How many pages of math theory can you absorb in one day?

[andytoh]

Let's suppose you have a full day, free from classes or work, and you wish to read something new from a math textbook (which is moderately paced and at your current level), reading every single definition, example, and every proof of every theorem in each page. How many pages of math theory can you absorb in one day?

I have found that I can easily read a whole chapter, but the whole chapter does not really sink in, even if I read every single word. However, if I restrict myself to 10-15 pages, it all sinks in and I can absorb and remember all the content in those 10-15 pages. I'm concerned that 10-15 pages is too little. But any more, and I cannot retain it. I admit I am not a learning machine. But I want to fully, fully understand what I read, and really grasp the heart of the matter, and not just memorize definitions and results without getting a strong feel for them. So I slow down my reading intentionally. What do the others say?

 

[fourier jr]

depends on the subject. I've heard of someone who was going to instruct a 1st-year calculus course & to get solutions (because the answers in the back had some typos apparently) she solved every problem in every section of the edwards/penney text which was on the syllabus. that must have been about 400-500 pages or more & it only took her 4hrs. I've never tried but i think i could do that. it would get easier as i get back into it I'm sure. with something new of course it would be slower going, especially if I'm not very interested & i have to do it just because someone told me to. (if i do it at all in that case) i think i used to be able to handle a section or two per day, but with no distractions like work, classes, etc maybe i could do as much as 5 or more. maybe it would also depend on how easily things like examples come to me.

 

[radou]

The answer to this question is extremely relative and varies from person to person.

But one holds - there is no 'fast' math learning, as far as I know. Often the book/tutorial isn't enough (although it may contain numerous examples etc.) and requires from the reader to do some (!) additional thinking.

 

[cristo]

depends on the subject. I've heard of someone who was going to instruct a 1st-year calculus course & to get solutions (because the answers in the back had some typos apparently) she solved every problem in every section of the edwards/penney text which was on the syllabus. that must have been about 400-500 pages or more.

That's a bit different though, as that's not learning as such; if she's about to instruct a course, then she's got to be a mathematician who has learned that in the past!

 

[fourier jr]

That's a bit different though, as that's not learning as such; if she's about to instruct a course, then she's got to be a mathematician who has learned that in the past!

i guess it was all in her brain somewhere but on the other hand i think she had forgotten & had to re-learn at least some of it.

 

[andytoh]

there is no 'fast' math learning, as far as I know. Often the book/tutorial isn't enough (although it may contain numerous examples etc.) and requires from the reader to do some (!) additional thinking.

Yes, it is precisely this "additional thinking" that forces me to slow down my reading. Reading a math textbook, including all the proofs of theorems, is not like reading a newspaper and just collecting the facts. There is a lot of reflecting required. Also, many proofs and explanations have intentional holes and that you must fill in yourself to fully absorb the content. And even after filling in the gaps and understanding the entire proof, I must still reflect again (how did it work?) and understand the implications (so what does this signify?) before I read on. These are what force me to slow down my reading to 10-15 pages.

Some people may read a sentence or a step, not understand it, and then say "Ah, who cares? Let's just move on." But that obviously does not qualify as ABSORBING.

 

[cristo]

i guess it was all in her brain somewhere but on the other hand i think she had forgotten & had to re-learn at least some of it.

Yea, I suppose so, but then re-learning is a lot easier than learning for the first time; especially if it's only first year calculus-- although she may have forgotten, experience with solving problems helps a lot!

However, 400-500 pages; that's still pretty brave! I'd just get the solutions of the guy that taught it last year

 

[quasar987]

11-15 is average for me but I can easily get stuck and spend a whole day on essentially one page. >.<

Interesting poll though.

 

[andytoh]

Yea, I suppose so, but then re-learning is a lot easier than learning for the first time; especially if it's only first year calculus-- although she may have forgotten, experience with solving problems helps a lot!

Remember, I said reading something NEW and at YOUR CURRENT LEVEL (or slightly above).

Also, perhaps 2nd year math students or higher should only participate in the poll. I remember reading ahead chapters from high school textbooks in one day easily and absorbing everything (because there were no proofs involved).

 

[andytoh]

Mathwonk, as that you who can absorb 30+ pages in one day?

 

[Mindscrape]

If I read something on my own then I won't do much more than 10-15 pgs, but for classes, and I think this is probably the case with a lot of people, that each subject will require about 4-5 pgs of mathematics every day.

I would be more interested to see how much physics reading everyone can do because physics readings generally demand both physical concepts and mathematics.

 

[DeadWolfe]

Depends on the topic and it depends on the day. Anywhere from being unable to do anything to being able to read a short book.

 

[quasar987]

Mathwonk, as that you who can absorb 30+ pages in one day?

mathwonk didn't vote because there were no "under 1 page" option. I remember him saying it sometimes took him 1 week to plough through 1 page of Riemann's original work.

I want to know what Gib Z and Tom1992 voted.

 

[Tom1992]

I want to know what Gib Z and Tom1992 voted.

hmmm... i never really kept track. let's see: since we can only count proof-based textbooks, i cannot count the textbooks i read up to calculus 1. after calculus 1, i read about 10 math textbooks = 5000 pages. this was done over 3 years, but take off one year because the physics textbooks i read don't count, and take off another 50% of the time spent on my other high school commitments. that's about 5000 pages in 365 scattered days or about 14 pages per day.

i'm not the one who voted 30+ pages per day. perhaps that's matt grime.

 

[Tom1992]

let's get this right: at 30+ pages per day, that's 1 entire textbook in less than 2 weeks. and this a textbook of new material which is at your current skill level, and this is fully understanding the entire content of the book. this to me sounds like more than the completion an entire course in under 2 weeks. at this rate, you can master about 30 courses in one year so in essence finish an entire university degree in one year and also hypothetically get A+ in every course, since this poll asks about fully absorbing the content.

i personally cannot learn this fast! and this is from a 14 year old in 1st year university. and why the empty gap before the 30+ category? it must be the professors who voted 30+ (i guess they have to be this good else they wouldn't have become professors, and they probably have to absorb material this fast when they do background reading for their projects), and the students the others.

 

[haiha]

My idea is how many pages of maths you can read one day depends on the difficulty of the pages. I remember when in university, linear algebra took me a lot of time, but derivative and integration did not take that much.

 

[Plastic Photon]

why doesn't this poll contain fractions?

 

[Crosson]

i personally cannot learn this fast! and this is from a 14 year old in 1st year university.

In this case your fallacy was the assumption that anyone would, or could, read 30+ pages in one day on consecutive days for a sustained period of time (2 weeks).

In the poll I claimed to be a 30+ math reader, but this is because I only read new material when I feel up to it, which is maybe 2 days a week. One can joke of the wise old sage who ponders a single half page in a week*, but in fact reading mathematics that has already been written and solving problems that have already been solved is easy.

*It is one thing to read 120+ year old works of Reimann, there the difficulty is historical as well as mathematical.

 

[Gib Z]

Quasar*somenumbers*: I want to know what GibZ and Tom1992 voted. I haven't voted just yet, but its easily more than 30. However, I am talking about things that I admit are relatively simple. Did anybody notice I didn't understand murshid_islams derivation of \int_{-\infty}^{\infty} e^{-x^2} dx=\sqrt{\pi}, but the next week was advising some Physics student for help with the Cross Product?

I finished a Multivariable Calculus textbook in about a week (in the holidays, so i had all day), and most people here will say its pretty easy. However I am sure I still can't do every single question out of the textbook. Also note I knew about the first 2 chapters before hand.

I doubt I would be able to reproduce the ...hmm, 86 pages I think it was, a day of learning with any other topic of mathematics. I already had a very strong base in single variable calculus and multivariable was merely an extension. O and maybe I should mention the pages were abit smaller, so its about 60 A4 pages I guess.

Essentially: New topic to me, maybe 40 pages a day, if I am free all day.

 

[mathwonk]

not me, i was going to say something like <1 page.

my old algebra teacher maurice auslander used to say that if you want to understand what you are reading you need to write out at least 5 pages oer page read, so reading 15 pages would require writing over 75 pages.

i spend a day on one proof, or one line in one proof, like the easy proof that in a noietherian ring every non zero non unit has a factorization nito irreducibles. I know how to prove it, but I want to really understand the proof, and find the best proof. And I waNT TO CONVINCE MY STUDENTS THAT A "proof" like that in dummit foote is incomplete.

But this is easy textbook stuff. If I am trying to read a paper where I am actualkly eklarning new ideas or new techniques I may spend much longer. I have spent about 20 years reading Mumfords paper on prym varieties. I did peruse Spivaks volume 2 on differential geometry in one or two days, but notice I said peruse, not learn.

And I once read KodaIRA MORROW ON COMPLEX MANIFOLDS AND THE vanishing theorem in 5 straight days, but that was under pressure, no sleep, and again I did not fully grasp all that stuff.

 

[J77]

And I once read KodaIRA MORROW ON COMPLEX MANIFOLDS AND THE vanishing theorem in 5 straight days, but that was under pressure, no sleep, and again I did not fully grasp all that stuff.

Apart from that extreme that's all sound advice.

These young guys need to slow it down a bit - they'll burn out before they're 20 at this rate!

 

[andytoh]

To those who selected 30+:

Are you certain that the 30+ pages (of NEW material AT YOUR LEVEL) you read in one day are CRYSTAL-CLEAR to you, in the sense that if you were to write a test on those 30 pages the next morning you would do well on that test?

 

[Crosson]

in the sense that if you were to write a test on those 30 pages the next morning you would do well on that test?

Yes, and even better I could pass an oral exam and convince a room full of people that I knew what I was talking about!

These young guys need to slow it down a bit - they'll burn out before they're 20 at this rate!

A key to avoiding this is not working on Math when you don't want to, so that you form a good association.

 

[andytoh]

Yes, and even better I could pass an oral exam and convince a room full of people that I knew what I was talking about!

Then I must say you are a true genius. As you can see, the majority of us can only absorb 1-5 pages per day. You should study 30+ new pages everyday and become a truly great mathematician. I wish I had your learning ability.

 

[J77]

Crosson - how old are you, and what level are you reading at?

Are you reading stuff which has been around, as has been gone over for a long time, or more cutting-edge stuff, in the form of research papers?

If the latter, I don't believe you can give a full account to people in a day; ie. these papers never contain all the info you need - for this you need to research back and back, through many past references.

 

[Gib Z]

Well what I am doing is definitely not cutting edge lol.

O btw I am pretty sure I would ace the test, if i had written it >.<"

 

[Crosson]

Crosson - how old are you, and what level are you reading at?

I'm 21, and the material I am currently reading Poizat's Model Theory.

Are you reading stuff which has been around, as has been gone over for a long time, or more cutting-edge stuff, in the form of research papers?

If the latter, I don't believe you can give a full account to people in a day; ie. these papers never contain all the info you need - for this you need to research back and back, through many past references.

I absolutely agree, I think a few posts up I mentioned the difference between material that has already been thoroughly digested and other things like obscure or cutting edge research. The only research papers I read are in quantum physics/dynamical systems, but that is the difference between work and (what is for now) play.

You should study 30+ new pages everyday and become a truly great mathematician.

Unfortunately the ability to read, solve, digest, recite and perform textbooks does not a mathematicians make. I see little value in producing obscure research for the academic system, e.g. "Super-Edge Magic Graph Labelings", but in America this is what is encouraged. I think integrating the knowledge we already have is a more important goal.

 

[complexPHILOSOPHY]

Do you guys have true photographic memories, or what's up? I have a pretty phenomenal memory and solid visualization skills but I can't imagine keeping up with you guys, so I imagine your memories are remarkably powerful.

 

[andytoh]

By the way, if you haven't figured it out yet. People like Crosson, who can absorb 30+ per day (and hypothetically ace the test immediately after) are learning 6 times faster than the majority of us!

 

[Crosson]

Do you guys have true photographic memories, or what's up? I have a pretty phenomenal memory and solid visualization skills but I can't imagine keeping up with you guys, so I imagine your memories are remarkably powerful.

Not myself, but I have a similar friend who gets the ability from his photographic memory (interestingly he cannot remember smells and tastes at all!).

Spinoza said, of the three forms of knowledge: sensory, deductive, and intuitive, that only intuitive knowledge is true knowledge. The sense in which I agree with this archaic statement is that I gain knowledge by studying the process and not the details, which is why I can remember a math text much better than a fantasy novel (ironically E.A. Poe critiqued fantasy literature as being analytical in the sense that once the rules of the fantasy realm are established it is a formulaic process to translate our world into the fantasy world according to the rules; where as mathematics is truly creative ).

When reading, strive to create intuitive knowledge. Don't worry about getting every detail, because that is a natural consequence of having an intuitive feel of the process. That said, until one is completely comfortable with the style of mathematical writing, the going is tough. But after this initial barrier, it becomes almost embarassingly easy.

"In mathematics we don't understand things, we just get used to them" - Von Neumann.

What the master meant is that the feeling we call understanding is actually a sensation of familiarity; this is the reason for the uniformity of style across mathematical literature: new definitions in a familiar style are immediately "understandable", with the lucidity being nearly too much to bear.

 

[quasar987]

The cool thing about Spinoza is that it is so impossible to get what the hell he's talking about that we can make him say whatever we want. :D

 

[complexPHILOSOPHY]

Not myself, but I have a similar friend who gets the ability from his photographic memory (interestingly he cannot remember smells and tastes at all!).

Spinoza said, of the three forms of knowledge: sensory, deductive, and intuitive, that only intuitive knowledge is true knowledge. The sense in which I agree with this archaic statement is that I gain knowledge by studying the process and not the details, which is why I can remember a math text much better than a fantasy novel (ironically E.A. Poe critiqued fantasy literature as being analytical in the sense that once the rules of the fantasy realm are established it is a formulaic process to translate our world into the fantasy world according to the rules; where as mathematics is truly creative ).

When reading, strive to create intuitive knowledge. Don't worry about getting every detail, because that is a natural consequence of having an intuitive feel of the process. That said, until one is completely comfortable with the style of mathematical writing, the going is tough. But after this initial barrier, it becomes almost embarassingly easy.

"In mathematics we don't understand things, we just get used to them" - Von Neumann.

What the master meant is that the feeling we call understanding is actually a sensation of familiarity; this is the reason for the uniformity of style across mathematical literature: new definitions in a familiar style are immediately "understandable", with the lucidity being nearly too much to bear.

Well, my friend, I understand what you mean about intuition but I can't relate to the notion of maths being embarassingly easy. That is certainly an awesome statement to be able to make.

 

[J77]

Do you guys have true photographic memories, or what's up? I have a pretty phenomenal memory and solid visualization skills but I can't imagine keeping up with you guys, so I imagine your memories are remarkably powerful.

I have the memory of a goldfish.

Being good in a field is not about learning everything which has been written about it.

 

[Crosson]

Being good in a field is not about learning everything which has been written about it.

Indeed, it is a necessary but not sufficient condition

 

[cepheid]

Indeed, it is a necessary but not sufficient condition

Ouch! You just contradicted him soundly. He said it was NOT about learning everything that has been written about that field. You said that that was required AND MORE.

 

[Gib Z]

lol. My memory is not that remarkable, i remember the first 10 primes, and various mathematical constants and physical constants to about 10 decimal places, but anyone could have done that if they bothered to do what i did. Every week i would say nothing but those digits off a piece of paper, over and over. Some came easily, eg e approx 2.718281828, nice repeating 1828's. My point is, to remember all the mathematics you learn, you sort of need to learn to feeling of it. If you can remember the "feeling" of how to do it, youve got it. I know I am not very clear, that's just all I can say lol.

Or take Newtons Approach, 1% Genius, 99% Perserverence.

 

[Gib Z]

Did you expect an even or a normal distribution?

And this is an internet forum, why in hell would anyone bother lying to other people here who they do not now, will probably never see, and are here to help them learn anyway?

 

[mathwonk]

boy are you naive. we are building a totally artificial persona here that we live with in in our fantasies. E.g. I have pretended for years here to understand tensors, whereas actually they scare me to death.

 

[Gib Z]

We'll perhaps I haven't reached that level yet where the mathematics I am learning is too abstract for me to grasp.

and andytoh, why did you delete you post..if looks like i double posted talking to no one...

 

[complexPHILOSOPHY]

I have never forced myself to remember anything, that I can recall. I only read through what I want to, when I want to and absorb whatever my brain decides to absorb. That is seriously that only way that I can learn. Forcing myself to memorize and learn things that I don't feel like absorbing, never works. I compartmentalize and organize information that I become consciously aware of during my reading (e.g. I decide it's interesting or might have a relationship with something else) and then construct my cognitive model of it. I think my memory is more cue oriented.

Does forced memorization, like what Gib Z does, work well for some of you?

 

[mathmuncher]

I think it's dependent on how terse the book can be. Books with exhaustive rigor, while often long in content, can be a breeze. Rudin-like terseness could be more challenging, and necessitates more of me to absorb.

 

[Gib Z]

I usually do not do forced memorisation for anything other than memorising digits lol

 

[complexPHILOSOPHY]

Does it really function as an aide to have those digits memorized? I try my best to only use what I know in my head without reference to external sources (if absolutely possible) but if it is something like the digits of some number, I do not trust my head (well I do, but I know I don't make conceptual mistakes, I make arithmetic mistakes or I copy the number down wrong).

Does it help you? I can't see that helping me with doing abstract algebras or anything. Is it more for Calculation? Even then, is it really that much more helpful?

I am ignorant dude, so help me out!

 

[Gib Z]

Yes its pretty much only for calculation. I own a calculator, but leave it at home and perform everything by hand. Square roots, sines, logs, you name it. But seeing as I am only in year 10, The most labourous thing I calculate is sines, not so bad. It doesn't help at all when doing anything other than arithmetic.

 

[andytoh]

my old algebra teacher maurice auslander used to say that if you want to understand what you are reading you need to write out at least 5 pages oer page read.

mathwonk is quite correct here. Just yesterday, I decided to add a footnote to every statement made in one page that needed further explanation. I typed out my FULL, RIGOROUS explanation for each footnote I inserted. The page had many footnotes, and my explanations for all the footnotes took up just about 5 pages.

With this thoroughness of absorption, I am now in the 5 or so pages per day category. Incidentally, each footnote I add serves as an exercise, so not only am I reading the pages with full understanding, but I am improving my fluency in the topic by doing (simple) problems.

For those in the 30+ category, are you fully absorbing everything by doing these footnote explanations (either by hand or in your mind?), or are you just accepting every single statement you read on faith?

 

[interested_learner]

You can't "accept everything on faith" in math. That misses the whole point. The real point is to start with the assumptions and develop the math through the proofs. When you understand the proofs and can work the problems then you understand the math.

Frankly I find it hard to absorb much math at a sitting. Unsually I have to leave it for a day or two and then come back. Then it is clear.

 

[Gib Z]

If I am really interested in the problem, I will try to prove it myself. Unsuccessful, I will read a small part of a known proof, see if I can go from there. If not, next part, so on so forth. That helps me remember the proof, and therefore the theorem.

 

[JasonRox]

I voted 30+ pages. I can pretty much read 30+ pages of mathematics in one day and do some questions that's for sure.

But the reality is, it hasn't fully sunk in yet. I can be pondering the ideas for a few days, and do more questions as the days go by.

To fully absorb material takes longer than a day in my opinion. Just like working out, you need to rest, and exercise again.

 

[mathwonk]

I would be very interested to have some feedback on how rapidly anyone here can absorb my notes on my webpage. E.g. I have linear algebar notes there shorter than 15 pages, that cover a whole semester's linear algebra. Can anyone here read them in one day?

I have other notes on the Riemann Roch theorem, about 30-40 pages in loength. Can anyone read them in a week? I have a book of algebra there about 300-400 pages long. Can anyone master those in a month?


If not, quit kidding yourself that you can absorb 10-15-20-30 pages a day.

 

[JasonRox]

I would be very interested to have some feedback on how rapidly anyone here can absorb my notes on my webpage. E.g. I have linear algebar notes there shorter than 15 pages, that cover a whole semester's linear algebra. Can anyone here read them in one day?

I have other notes on the Riemann Roch theorem, about 30-40 pages in loength. Can anyone read them in a week? I have a book of algebra there about 300-400 pages long. Can anyone master those in a month?


If not, quit kidding yourself that you can absorb 10-15-20-30 pages a day.

He said a full day of free time. How rare is that? Quite rare.