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Foundations How to effectively study a mathematics textbook?

Hello

I am looking to find advice on the most optimal and efficient way to properly and deeply study a mathematics textbook, such that I completely understand everything and thus giving me good foundations for future study of more advanced mathematics. However, I want to change my ways of studying, such that I am not wasting my time using woefully inefficient methods, which I believe I may be using currently in my studies.

Currently I am reading and self studying a basic high-school level algebra textbook. However, I am taking much longer than I want to and feel I should be taking.

Mathematics is cumulative knowledge, and so I fear, should I not make sure I understand everything in the textbook 100%, I will fail in the future when I try to study further advanced mathematics, because I might have gaps in my knowledge and understanding of more foundational mathematics. Such as basic algebra.

Therefore, I have acquired a bad habit of highlighting, underlining anything and everything I deem even remotely important that I might read in a page in the textbook. In every single page of my textbook, some portion of text is underlined or highlighted. I have fear that if I don't write down what I've just learned and understood, on a page, I will forget it in the future and won't be able to understand mathematics anymore.

But I actually feel this may be counter productive. Instead maybe I should read a page, make sure I completely understand it mentally, then instead of writing down notes and highlighting things in the textbook, I should actually just tackle the problems in the book about the topic I just learned, which will then actually consolidate my knowledge, understanding and memory of the topic I just read and learned from in the textbook, rather than making notes before I do the problems.


I may simply read that a+b = b+a for instance, and I'd highlight it. Then I might write something in the margins of the pages in the textbook about the new information I just read of, a+b = b+a. I'd think about it a little longer and come up with a new way to think about it or visualize it more intuitively and then I'd instead write a new stream of consciousness into my actual note book and potentially write a page, 2 pages or 3 pages to explain my new way of visualizing or thinking about something I just learned in the textbook. I usually never ever read these notes again that I've written and I feel like they are a waste of time almost. But I'm obsessed with making sure I completely 100% understand everything I read in a textbook in mathematics such that I don't have any possible gaps in my knowledge. I am scared I will not be able to learn or understand more advanced mathematics in the future, if I don't make sure to understand everything now, to give myself solid foundation knowledge of mathematics.


I am wondering whether, instead, to read a mathematics textbook efficiently, should I: Read the page in the textbook, think about it, make sure I understand it. Then when I've made sure I completely understand what I've just read/learned in the textbook, I will then solve the set of problems in the textbook to consolidate my knowledge and understanding. After that, I'll simply move on to the next page/topic in the textbook and repeat the same process.

Instead of taking many notes, writing in the margins of the book, highlighting and underlining everything. Then after all those notes, I would only then actually attempt to solve the problems in the book. But maybe, I should just read, make sure I understand what I've just read/learned, then attempt the problems in the book. Rather than, read the book, highlight and underline information, then make sure I understand, write notes, then do the problems. This process I feel like takes me 4 or 5 times longer than it would if I were to just read the book, make sure I understand the topic, do the problems and move on to the next page/topic in the textbook.

I feel like, if I understand something currently as I am reading it, if I don't write it down in the form of notes on the book and in my note book, I will forget it in the future. But I think that, by instead, making sure I understand it after reading it, but not writing any notes, instead keeping it in my brain, then doing the problems, will actually consolidate my knowledge much better than writing many notes?
 

Stephen Tashi

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Mathematics is cumulative knowledge, and so I fear, should I not make sure I understand everything in the textbook 100%, I will fail in the future when I try to study further advanced mathematics, because I might have gaps in my knowledge and understanding of more foundational mathematics. Such as basic algebra.
Students who study mathematics in schools and colleges usually don't understand everything presented in their textbooks. They also forget things that they once understood. Yet the educational system produces many competent mathematicians. So, as a generality, it is not true that the cumulative knowledge of mathematics must built up in completely systematic fashion.

It may be that your personality demands that you understand everything up to page N before proceeding to page N+1. That's a pleasing way to study, if you aren't concerned with any deadlines. However, if you are self-studying with goals that involve preparing for certain future courses by a certain time, that methodical style of studying will be too slow.


Instead of taking many notes, writing in the margins of the book, highlighting and underlining everything. Then after all those notes, I would only then actually attempt to solve the problems in the book. But maybe, I should just read, make sure I understand what I've just read/learned, then attempt the problems in the book. Rather than, read the book, highlight and underline information, then make sure I understand, write notes, then do the problems. This process I feel like takes me 4 or 5 times longer than it would if I were to just read the book, make sure I understand the topic, do the problems and move on to the next page/topic in the textbook.
Personally, I find highlighting text and writing notes is useful as a way to "make myself study" and avoid distractions, but as an aid to comprehension, it isn't very useful. Why hightlight something that you already understand? If you pick up a book you studied 5 years ago, you might find your highlighting and marginal notes useful. Examine your own psychology. Are you highlighting text mainly to give yourself concrete evidence of progress or to avoid distractions?

I you want to progress at a rate comparable to students taking a formal course, then, yes, get to the problems as soon as possible. Tackling problems reveals things we didn't understand from reading expositions. Students taking courses must get used to being embarrassed in this way.
 
Students who study mathematics in schools and colleges usually don't understand everything presented in their textbooks. They also forget things that they once understood. Yet the educational system produces many competent mathematicians. So, as a generality, it is not true that the cumulative knowledge of mathematics must built up in completely systematic fashion.

It may be that your personality demands that you understand everything up to page N before proceeding to page N+1. That's a pleasing way to study, if you aren't concerned with any deadlines. However, if you are self-studying with goals that involve preparing for certain future courses by a certain time, that methodical style of studying will be too slow.




Personally, I find highlighting text and writing notes is useful as a way to "make myself study" and avoid distractions, but as an aid to comprehension, it isn't very useful. Why hightlight something that you already understand? If you pick up a book you studied 5 years ago, you might find your highlighting and marginal notes useful. Examine your own psychology. Are you highlighting text mainly to give yourself concrete evidence of progress or to avoid distractions?

I you want to progress at a rate comparable to students taking a formal course, then, yes, get to the problems as soon as possible. Tackling problems reveals things we didn't understand from reading expositions. Students taking courses must get used to being embarrassed in this way.
Thank you for the reply.

I am self studying, such that I can gain enough knowledge, to enter a mathematics degree at university. The knowledge I must gain is that of high school mathematics, which goes up to about calculus. Once I am able to pass the UK university exams, that being A-levels which are completed in secondary school in the UK(In the US I believe they are called AP courses?). Thus I have to learn all of the mathematics curriculum up to calculus such that I can sit the A-level exams to be able to apply to university to then start a degree in mathematics. Therefore, I certainly am on a time constraint and I know that if I continue this method of writing notes all the time, I might not be able to learn all of the mathematics in a timely manner such that I can go to university to study mathematics.


I think then, since I must get to university as soon as possible, that I shall mostly focus on reading the book, understanding what is in the book, doing the problems, reading the solutions, then moving forward. Rather than taking many useless notes that may only slightly aid in my understanding. Most of which I already have gained by reading the book and doing the problems. And may actually become a hinderence because I am not learning or progressing anymore as I make notes, I am simply consolidating what I've already learned. Much of which can be consolidated by doing mathematics problems in the textbook. As you have said, the courses usually never teach to the depth or quantity as the actual textbooks, rather teaching certain parts and avoiding others, also maybe not going in depth as much.


I ask of you, how do you personally study a mathematics or science/physics textbook? How much of your time is spent reading and doing problems, versus, writing in the margins or highlighting? Or what techniques or methodologies do you have?


One of the reasons I spend so much time writing notes is because, first, I read what is in the textbook. I then lift my eyes away from the book and just sit and think for a couple minutes about what I've just learned/read. More often than not, I will find a way to intuitively think about and understand what the textbook just taught. Then, in this is moment that I find a new inuitive way to understand it, I feel the urge to document this new discovery I have made and to write it down in my notebook such that I don't forget it in the future. Although I think it is rather unnecessary as when I come up with these leaps in intuition, it sticks in my brain anyways, and doing the problems in the textbook consolidates that new understanding and intuition formed. So the notes are almost useless, 97% of the time I never read them again.

I feel like 50% of my time is actually writing in my note book. Then 10% highlighting/underlining and writing in the margins.

Doing away with writing big notes in my note book, and sticking to just writing the occasional thought or underlining the occasional thing in the textbook itself isn't so bad, as it doesn't take nearly as much time as writing multiple full sized pages of notes does. It's mentally draining to be honest. It's a habit, and I don't think it's a particularly useful habit, at least in terms of efficiency. Sure, if I had infinite amount of time it would be beneficial to write all my thoughts down onto paper and explain and diagram all of my ideas and intuitions onto a notebook. But it simply takes too much of my time, and actually subsequently stops me learning as much as I would otherwise. Halting progress. Than simply if I were to read, do problems, read solutions and move on.

So that is what I shall force myself to do and at least, try. Actually, I can't try, I must, I am aiming to learn calculus within about 2 to 2.5 years. But given that I am only learning prealgebra and algebra level mathematics, at the rate I am going, it would be closer to 5 or 5.5 years really. So I will need to change my study habits and methodology of studying mathematics, and mathematics textbooks.

No one has ever told me or showed me the proper ways to study, so this is what I had devised as I started reading mathematics textbooks a couple years ago.
 

Stephen Tashi

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I ask of you, how do you personally study a mathematics or science/physics textbook? How much of your time is spent reading and doing problems, versus, writing in the margins or highlighting?
I'm a retired guy and it's been over 30 years since I studied mathematics in college. In those days, I spent almost all my study time doing problems. However, "problems" included figuring out aspects of the material that seemed logically inconsistent to me. (I learned the formal and legalistic approach to math while a teenager, so I knew the expositions in some texts were inexact.) Trying to do problems led to reading the textbook and notes, especially reading definitions and looking at examples. So reading the text and sometimes highlighting and making marginal nodes did occur, but only happened as a "side effect", not as a preliminary. Of course, I attended lectures and in undergraduate courses, they gave satisfactory introductions to topics.

Nowadays, insofar as I study a topic in math, I pursue it as a recreation, without any deadlines. I make use of the internet and hardly every open a printed book. If I were still a student, I think that I'd make heavy use of online math course notes and videos. (They didn't exist 30 years ago.)
 
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Stephen Tashi

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One of the reasons I spend so much time writing notes is because, first, I read what is in the textbook. I then lift my eyes away from the book and just sit and think for a couple minutes about what I've just learned/read. More often than not, I will find a way to intuitively think about and understand what the textbook just taught. Then, in this is moment that I find a new inuitive way to understand it, I feel the urge to document this new discovery I have made and to write it down in my notebook such that I don't forget it in the future.
Intuition can be sophisticated or it can be un-sophisticated to the point of being useless or misleading. I don't know whether you are forming intuitions that are in harmony with the formal definitions of mathematics or whether you are constructing private interpretations of "what the book is really trying to say" that aren't what the book is really saying. Don't neglect the legalistic aspect of mathematics.
 

Dr Transport

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sitting down and doing problems will teach you more than reading and rereading a book, highlighting is fine, but nothing sets a concept more than the pen and paper approach.
 
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When I was a kid I liked to look at two or more textbooks for a given subject. Nowadays, as @Stephen Tashi pointed out, you can use the net to find multiple approaches to any topic. I think that having recourse to an alternative presentation of the material covered in an assigned textbook can help to provide perspective that can aid acquisition of firmer understanding.
 

Doc Al

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Let me echo some of the advice given by @Stephen Tashi and others.

sitting down and doing problems will teach you more than reading and rereading a book, highlighting is fine, but nothing sets a concept more than the pen and paper approach.
Exactly!

What really locks something into your brain is trying to pull it out. What they call retrieval practice. Read something that you want to really learn? Instead of just rereading it over and over (which can give you a deceptive illusion of mastery), get a blank sheet of paper and write down what you recall without looking at the book. Try to duplicate the proofs or principles that you are learning. And don't just do the same thing for six hours straight. Mix it up and space out your practice sessions. Test yourself by solving problems without looking up the solutions. (At least not until you've busted your brain and really tried to solve it.) Solve problems from two chapters back as well as the current chapter.

Back in my day, there was no internet or Google (yeah, I'm old :H). One advantage of that is that you couldn't "cheat" and just give up and Google a solution. I remember working on a problem for days (and arguing with my classmates over our solutions). Making things too easy can actually work against you.

Here's a book I recommend that has good advice on study strategies: A Mind for Numbers by Barb Oakley.
 
Let me echo some of the advice given by @Stephen Tashi and others.


Exactly!

What really locks something into your brain is trying to pull it out. What they call retrieval practice. Read something that you want to really learn? Instead of just rereading it over and over (which can give you a deceptive illusion of mastery), get a blank sheet of paper and write down what you recall without looking at the book. Try to duplicate the proofs or principles that you are learning. And don't just do the same thing for six hours straight. Mix it up and space out your practice sessions. Test yourself by solving problems without looking up the solutions. (At least not until you've busted your brain and really tried to solve it.) Solve problems from two chapters back as well as the current chapter.

Back in my day, there was no internet or Google (yeah, I'm old :H). One advantage of that is that you couldn't "cheat" and just give up and Google a solution. I remember working on a problem for days (and arguing with my classmates over our solutions). Making things too easy can actually work against you.

Here's a book I recommend that has good advice on study strategies: A Mind for Numbers by Barb Oakley.

Ah I see, so what exactly is the "pen and paper approach"? Is that taking notes? Or is that a different thing.

When I think of notes, I think of writing a stream of consciousness, pretty much writing your thoughts out onto paper. Then in my case, never really reading them again. Not really a retrieval practice as such, that is stated by you.

But it would seem that, the pen and paper approach is about recreating proofs and trying to understand how one got to the solutions and rewriting them and recreating them in your own personal format to understand the proof better, until it is intuitive to one's self?

I haven't exactly gotten to studying proofs as of yet, so this retrieval practice and pen and apper approach doesn't make as much sense to me currently.


Yes that does sound pretty advantageous in a way, not having access to google for solutions! But for me, it probably would be counterproductive due to the fact I am studying in isolation, so I would need some form of contact to someone to verify if my solutions I have created is valid or not. As I have no class mates or professors to discuss with! However, I will make an effort now in the future, to not give up on a problem easily. I'll stick with it for multiple days, or, just maybe never give up. Even if that means I don't solve it until I've already progressed further enough in my knowledge to then understand it months later. That is probably the best solution. I might even, try to find multiple solutions to a problem, and not check for answers/solutions until I am confident in my solution and that I believe it is correct, before checking the solution. Sometimes, if not many times, I may come up with a first solution that I absolutely believe is correct. Then I check it and it is wrong. However, I then realise, had I given myself more time, and maybe tried to make myself forget about my first initial solution, I might have been able to realise the actual correct solution. So it is hard to make yourself forget your first initial solution, or to stop thinking along the same path that got you to that initial solution. But rather to instead explore other avenues of thought and methodologies to a solution.


I actually have the book, "A mind for numbers" which I am currently reading. I had gotten about half way through it about a year ago, then forgot to continue it. So I am re-reading it, except in physical form, instead of how I had previously read it, on my iPad.


Last night, when I was searching online for the best methods to study mathematics and textbooks, someone on reddit suggested this method called the asterisk method for studying:


Is this a recommended approach?
 

PeroK

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I am self studying, such that I can gain enough knowledge, to enter a mathematics degree at university. The knowledge I must gain is that of high school mathematics, which goes up to about calculus. Once I am able to pass the UK university exams, that being A-levels which are completed in secondary school in the UK(In the US I believe they are called AP courses?). Thus I have to learn all of the mathematics curriculum up to calculus such that I can sit the A-level exams to be able to apply to university to then start a degree in mathematics. Therefore, I certainly am on a time constraint and I know that if I continue this method of writing notes all the time, I might not be able to learn all of the mathematics in a timely manner such that I can go to university to study mathematics.
If you are studying for A-Levels, I would take a look at this site:


My advice is to focus on the syllabus. If you don't understand a specific topic, then you need to tackle that. It's difficult to give advice without seeing what you can and can't do. You should be measuring yourself against how many of the exam questions you can and can't do.

The first thing I'd do is go through the syllabus and make an honest assessment of how much you can do and what you can't do.

To take a random example. Can you do this problem?


PS when are you planning to sit your A-Levels?
 
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Doc Al

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Ah I see, so what exactly is the "pen and paper approach"? Is that taking notes? Or is that a different thing.

When I think of notes, I think of writing a stream of consciousness, pretty much writing your thoughts out onto paper. Then in my case, never really reading them again. Not really a retrieval practice as such, that is stated by you.

But it would seem that, the pen and paper approach is about recreating proofs and trying to understand how one got to the solutions and rewriting them and recreating them in your own personal format to understand the proof better, until it is intuitive to one's self?

I haven't exactly gotten to studying proofs as of yet, so this retrieval practice and pen and apper approach doesn't make as much sense to me currently.
The "retrieval practice" I'm talking about can be applied to just about anything. Say you've just spent the last hour learning about trig functions and their interrelationships. Rather than merely highlighting text or rereading things over and over, grab a sheet of paper and see what you can remember about what you just learned. Ask yourself questions. See what you can pull out of your head. You'll have a lot of gaps, so you'll then want to go back to your source and patch up those holes.

Yes that does sound pretty advantageous in a way, not having access to google for solutions! But for me, it probably would be counterproductive due to the fact I am studying in isolation, so I would need some form of contact to someone to verify if my solutions I have created is valid or not.
Absolutely nothing wrong with checking your solutions, in fact, it's crucial to get such feedback. But don't cheat yourself by trying a problem, giving up after 30 seconds, then looking up the solution. You really need to struggle a bit.
 

symbolipoint

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Much of what you described seems mostly right. Textbooks should have example problems in each section. What you should do is read, think, reread, the topic section, and then try the example problems yourself, and compare your answer and your method with the solution as the example presents. Find and rethink your efforts for the example. Do not be afraid to then reread the topic discussion or part of it again.
NOW, do some, many, most or all of the topic exercise problems. Check each solution in the manual only when you are stuck. You really must try each exercise first, and check (1) if you are stuck, and (2) when you have finished the exercise problem. Slow maybe , but powerful.

Next, move on to the next textbook section. Review in the same way as the section you previously finished.
 
If you are studying for A-Levels, I would take a look at this site:


My advice is to focus on the syllabus. If you don't understand a specific topic, then you need to tackle that. It's difficult to give advice without seeing what you can and can't do. You should be measuring yourself against how many of the exam questions you can and can't do.

The first thing I'd do is go through the syllabus and make an honest assessment of how much you can do and what you can't do.

To take a random example. Can you do this problem?


PS when are you planning to sit your A-Levels?
What I am trying to do is to first give myself a well rounded general education before going to university. To make it much easier when I go to university. I.e, exposing myself to proofs, number theory and other mathematics not taught on the A-Levels really.


A-Levels are quite far away for now. I am only just studying basic algebra currently. I am not advanced as of yet. I guess you could say I am at GCSE level. However I am learning it much more deeply and making sure I understand it, rather than being able to do it, but not understand it. So about 2 years or 1.5 years would be a good estimate for learning the A-Level syllabus, and some more.


A-levels aren't particularly the goal. They are more of a side goal or a byproduct of the topics I'd like to learn before going to university. That does include calculus and proofs. Linear algebra and topology are on my list too. But I think it'd be best to learnt those while at university maybe? Since they are so advanced. It would be beneficial to have a mentor and peers to help with studying it.

However, I estimate I will sit them in 2 or 3 years. However, maybe with my new found studying method, which is getting rid of my note taking habit that slowed me down. I might do them in 2 or 1.5 years. Time will tell.


I will have to read spivaks calculus during that time frame. So maybe more like 2 or 2.5 years.

I also have to complete my current, part time, computer science degree with the open university. That'll be done in 1.5 years.


I'm 18 years old currently, so I will be maybe 20 or 21 when I go to university for my mathematics degree. I wish I could go sooner, so I wouldn't be so much older than my class mates, but not much I can do. Learning takes time. However, I do hope I can speed up this process with the new study methodology I'm trying to develop with the help of you people on the physics forums.


What I am understanding is that doing problems is better than note taking, and reverse engineering or rewriting proofs if also a good idea. Not giving up on a problem and preserving is also something that should be done. Rather than giving up quickly and then promptly reading the solutions.


I think that my note taking was mostly useless as I am not studying for a specific exam, as I am not needing to document specific details of any things to remember. Nor do I ever read them again.


Problem solving is much better for learning and consolidating new found knowledge. To learn mathematics, one must do mathematics I guess. So that also is reconstructing and thinking about how an author made and reached a conclusion in their proof.
 

PeroK

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What I am trying to do is to first give myself a well rounded general education before going to university. To make it much easier when I go to university. I.e, exposing myself to proofs, number theory and other mathematics not taught on the A-Levels really.

A-Levels are quite far away for now. I am only just studying basic algebra currently. I am not advanced as of yet. I guess you could say I am at GCSE level. However I am learning it much more deeply and making sure I understand it, rather than being able to do it, but not understand it. So about 2 years or 1.5 years would be a good estimate for learning the A-Level syllabus, and some more.
If you want to study mathematics at university in the the UK, then you need to have passed A-level maths. You also need to contact prospective universities to find the full entrance requirements.

I think you need to focus on what you need to study to go to university. The core examinations that you'll need to pass should take all your time.

The site I linked to above also has GCSE maths. You are at GCSE level if you can pass the GCSE exams.
 

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