, 2016 at 16:42

I use that way in my studying but that's not a right way. you do not need to make proofs by yourself for every theorem you met, learn how to use theorems is better, read the proofs rewrite the proofs in your words if you want.

Every time I see a question about mean value theorem I try to prove it and I do not SOLVE the question and when I read the proof, it used extreme value theorem so I start to think how to prove it and so on.

This way in studying is not suitable for students it is slow and you spend much time in it.

At the end we need to know how to drive a car but we do not need to know how the mechanism works.

I would suggest this :

(1) Always make sure you are able to read the symbols and the sentences in spoken mathematics

Example :

$$f(a) = 5$$ --> " the image of $$a$$ under function f is 5".

$$b=a^{-1}$$ --> " b is the multiplicative inverse of a"

$$b=a*c$$ --> " b is the image of the ordered pair (a,c) under the operation $$*$$"

{{1,2},3 } --> " the set having as elements the set {1,2} and the number 3 " ( this is therefore a set with 2 elements)

"(A-->B)-->C" --> " IF (A implies B) THEN C (is true)"

(2) Always make sure you know to what category every object involved in the sentences belong ( is it a sentence, number, a function, a relation, an operation)

For example: AInterB is a set ( not a sentence alledgedly meaning that set A and set B intersect or have anything in common) ; provided function f is differentiable at 2, $$f'(2)$$ is a number ( not a function); (4 - 4) is something, namely, the number 0, and not a " nonentity", or a " nothing", " the (indefinite integral) from 2 to 5 of (t²)dt" is not a function, but a number , etc.

(3) Reword every theorem as a practical rule , that is, rephrase every theorem ( if possible) as a method to prove things : " IF i know that .... THEN I know that ...", or " TO PROVE ... IT IS SUFFICIENT TO PROVE ...".

(4) find the contrapositive of each theorem ( everytime the theorem is an "if..then.." statement, or can be turned into such a conditional statement ) and express the contrapositive - in the same way as in (3) - as a proof method.

The real problem is the way that math books are written. Usually, they are written proof first. Then, maybe, something about the intuition behind it. When possible, aim for better books. Math books should be intuition-first, then the proof. This makes the proof much easier to follow. Otherwise you are looking at lemmas and theorems that are not connected to anything (and therefore harder to learn/remember). If, instead, we could see where we are going, learning it would be much easier. I say this in case there is someone writing math books - help us out by stating the intuition first, lay out the path of the proof, and then do the proof! We will all be thankful for it.

So, in short, the answer is find the best books. I will say that, if time is not an issue, what I tend to do is to read a book until I stop understanding it. Then, I put it back on the shelf for a time (month or year). Then, I pick it back up and read some more. Usually, by this time, the ideas have formed themselves better in my head, and I get a fresh look. I'm not stuck on the same thought pattern as before, and my brain has been working on it while I'm doing other things. This makes me much more likely to get past my prior issue.

If time is an issue, here is my suggestion - do multiple readings. The first time through, try to go through the whole book even if you don't understand it. This will help you see where the author is trying to go. Then, go through it to try to find the author's big picture points. Finally, go through it slowly to pick out the finer details. I think this will wind up being paradoxically much faster than just slogging your way through a page at a time.