# The best way of learning

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
This thread is dedicated to discuss the best methods of learning . Every one can post his approaches of of learning new things (not just Mathematics ) . You have to specify why you chose these methods and mention any drawbacks , if you think there are .

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
First , if the subject is too broad and new for me , I have to get someone to give me a general idea about it to know how to start . After that , I try to get a book that is suitable for me level and show it to an expert to get a recommendation . If the subject is not too broad I start by searching the internet to get some useful PDF files . Or I use YouTube for Interesting lectures . I am currently using these two approaches and they are working fine for me . The problem is if you get stuck at a concept or certain problem and you have no body to help . Solution : post it in MHB .

#### Ackbach

##### Indicium Physicus
Staff member
I usually use a combination of Amazon reviews and word-of-mouth to find out what the best textbooks are. Then I read straight through the textbook, doing all problems. I find this method works great for me. If I get stuck, I post on MHB (common theme here).

#### Petrus

##### Well-known member
Hello,
For some reason I can't really study good at morning... I can really focus when it's late and dark out side. Is it only me?

Regards,
$$\displaystyle |\rangle$$

#### MarkFL

Staff member
Hello,
For some reason I can't really study good at morning... I can really focus when it's late and dark out side. Is it only me?

Regards,
$$\displaystyle |\rangle$$
I'm the same way...I get my best work done when "decent" folks are asleep.

For me to truly learn something mathematical, first I need to understand the derivation or proof, then it's a matter of being able to apply it to as broad a scope of problems as possible.

#### Petrus

##### Well-known member
I'm the same way...I get my best work done when "decent" folks are asleep.

For me to truly learn something mathematical, first I need to understand the derivation or proof, then it's a matter of being able to apply it to as broad a scope of problems as possible.
Hello MarkFL,
I agree with you but if I am honest. I know this is not really something good but I bet many student can admit. In my school we read math really fast, and I can't really do that fast so I will be a litle behind, then their is moment when you get close exam and you got some part left to understand and you don't really got time to fully understand and want to know the 'method to solve' because you wanna succed exam. Well those part I done I really understand and feel safe but their is part that take more time.. That is why summer will be a great time where you can really careful read and understand without have any pressure

Regards,
$$\displaystyle |\rangle$$

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Standard procedure when learning new stuff.

1. Skim through all the bold and/or italic words and try to absorb what they mean (and/or write them down).
With little time this is all I do.
It's the difference between being considered an expert and being considered someone who shouldn't be present in a meeting.
2. Skim through any examples or other concrete stuff (often at the end of a chapter).
3. Try a couple of exercises if present (or think up my own).
4. Try to make sense of propositions.
5. Skim through proofs, although I usually don't get that far until confronted with the need to do so.

Whenever I'm teaching, the first and foremost thing as far as I'm concerned, is definitions, definitions, definitions.
If someone doesn't know the words and symbols, there is nothing to talk about in a sensible manner - it's like trying to make sense of Chinese (assuming you don't know the language).

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
In mathematics ,theorems and lemmas are all over the place , at what level do you think it is important to understand the proofs ?

For me , when I first study something I take theorems for granted . But if I want to go deep in a certain subject then I have to understand the proof and be able to derive it .

#### Klaas van Aarsen

##### MHB Seeker
Staff member
In mathematics ,theorems and lemmas are all over the place , at what level do you think it is important to understand the proofs ?

For me , when I first study something I take theorems for granted . But if I want to go deep in a certain subject then I have to understand the proof and be able to derive it .
Yep. For most purposes there's no need to know or understand the proofs.
That usually only becomes important if you're actually studying math with an accent on the theoretical side.
IMO, in math it is important to know which methods you can use to proof something (mostly deduction and proof by contradiction).

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Yep. For most purposes there's no need to know or understand the proofs.
That usually only becomes important if you're actually studying math with an accent on the theoretical side.
IMO, in math it is important to know which methods you can use to proof something (mostly deduction and proof by contradiction).
Absolutely , I think everyone should take a course in logic and methods of proofs . It will make your life just easier .

#### ModusPonens

##### Well-known member
I have to read the proofs of the theorems, propositions and lemmas and understand them. Only after that I solve problems.

#### mathmaniac

##### Well-known member
For me , when I first study something I take theorems for granted . But if I want to go deep in a certain subject then I have to understand the proof and be able to derive it .
I think the best way to study a theorem is by keep deriving it(rather than simply byhearting).

#### Fantini

MHB Math Helper
I have to read the proofs of the theorems, propositions and lemmas and understand them. Only after that I solve problems.
I try to do that as well - the problem is that it takes too much time, and when you're hard pressed it's quite tough to digest everything properly.

I think the best way to study a theorem is by keep deriving it(rather than simply byhearting).
That also happens, but when you have major theorems (whose proofs take pages) it's somewhat impractical to continuously keep deriving it.

I'm not sure my way of learning has been established it, but what I do know is that it doesn't match tests here. I'm so frustrated about my bad results, despite often knowing the material.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
I'm not sure my way of learning has been established it, but what I do know is that it doesn't match tests here. I'm so frustrated about my bad results, despite often knowing the material.
Yes , it sometimes happens to me . When you get a bad grade in a subject which you think you understand, your confidence becomes less.
But , I think it is difficult to measure how well we understand a certain topic. Exams don't usually measure that , they , for the most part, measure how well you read the book or listened to your professor . Sometimes , exams measure how many problems you solved (this is the worst for me ).

#### Fantini

MHB Math Helper
Yes , it sometimes happens to me . When you get a bad grade in a subject which you think you understand, your confidence becomes less.
But , I think it is difficult to measure how well we understand a certain topic. Exams don't usually measure that , they , for the most part, measure how well you read the book or listened to your professor . Sometimes , exams measure how many problems you solved (this is the worst for me ).
Everything you said is so true; yet, how can counteract the fact that you will, in the end, be measured by people who don't know you by those same grades?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Exams are often carefully set up in different difficulty levels.

Questions verifying you know the definitions.
A couple of checks if you can reproduce an example.
If you can do both, you may already have a sufficient grade.
Then more questions for which you may need to know a proof.
And more questions, requiring deeper understanding of the matter.

You can only get a perfect grade if you can do all.

#### Fantini

MHB Math Helper
Exams are often carefully set up in different difficulty levels.

Questions verifying you know the definitions.
A couple of checks if you can reproduce an example.
If you can do both, you may already have a sufficient grade.
Then more questions for which you may need to know a proof.
And more questions, requiring deeper understanding of the matter.

You can only get a perfect grade if you can do all.
Interesting, you should give that piece of advice to my teachers. I haven't seen such a set up for my exam in a LONG while. Right now, most of my examinations require fierce memorization of lists of exercises set forth, and those who fail to memorize all the details score poorly. In fact, that happened in cases where we had the teacher choose a set of problems.

When I took multivariable analysis the set was "all exercises from 'Calculus on Manifolds' by Spivak"; when I took real analysis (measure theory) the set was "all problems from 'Elements of Real Analysis' by Bartle". It also happened that both teachers did not know how to solve said exercises. The first constantly used the solutions manual found in the internet, the second didn't care. We didn't know topology when we took measure theory, and the teacher made constant use of it. At some point he realized people weren't exactly following his 'explanations' (read: copying the book on the board). He promptly asked: "Everyone knows point-set topology, right?" The answer was an unanimous "no." He replied with "Well, Bartle does," followed by a shrug and continuing to copy the book on board.

I am taking Metric Spaces Topology this semester. My lecturer has decided that this course is useless and instead chose to use Munkres' General Topology as the textbook. Frequently he does not know what he is doing in front of the class and claims absurdities until somebody points out the huge flaws in them (he once said that projections weren't continuous in the product topology). He confessed that while he chose some problems of the book he didn't know how to do most of them.

Yet, can you guess what happened on the test? That's right: rote memorization of certain problems he had chosen on the lists. And not quite so easy ones: one of the questions reminded you about the uniform topology on $\mathbb{R}^{\omega}$ (the cartesian product of the real line with itself "real" times), and defined the metric as $$d(x,y) = \left( \sum_{i=1}^{\infty} (x_i - y_i)^2 \right)^{\frac{1}{2}},$$ assuming that $\sum_{i=1}^{\infty} x^2_i \leq \infty$. The problem: show that this defines a metric on $\mathbb{R}^{\omega}$ and that it produces a topology finer than the uniform topology but coarser than the box topology.

There were 4 other questions to be done within a period of 2 hours. Can you imagine proving this in necessary detail and still having feasible time to answer the other four? I could tell you them, but I believe this already gives quite an idea of what is common around here.

tl;dr All this ranting is to say one thing: if the exam isn't "carefully set up in different difficulty levels" but rather "sloppily set up with arbitrary criteria", how does one get by? I have a hard time not getting pissed off about all of this. It goes against my nature to just see things hitting the fan and going unpunished. Later, the same incompetent professor will be the one to evaluate your skills, and when he sees the numbers without the accompanying background he'll be likely to say "well, not so skilled, are we?"

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Hi Fantini , I know what you are feeling but criticizing will not make things better. You have to find a way to cope with the situation . Sometimes we are forced to do things that we hate , just because failing to do them will affect us at the end . I know that If there were now grades I would learn better , at least , I will choose how to learn and be free of pressure. I confess , I learn myself better than in the lecture , not because the teacher can't teach it is because I read what I like and am passionate about . We cannot make education better but someone should know his potentials and learn how to improve them . Failing to do so means you are following blindly what others are telling you and hence live as an ordinary person with no dreams .

#### Fantini

MHB Math Helper
I'm sorry for venting out, but I was under the impression that failure to get good grades meant a failure to understand the concepts and lay them in the way I Like Serena has pointed out. I merely stated so much of what has happened to me to get the point across that not all tests are fair and therefore we shouldn't take the grades blindly.

I apologize if I have sounded rude to you, I Like Serena. It's just that it hit my berserk button pretty hard that one adopts the position that tests are carefully set up the way you mentioned and therefore failure to acquire the desirable grade was basically a failure to understand the concepts and prepare yourself for such standard test. That, unfortunately, seems to be pretty far from my frequent reality that it's not even funny.

Hi Fantini , I know what you are feeling but criticizing will not make things better. You have to find a way to cope with the situation . Sometimes we are forced to do things that we hate , just because failing to do them will affect us at the end . I know that If there were now grades I would learn better , at least , I will choose how to learn and be free of pressure. I confess , I learn myself better than in the lecture , not because the teacher can't teach it is because I read what I like and am passionate about . We cannot make education better but someone should know his potentials and learn how to improve them . Failing to do so means you are following blindly what others are telling you and hence live as an ordinary person with no dreams .
Addressing your points, Zaid, I want to point out that I try hard to conform to what reasonable testing should be. I work a lot to maintain subjects as close to continuous practice as I can (although I may not always be successful), and it breaks me when I get surprised with unfair or completely off the rails tests.

It's pretty much the rule that most people get close to zero out of lectures. In the video I posted of Eric Mazur there is a discussion about that and it clarifies many points regarding this issue. The main problem seems to lie in that lecturing almost always implies passivity from us students, and that makes the transfer of knowledge way harder. It's not unusual to realize that when you participate in class by asking questions or debating with teacher generates greater comprehension of the material. That happens because the stance adopted requires active work on our part and thus our brains manage to retain more information.

Anyway, the thread is about the best way of learning. Regardless of the events I mentioned, there is much to learn about how to study. Let's keep the dice rolling!

#### Ackbach

##### Indicium Physicus
Staff member
It's pretty much the rule that most people get close to zero out of lectures. In the video I posted of Eric Mazur there is a discussion about that and it clarifies many points regarding this issue. The main problem seems to lie in that lecturing almost always implies passivity from us students, and that makes the transfer of knowledge way harder. It's not unusual to realize that when you participate in class by asking questions or debating with teacher generates greater comprehension of the material. That happens because the stance adopted requires active work on our part and thus our brains manage to retain more information.
I totally agree with this one. Which is why Socratic dialogue is, in my opinion, one of the most effective ways to teach. I ask slightly leading questions, hoping the students will do the heavy lifting in their thinking. I don't go so far as the modern education fads, which say that the teacher is an equal. No, the teacher knows the subject or art or lesson to be taught, and has to know it. But if the teacher can get the student to rediscover it, then the student owns that.

There's an old book, originally published in 1884, called The Seven Laws of Teaching. I have one quote from it in my signature. The entire book is filled with pithy statements like that. The gold in that book, are all the violations of the laws that John Milton Gregory points out. You won't find a single violation he mentions that does not occur in many schools.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
I apologize if I have sounded rude to you, I Like Serena. It's just that it hit my berserk button pretty hard that one adopts the position that tests are carefully set up the way you mentioned and therefore failure to acquire the desirable grade was basically a failure to understand the concepts and prepare yourself for such standard test. That, unfortunately, seems to be pretty far from my frequent reality that it's not even funny.
I guess I should have added that it has been that way in my experience.
On high school, on university, and for all the people I've been tutoring.

What you write sounds pretty bad.
Rote memorization? IMHO, that's no way to learn math!

It's pretty much the rule that most people get close to zero out of lectures. In the video I posted of Eric Mazur there is a discussion about that and it clarifies many points regarding this issue. The main problem seems to lie in that lecturing almost always implies passivity from us students, and that makes the transfer of knowledge way harder. It's not unusual to realize that when you participate in class by asking questions or debating with teacher generates greater comprehension of the material. That happens because the stance adopted requires active work on our part and thus our brains manage to retain more information.
I totally agree with this one. Which is why Socratic dialogue is, in my opinion, one of the most effective ways to teach. I ask slightly leading questions, hoping the students will do the heavy lifting in their thinking. I don't go so far as the modern education fads, which say that the teacher is an equal. No, the teacher knows the subject or art or lesson to be taught, and has to know it. But if the teacher can get the student to rediscover it, then the student owns that.
Interesting.

I have found most lectures in my higher education near useless.
Now work group sessions - those were useful!

My own favorite way of teaching is indeed the Socratic dialogue.
But I can see that it's not feasible to do on a great scale.

#### Fantini

MHB Math Helper
I totally agree with this one. Which is why Socratic dialogue is, in my opinion, one of the most effective ways to teach. I ask slightly leading questions, hoping the students will do the heavy lifting in their thinking. I don't go so far as the modern education fads, which say that the teacher is an equal. No, the teacher knows the subject or art or lesson to be taught, and has to know it. But if the teacher can get the student to rediscover it, then the student owns that.

There's an old book, originally published in 1884, called The Seven Laws of Teaching. I have one quote from it in my signature. The entire book is filled with pithy statements like that. The gold in that book, are all the violations of the laws that John Milton Gregory points out. You won't find a single violation he mentions that does not occur in many schools.
I think you'd be amazed if you watched part of the video I posted in the thread Peer Instruction. While the advice is sound for all, it's mostly directed at physics. I agree with what you said about the teacher not being an equal: someone needs greater experience than the rest so he can guide the discussion in a fruitful manner, or else venues that are a waste of time will be explored rather than interesting problems.
Also, I didn't get the "violations" paragraph.

I guess I should have added that it has been that way in my experience.
On high school, on university, and for all the people I've been tutoring.

What you write sounds pretty bad.
Rote memorization? IMHO, that's no way to learn math!
I agree! I wish I had a similar experience, I'm sure I'd be learning more.

Interesting.

I have found most lectures in my higher education near useless.
Now work group sessions - those were useful!

My own favorite way of teaching is indeed the Socratic dialogue.
But I can see that it's not feasible to do on a great scale.
That's because most lectures are designed as simple transfer of information from the professor speaking to the student. The problem is that it doesn't work as effectively as we hoped. Eric Mazur even mentions that he once saw someone define lecture as the process where the notes get transferred from the professor's book to the student's. Work group sessions are amazing - if done right. What I mean is that I often don't get much from those unless I'm levelled with others in terms of material, even if I'm not confident about it.
About your great scale, I must point out the peer instruction link above. I think you may find it an interesting option for your classes.

#### Poirot

##### Banned
That seems like a lazy way of lecturing and setting exams Fantini. Maybe I can get a job there...