How long does it takes to completely understand a page?

[kant]

How long does it takes to fully understand completely a "page" of math or physics? Now, I have a teacher that tell me that there are in general two style of reading: 1) 15 minutes of reading without writing, and the objective is to get the main points, 2) 45- x hrs of reading with writing, where the person write down each and every step( and perhaps fill in missing steps) to a the chain of reasoning in the page.

i asked 3 professors, and 2 said one should speed 45 min to x hrs on step 2, and one said it depends on the book. I pressume the two professor that gave a greatest lower bound estimate made the implicite assumption that the book is of averga quality, and difficulty.


The second question i asked is " how do we know that we know after reading a section?". The first two professor said that we should do a lot of exercise and problems. the last professor said that one should imagine giving a lecture to a imaginery audience, or to have someone to lecture to.

There are two question i want to ask in this thread.

Question 1: how much time does it take you to read an average difficulty "page" of math( subjective), and do you read it by writing down the chain of reasoning as you go along? Do you do some light reading before the pencil-paper intensive reading?


Corrollary question: Now, it seems that from the responds of my sample of 3 pro, reading a page of math does consume a bit of time, and that is only a single page. The are two 2 factors here. Factor 1) time complexity: you speed a lot of time in reading. 2) sustain concentration to read the page, or the whole section. My corr question is how do you sustain you concentration after reading a 45 min page of math?



Question 2:
How do you know that you know fully the content of a single "section"( not a page) of math? Do you do a lot of problems, and if so, how many problems do you do? Do you try to explain it to an "imaginary", or "real" audience? How do you know that you know?

 

[theperthvan]

That entire post took me about 45 minutes to read so don't anticipate many responses.

Just hit up the exercises in the end of the chapter. Invent some questions. Listen to Pink Floyd and power through it.

 

[kant]

That entire post took me about 45 minutes to read so don't anticipate many responses

1 reply for every 46 view. so far, the stat is not bad.

Just hit up the exercises in the end of the chapter. Invent some questions. Listen to Pink Floyd and power through it.

Well, i guess you are trying to be fun, because the otherwise option is absurdity.

 

[leright]

I agree with the prof that lecturing to an imaginary audience helps a lot. I do this all the time. Teaching and explaining things in your own way helps you understand things a lot.

 

[mathwonk]

pay no attention to how much time is passing. try to find a key idea and think about it until you understand what it says, what are some basic examples of it, what are its limitatiions, how to compute it ,etc,...

if there is even one idea on the page it may take avery long time for it to sink in. if the page is just historical discussiion, or a single worked example, it may take much less.

but note for example that there are only 4 basic theorems in calculus, and even these are related.

1) intermediate value theorem: the vakues of a continuious function defiend on an interval also form an interval.

2) extreme value theorem: the values of a continuous function defimned on a closed bounded interval, is also closed and bunded.

3) Rolles theorem: A continuous function which takes the same value twice on an interval, say f(a) = f(b), assumes an extreme value (max or min) for the interval [a,b] at a critical point in the interior (a,b).

4) mean value theorem: a continuous function on the interval [a,b] differentiable on (a,b), has derivative somewhere on (a,b) equal to the slope of the secant line joining the points (a,f(a)), and (b,f(b)).Hence a page containing one of these statements takes much longer to read than a page containing the year of death of fermat.

Even these 4 statements are highly repetitious logically, since 2) implies 3) which implies 4), and the method of proof of 1) is used again to prove 2).

But the main job is not how much time to spend on the avergae page, the main job is to understand these 4 statements, their meaning, application, proof, and limitations. Oh, and how do you know which pages contain important ideas?, read the introduction and chapter summaries. then look for the key topics mentioned there.

 

[mathwonk]

a philosopher once paid a man to translate the bible to him every day. they came to the scrpture where the writer says, "consider the tongue, it is one of the smallest members in the body but causes the most trouble."

The philosopher motioned the reader to stop, and said that is enough for today. He returned only after 6 months. The translator asked why he was absent so long, as he expected him to come every day. The philosopher said, "I meant to, but it took me this long to get that much."

At the end of the course on calculus, it becomes sadly obvious that some students have never gotten "that much" in the whole semester, and cannot state or use any of the 4 basic principles above.

 

[mathwonk]

If you want a simple average formula, take the number of pages needed for the semester, and divide by the number of days available, and that's how many to read per day. but allow also for rereading. so 300 pages in a 15 wek semester means one must master what? 20 pages a week? if it takes an hour per page on average, that 20 hours a week.

But I recommend taking breaks and discussing the reading with a friend to help it sink in. reading it helps only a little, reflecting on it,a nd putting it to work is crucial.

and my best teacher said we should write 3-5 pages per page we read, always, or else we had not learned anything. so think about how long it takes to write out 5 pages of work and examples, when measuring how long per page.

 

[mathwonk]

you may be talking abiut engineering, but if it is a math course you are interested in, and you will state the topoic, i will try to summarize the main ideas for you, if i know them. just so you will know which ideas are crucial to understand and focus on.

when i was first teaching claculus of several variables, i felt i did not understand stokes and greens and gauss theorems myself, so i did not know how to explain them. so i worked at it thinking abut what they emant until i saw how to use greens theorem to prove that a unit circle could not be pulled away from the origin without opasing across the origin.

then i realized that those theorems had strong topological implications, and i ued them to prove the fundamental theorem of algebra and the theorem on vector fiuelds on a sphere.

so being able to se how to use the theorems was key for me to understanding, i,.e. knowing when you know something. can you use it to do something else?

shortly after i did this i ran across an article in the math monthly doing the same thing but making it look harder.

i.e. in brief, the fact that the punctured plane differs topologically from the plane itself, is demonstrated by the existence of the angle form dtheta. this implies the fundamental theorem of algebra.

the thing that makes it computable and not just intuitive is greens theorem, that let's you draw a conclusion about whether the circle surrounds the origin, from just computing the integral of dtheta around the circle. so the whole fundamental theorem of algebra boils down to the fact that 2npi does not equal zero, unless n = 0, where n is the degree of the polynomial.

i.e. only polynomials of degree zero do not have roots.

 

[Equilibrium]

i heard from one of the threads here that there was professor who completed a math book in just a week

 

[complexPHILOSOPHY]

Honestly, listen to whatever mathwonk says, he speaks gold.

 

[Office_Shredder]

i heard from one of the threads here that there was professor who completed a math book in just a week

if someone gave you a book on arithmetic all the way through algebra (that's kindergarten through high school... 9 years worth of math!) how long would it take you to get through it? Two days?

 

[Equilibrium]

if someone gave you a book on arithmetic all the way through algebra (that's kindergarten through high school... 9 years worth of math!) how long would it take you to get through it? Two days?

no
and besides.. i just read it from another thred...
edit
what ? its not like I am boasting here... read my post...

 

[cristo]

no
and besides.. i just read it from another thred...

I think the point was that there is a difference between refreshing one's mind of material that one has learned in the past, and learning new material from scratch.

 

[kant]

If you want a simple average formula, take the number of pages needed for the semester, and divide by the number of days available, and that's how many to read per day. but allow also for rereading. so 300 pages in a 15 wek semester means one must master what? 20 pages a week? if it takes an hour per page on average, that 20 hours a week.
.

I will take your advice next quarter. With slight variation, i will divide the number of pages by the next h.w due day minus two.


It would really help me a lot if you could answer question 1 and 2 of my original post.

 

[Chris Hillman]

It would really help me a lot if you could answer question 1 and 2 of my original post.

Kant, I think it would be impossible to honestly answer those questions because there is no reason whatever to think that our situations are comparable to your situations in any way. Some of us have a lot of experience reading math/physics, which means we are much more efficient, but we are probably also "on average" reading much more challenging stuff, well beyond (as I guess) undergraduate textbook stuff that we learned years many ago (but might need to review from time to time).

Now, if you gave a URL giving the official syllabus for the course which is troubling you and asked for our recollections of how much time we spent on a similar course when we were students, we might come up with something, but even then, it seems clear to me that your assumption that there exists some useful rule of thumb for how much time you can expect to spend, is illusory.

I think the bottom line is: don't worry about what you can't control, and how much time you spend on each course is one of those things. Sometimes it happens that something you expected to take time T winds up taking 20T. Successful students/whatever adjust accordingly.

 

[Moonbear]

I agree with Chris that there's simply no way to answer the question for you. It will take however long it takes you to read a page, and will also depend on your ability in the subject, how fast of a reader you are, how dense the page is with information, etc.

As for how to know you learned what's on the page when you're done reading it, as others have suggested, practice the exercises at the end of the chapter or try to explain it to someone else (this is a reason study groups can be beneficial if used properly...you can explain to others your understanding of the subject material, which will help you see if you really understand it well enough to explain, as well as getting feedback from other group members if you explained it according to their understanding).

Not everyone learns the same way or works at the same pace. That's why you get different responses by asking different people.

 

[kant]

Kant, I think it would be impossible to honestly answer those questions because there is no reason whatever to think that our situations are comparable to your situations in any way. Some of us have a lot of experience reading math/physics, which means we are much more efficient, but we are probably also "on average" reading much more challenging stuff, well beyond (as I guess) undergraduate textbook stuff that we learned years many ago (but might need to review from time to time).

Now, if you gave a URL giving the official syllabus for the course which is troubling you and asked for our recollections of how much time we spent on a similar course when we were students, we might come up with something, but even then, it seems clear to me that your assumption that there exists some useful rule of thumb for how much time you can expect to spend, is illusory.

I think the bottom line is: don't worry about what you can't control, and how much time you spend on each course is one of those things. Sometimes it happens that something you expected to take time T winds up taking 20T. Successful students/whatever adjust accordingly.

I think i know myself more than you. For me, i find that i could understand the matter better and require less time than a lot of the people in my classes on the subject. I asked this question because i think it could help me understand myself better.

 

[kant]

Not everyone learns the same way or works at the same pace. That's why you get different responses by asking different people.

i disagree, I have talked to one field metalist, couple of ph.d s and professors in mathematics, and one math olympaid winner, and all seems to say pretty much the samething. They all seem to agree that learning math is hard work, and a lot of hand job. I am sure it is "hard work" to learn math, but i am sure there is a set of methods or tricks that one can use to be as efficient as possible. I actually share your thinking back in high school.

 

[cristo]

I think the bottom line is: don't worry about what you can't control, and how much time you spend on each course is one of those things. Sometimes it happens that something you expected to take time T winds up taking 20T. Successful students/whatever adjust accordingly.

I think Chris makes a very good point here. I recall various occasions where I set aside a certain amount of time to learn a topic, or perform a calculation, and it taking far longer than the time I set aside to do it in. This is not a problem as long as one takes account of this and adjusts future plans; for example, noting which topic took longer than anticipated to learn, and accounting for this when planning revision.

I know this is just reiterating Chris' advice, but it is important to remember that occasions like this should not be taken as set-backs, but rather as important points to remember for the future.

I think i know myself more than you.

So.. you ask for advice, then go on to slate a guy who gives you experienced advice...:uhh:

i disagree, I have talked to one field metalist, couple of ph.d s and professors in mathematics, and one math olympaid winner, and all seems to say pretty much the samething. They all seem to agree that learning math is hard work, and a lot of hand job.

Well, I think anyone who's studied maths to a high enough level will agree that it's hard work, so this point doesn't really help you when looking for "ways in which to learn."

I am sure it is "hard work" to learn math, but i am sure there is a set of methods or tricks that one can use to be as efficient as possible.

Well yes, there are, but they are different for each person. For example, if you asked people whether they worked better, say, in silence or with background music, you would get different answers depending upon the person. If you asked whether one works more efficiently in the morning, or in the evening, again, you would get answers depending upon the person. This question which you seek an answer to "how long does it take to completely understand a page of maths" is, like the above questions, subjective, and therefore you will not be able to obtain one definitive answer. Try different techniques, and see which one works best for you.

 

[Office_Shredder]

I think the best way is to read through a section, and say to yourself 'what did I learn?' Then re-read the section and see what you didn't learn. Multiply how long it took you to do the section by the ratio (total stuff to learn)/(stuff already learned) and you know how long it will take

That's actually quite an elegant formula

 

[mathwonk]

It takes me a really long time to read even one page which has any ideas on it at all. But Auslander was a really good teacehr and his formula was to write 5 pages per page read. So read the page then write down as you suggested, the key line of reasoning on that page, thenr read it and think about it, and se if you can do the problems from that section. Then try to remember what you have read. if you cannot even write out the statement of the theorem in the section then you sureloy have not mastered it.

it is very ahrd to be sure you have mastered a given section. I like my criterion of can you use it to do something else? Also I really like discussing it with friends. See if you get it better than your friend. if so you feel beter and you have helped your friend. if not your friend has helped you.

To me math is elaned mostly by discussion with friends. In fact my main pleasure is discussing it which is wy i am here. I enjoy discussing math, and the mroe I try to explain it to others the beter I understand it. So also try to explain it to someone else. teaching is the best way to learn.

 

[mathwonk]

kant,

i want to put you somewhat at ease, but not completely answer your question. You are obviously a bright student, and i am sorry we perhaps got off on the wrong foot.Anyone trying as hard as you are to master the art of learning is sure to succeed, if you just hang in there. But it is more of a process than an algorithm, if that makes sense.

On one hand, it is true that there is no way to be sure you have understood what you have read. But on the other hand there are guidelines, like Auslander's rule about how much time you have spent trying.
My advisor told me that improvement is almost proportional to the amount of time spent studying. This is the basic rule. But try to remember to ask yourself what you have understood periodically.

I am sorry this is vague. read Courant's preface in his calculus book, or look in my "who wants to be a mathematician" thread for reproductions of some of it.

With your desire, you WILL succeed. Do not be discouraged, and try to make the acquaintance of some profs in your department. When they realize you are really interested in learning they will help you, by suggesting the best books, e.g.

The most valuable advice I ever received was this: "Attention will get you teachers". Do you understand this? You probably do, but to me it means that teachers will teach anyone who really wants to learn and shows it by listening.

Hang in there. You are right that there are lots of terrible books and bad teachers, but the good ones are just waiting for a receptive student. Good hunting my friend.

 

[theperthvan]

1 reply for every 46 view. so far, the stat is not bad.
Well, i guess you are trying to be fun, because the otherwise option is absurdity.

Nope. Inventing questions is quite good because it makes you think.
And you could make a question that is impossible to solve, work out why, and you've learned something new.

 

[kant]

I think Chris makes a very good point here. I recall various occasions where I set aside a certain amount of time to learn a topic, or perform a calculation, and it taking far longer than the time I set aside to do it in. This is not a problem as long as one takes account of this and adjusts future plans; for example, noting which topic took longer than anticipated to learn, and accounting for this when planning revision.

I know this is just reiterating Chris' advice, but it is important to remember that occasions like this should not be taken as set-backs, but rather as important points to remember for the future.

No. I don t think it is a set back.

So.. you ask for advice, then go on to slate a guy who gives you experienced advice...

simply restating that different people learn differently, or take different time is not something that interest me a whole lot. The questions of how much time does it take to understand a average difficulty page of math is for the most part subjective, but a sort of subjective evaluation onto one self is not that fuzzy of a notion.

 

[daniel_i_l]

I don't know if this will work for you but I look at every theorem, "left to the reader"... as a riddle and have fun trying to work them out by myself. I find that the more time you spend trying to prove something the better you remember and understand it.
For example, I remember when in 11th grade we learned basic integrals and our teacher mentioned that in addition to calculating the area under a curve you can also calculate the length but that he wasn't going o show us because he didn't have time. From the time that he said that I tried for about 1/2 an hour until i figured it out, and from then on I've always had a "special place" for that equation...
It also makes the reading much more fun if you take an active part in the proofs.
Also, after finishing a chapter I try to go through it in my head step by step and make sure that I understand the motive for each one.
After that do practice problems until your hand gets numb:)

 

[mathwonk]

That's a very good habit Daniel. In reading most amth books there are things left to the reader in almost every line, but they do not bother to say this. I.e. every time they say something you do not know the reason for, that is left to you the reader. Just acept it, and do not expect to read every sentence in the time it took to write it. Just ask your self in every line, why? why? and sit down with paper and pen or pencil to answer your self.

 

[JeffN]

but note for example that there are only 4 basic theorems in calculus, and even these are related...

hey, I've seen you say this before, could you explain to me why these are the four big theorems in calculus?