What is the proper way to study Mathematics? | IIT prof's tips

Summary

The video script emphasizes the importance of self-study in mastering mathematics, as opposed to an over-reliance on teachers for theoretical knowledge and examples. The speaker suggests that students should select a comprehensive textbook and engage deeply with the material, including understanding theorems, reproducing them independently, and contemplating their implications. Solved examples should be used to apply this knowledge, and students are encouraged to attempt problems independently before consulting solutions. The process involves a postmortem analysis of both successful and unsuccessful problem-solving attempts to identify shortcomings and reinforce learning. The speaker also highlights the significance of starting with basic problems to build a strong foundation before tackling more complex ones. The summary concludes with a congratulatory note to students who successfully solved a posted problem, underscoring the value of a methodical and comprehensive approach to mathematical study.

Takeaways

• 📚 Mathematics requires self-study for true understanding beyond classroom learning.
• 📊 A significant number of students overly rely on teachers for theory and examples.
• 📝 Achieving mathematical maturity is essential for mastering mathematics.
• 🔍 Effective self-study involves understanding theorems deeply and reproducing them from memory.
• 📖 Select a comprehensive textbook for self-study rather than relying on books filled with solved problems.
• 🔄 Treat solved examples as opportunities to apply knowledge, not just pattern matching exercises.
• 💡 Approach problems independently before looking at solutions to reinforce learning.
• 🧩 Solving easy problems is crucial for building a foundation before tackling more complex ones.
• 🧠 Postmortem analysis of why a problem was solved or not is essential for improvement.
• 🎉 Celebrating successful problem-solving builds confidence and reinforces correct approaches.

Q & A

What is the main issue the speaker identifies with students' approach to studying mathematics?

The speaker identifies an over-reliance on teachers for theory and examples as the main issue, which hinders students from achieving mathematical maturity and truly understanding the subject.

What does the speaker suggest as an alternative to relying solely on teachers?

The speaker suggests self-study as an alternative, emphasizing the importance of deep self-study in addition to practice for a proper grasp of mathematics.

What is the speaker's recommendation for selecting a textbook for self-study?

The speaker recommends picking up a very good textbook that presents the theory comprehensively, written by a proper expert in the subject, rather than commercial books that focus on solved problems.

How does the speaker propose students approach the process of self-study?

The speaker proposes that students should study theorems on their own, reproduce them without looking, think about the implications, and then apply this knowledge to solved examples and problems, ensuring they understand the core concepts.

What is the significance of reproducing a theorem on a piece of paper after studying it?

Reproducing a theorem helps to create connections in the brain with the studied material, reinforcing the learning process and aiding in understanding the core concepts more deeply.

How should students approach solved examples during self-study?

Students should not immediately look at the solution but first try to apply the theorems they've learned to the example, think about possible approaches, and only then consult the solution to reinforce their learning.

Why is it important for students to attempt easy problems as well?

Easy problems are crucial for building a strong foundation in mathematics. They help students become proficient with basic concepts and theorems before moving on to more complex problems.

What is the speaker's advice regarding the use of solutions and resources when solving problems?

The speaker advises students to first try problems on their own, exhaust all possibilities, and only then consult solutions. They should also perform a postmortem analysis to understand why they couldn't solve the problem and learn from it.

What is the role of a teacher in the learning process according to the speaker?

According to the speaker, a teacher should serve as a guide, providing introductions to subjects and pointing students in the right directions. However, it is up to the student to follow these directions and discover the subject matter independently.

What is the significance of mathematical maturity in learning mathematics?

Mathematical maturity is essential for truly grasping mathematics at its core. It involves the ability to think independently, make connections across different theorems, and understand the implications of mathematical concepts.

How does the speaker evaluate the performance of students who attempted the problem posted in the community?

The speaker evaluates the students' performance based on the correctness and completeness of their solutions. They congratulate the students who successfully solved the problem and appreciate those who thought in the proper direction, even if they missed some parts of the argument.

Outlines

00:00

📚 Overreliance on Teachers in Mathematics Learning

The speaker discusses the common mistake students make in mathematics by relying too heavily on their teachers for theory and examples. They emphasize the importance of self-study for achieving mathematical maturity and understanding the subject deeply. The video aims to guide students on how to improve their study habits in mathematics, contrasting it with physics and chemistry. It highlights that while practice is crucial, a significant part of learning mathematics involves deep self-study, which is often overlooked. The speaker also points out that a teacher's role should be as a guide rather than the sole source of knowledge.

05:02

🧐 Strategies for Effective Self-Study in Mathematics

The paragraph outlines a structured approach to self-studying mathematics. It advises students to choose a comprehensive textbook and to study theorems and examples actively. The process involves understanding core concepts, reproducing theorems without looking at the book, contemplating the implications of theorems, and considering possible corollaries. Solved examples should be treated as opportunities to apply theorems rather than memorizing patterns. The speaker also emphasizes the importance of attempting to solve problems independently and conducting a postmortem analysis when solutions are consulted. They stress the significance of respecting every problem, regardless of difficulty, and learning from both successful and unsuccessful attempts.

10:04

📈 Sequential Problem-Solving and Respect for All Problems

The speaker insists on the sequential nature of problem-solving in mathematics, advocating for proficiency in easier problems before tackling more complex ones. They share personal experiences from their preparation, emphasizing the importance of not disregarding easy problems. The speaker advises against immediately looking at solutions and instead encourages a thorough attempt at solving problems independently. If a solution is consulted, it should be analyzed to understand why the problem was challenging and how the correct approach was identified. The speaker also mentions a community post where students attempted a problem, congratulating those who solved it and encouraging others who were close to the correct solution.

15:06

🎓 Final Thoughts on Self-Studying Mathematics

In the concluding paragraph, the speaker offers well wishes to students on their journey of self-studying mathematics. They encourage a comprehensive approach to learning mathematics, emphasizing the importance of a proper and thorough understanding of the subject. The speaker thanks the audience for their patience and reiterates the significance of a structured and respectful approach to problem-solving in mathematics.

Keywords

💡Mathematical maturity

Mathematical maturity refers to the ability to understand and apply mathematical concepts independently. In the video, it is emphasized that relying solely on teachers for theoretical knowledge does not lead to true mathematical maturity. It is achieved by engaging in self-study, which allows students to grasp mathematics at its core, rather than just memorizing patterns for solving problems.

💡Self-study

Self-study is the process of learning independently without direct guidance from a teacher. The video stresses the importance of self-study in mathematics, as it involves deep understanding and application of concepts. It is a crucial aspect missing from many students' study habits, and the video suggests that self-study should be combined with teacher guidance and problem-solving for a comprehensive approach to learning mathematics.

💡Theorems

Theorems are statements that have been proven to be true through mathematical reasoning. In the context of the video, studying theorems is a fundamental part of self-study. The speaker advises students to study theorems on their own, understand their implications, and then attempt to reproduce them without looking at the book, which helps in reinforcing the understanding of the core concepts.

💡Solved examples

Solved examples are worked-out problems that demonstrate the application of mathematical theorems. The video suggests that students should not just memorize these examples but use them as opportunities to apply the knowledge they have gained from studying theorems. It is important to attempt to solve the examples independently before looking at the solution to reinforce learning and understanding.

💡Problem-solving

Problem-solving in mathematics involves applying learned concepts to find solutions to mathematical problems. The video emphasizes that students should not disregard easy problems, as they are essential for building a strong foundation. It also advises against immediately looking at solutions, but rather to attempt problems independently and then analyze why a particular approach worked or didn't work.

💡Pattern matching

Pattern matching is a skill where students recognize and apply similar methods to solve problems based on previously seen examples. While the video acknowledges that this is a good skill to develop, it cautions that relying solely on pattern matching does not equate to truly learning mathematics. The focus should be on understanding the underlying concepts and theorems.

💡Commercial textbooks

Commercial textbooks are those that focus on providing a large number of solved problems, often aimed at helping students achieve good grades in exams. The video criticizes the over-reliance on such books, as they may not foster a deep understanding of mathematics. Instead, the speaker recommends textbooks that present theory comprehensively and are written by subject experts.

💡

💡Postmortem analysis

Postmortem analysis refers to the process of reviewing and analyzing why a particular approach or solution was successful or unsuccessful. In the context of the video, it is suggested that students should perform a postmortem analysis after attempting a problem, especially if they were unable to solve it. This helps in identifying shortcomings in their understanding or application of mathematical concepts.

💡Competitive examinations

Competitive examinations are high-stakes tests that students often prepare for, aiming to qualify for prestigious institutions or programs. The video mentions that some students may overlook easy problems in their preparation for these exams, focusing only on complex problems. However, the speaker argues that a solid foundation in basic problems is crucial before moving on to more advanced levels.

💡NCRTL books

NCRTL (National Council of Educational Research and Training) books are textbooks published by the Indian government that are known for their comprehensive presentation of subject matter. The video recommends these books as a good resource for self-study in mathematics, as they are written by experts and cover the theory in depth.

💡Logical and geometrical problems

Logical and geometrical problems are types of mathematical problems that require logical reasoning and understanding of geometric concepts. The video mentions a community post where a problem of this nature was posted, and students were encouraged to solve it. This serves as an example of the kind of problems that can be encountered in mathematics and the importance of a comprehensive approach to solving them.

Highlights

The video discusses the common mistakes students make while studying mathematics and provides tips for improvement.

A poll revealed that over 50% of students rely on teachers for theory and examples, indicating an over-reliance on teachers.

The speaker emphasizes that relying too much on teachers does not lead to mathematical maturity.

Self-study is crucial for a proper grasp of mathematics, involving deep understanding and not just pattern matching.

Mathematics cannot be fully taught; a teacher can only guide, and students must follow the directions independently.

An ideal study situation involves a teacher's introduction, self-study, and then problem-solving.

Selecting a good textbook that presents theory comprehensively is essential for self-study.

The process of self-study involves understanding theorems, reproducing them without looking, and contemplating their implications.

Solved examples should be treated as opportunities to apply theorems rather than memorizing patterns.

When solving problems, start with easier ones to build proficiency before moving on to more complex ones.

Avoid looking at solutions directly; try to solve problems independently first, then analyze why you couldn't solve it.

After solving a difficult problem, analyze what led to the successful solution to reinforce learning.

Every problem, regardless of difficulty, should be treated with respect and attempted with full effort.

The speaker congratulates students who successfully solved a problem posted in the community, highlighting the method of contradiction.

The importance of a comprehensive and proper approach to self-studying mathematics is emphasized for long-term success.

The video concludes with well wishes for the viewers' mathematical studies and self-study journey.

Transcripts

hello everyone in my previous two videos

I have discussed the biggest mistakes

which students make while studying

physics and chemistry and I've also

shared a few tips on how to improve

their study habits and in this video I'm

going to do the same for mathematics now

because mathematics is inherently

different in its nature from physics and

chemistry I first decided to make a poll

uh through a community post and uh I am

not actually surprised by the results of

this poll so uh the question was what

system do you follow when preparing

mathematics and more than um close to

50% of the students said that they

relied on teacher for Theory and

examples and then they tried the

problems on their own and uh a certain

fraction of the students said that they

relied on the teacher for theor and

examples and then they went about

learning the problems from the solved

examples

so uh almost 60% of the students uh from

the response it is clear that there is a

clearcut Reliance or even I would say an

over Reliance on the teacher for the

theory and the uh the introduction to

the subject the the discussions so all

the theoretical kinds of discussions the

students are relying on the teacher so

this kind of an over Reliance on the

teacher is something which is not

actually good and this is one of the

main points of this uh of this video

that this kind of Reliance on the the

teacher is not something which is going

to help you to ultimately achieve

mathematical maturity and unless you

achieve mathematical maturity you are

not going to be able to actually grasp

mathematics at its very core you may be

going from one class to at higher class

but actually you are not learning

mathematics as it should be learned so

what should be done so instead of overly

relying on the teacher you should be

doing self-study so yes in mathematics

also even though many people say that

mathematics is about practice I would

say that a significant portion of

mathematics involves some very very deep

self-study this is one crucial aspect

which is missing from many of the

guidance that is given by our school

teachers this is unfortunate but this is

true and Mathematics is such a subject

which really cannot be taught by Any

teacher a teacher can only be there as a

guide can only give pointers to you in

certain directions but it is you who

yourself have to follow these directions

on your own path and figure out things

for yourself so self- study is the first

step which you need to actually do in

order to uh have a proper grasp of

mathematics and this is something which

is actually very much missing in the

vast majority of the students as the

poll has clearly shown mind you that

there is a certain fraction of students

in the poll who have mentioned that they

do completely self-study but that also

has certain issues that when you

completely do self-study you you miss

out on a certain perspective from a more

mature person so that also is request so

an ideal situation an ideal situation

would be when a teacher gives you an

introduction to the subject and then you

do self-study and then you go about

solving problems so all of these three

aspects in conjunction are what

contributes to making your study of

mathematics your preparation of

mathematics complete and

comprehensive now so in this video what

I'll do is first of all I'll point out

certain things what you can do in

regards uh with regards to self-study so

first and foremost pick up a very good

textbook not the commercial kind of

books which only focus more on giving

you lots and lots of solved problems so

these books are very attractive from the

uh Viewpoint that they help you get good

marks but remember that getting these

good marks in the board exams and the

school LEL

exams um this is certainly good but just

because you are getting good marks

doesn't actually mean um and and this is

not your fault at all doesn't actually

mean that you are learning mathematics

so

uh if you are time and again getting

almost 100 out of 100 in mathematics but

this score is based on your preparation

which is very fast very comprehensive

but this comprehensiveness is based only

on some kind of a pattern matching

exercise then you are not really

learning mathematics what you're doing

is basically developing a Proficiency in

pattern matching which is a good skill

to develop I certainly agree to that but

mathematics is not all about that so

pick up a very good textbook where the

theory is presented in a comprehensive

fashion by a proper expert of the

subject there there are some excellent

School teachers who have written

textbooks their Boards of teachers the

ncrt books are written very well uh uh

then at the higher levels also the

university teachers for class 11 and

Class 12 uh the university

teachers they uh they have written some

very good books uh and again the ncrt

books are there so pick up a good

textbook don't go for too many textbooks

pick up one very good textbook and

follow that

religiously now what do you do actually

in self-study so this discussion is more

about how you go about doing things I'm

not going to suggest this book or that

book rather I'll uh tell you what you

can do in the actual process of

selfstudy so the first thing is that

actually you need to study the theorems

on your own and you need to go through

the solv examples and even before that

the discussions the explanations on your

own line by line trying uh very hard to

really understand understand and grasp

The Core Concepts from it but that is

not the end of it because mathematics is

so very different from math from physics

and chemistry what you have to do is

after you have studied the theorem you

have to close your book and without

looking you have to try to reproduce the

theorem on a piece of paper so that is

how you exercise the theorem it may seem

like ratification as some people call it

like Road learning but it is not when

you do the when you reproduce the

theorem on your own it creates certain

Connections in your brain with what you

studied earlier and this has some very

serious implications on what you're

going to uh study

ahead uh after you have done these two

steps think very hard about the

implications of the theorem unless you

have done the theorem yourself on the on

a piece of paper you'll not be able to

do it so think about the implications of

the theorem what I mean by that is so

usually after a certain theorem or after

certain theorems there are certain

corollaries to

it but in certain places there are no

corollaries there are no subsequent

discussions so you have to think for

yourself what kinds of possible

corollaries or some kind of adjacent

explanations could be possible from a

theorem now this is where you need I

mean this is the place where you develop

your mathematical maturity and this is

also the place where the tips and

pointers and the little guidance from

the teachers actually help you out a

little bit I I mean a

lot

so next is the solved examples so when

going through the solved examples don't

just treat the solved examples as some

kind of different patterns which you

have to learn and later on use in the

unsolved examples rather treat the

solved examples as opportunities for

applying the knowledge of the theorems

the knowledge that you have grasped from

the explanations and the discussions

don't look at the solution immediately

first of all what you do is close close

the solution read the solved example and

then think for yourself how the various

theorems that you have just studied how

they apply in this uh in this solved

example how they would apply in this

kind of a in this question uh you must

also try to think about possible

connections with the theorems of the

previous chapters unless you do this and

if you just look at the solution

immediately this thinking process that

you absolutely must go through will not

be there and your learning will not be

reinforced your learning of the theorem

will not be reinforced however if you do

this by closing the solution and first

thinking about the about the possible

modes of going ahead with the solution

even if you do not make any Headway into

the solution yourself still you'll be

able to make a clear cut reinforcement

of your learning of the theorem so those

things will actually get embedded it in

your brain this is very very important

so next we are going to solving problems

and by the way after I have discussed

this solving problems I'm going to

announce the names of um the students

who were able to solve

the um the problem that I had posted in

my community post uh that successfully

solved that problem it was a little

logical kind of geometrical problem so

solving problems uh so many students

make the mistake of treating certain

easy problems as being beneath them yes

they think that solving such kinds of

easy problems will be like an insult to

their intelligence this is kind of

common um perhaps uh most common in

among students who are preparing for

some kind of competitive examinations

like J because they think that they have

to go for such a high level will they

waste their time doing these kinds of

simple problems but mind you mathematics

is something and this I had emphasized

um in my earlier video on problem

solving and I I really insist that you

please make it a point to go through

that that rather lengthy video on

problem solving where I have discussed

all of these things in great detail so

mathematics is such a subject where you

really cannot proceed but without

leveling up so you H really have to

level up one after another so without

first being absolutely proficient and

expert in the very easy level problems

how can you think of going to the higher

level problems so only after you have

you have convinced yourself that you are

perfectly perfectly comfortable with the

very easy level problems then only go to

the certain certain higher level

problems and then only go for more

competitive level examination problems

that that level and a Frank confession

here during my own preparation when I

was first studying in my class 11 12 I

used to study problems or I used to do

problems which were so easy that they

would not even come in the board

examinations they were so very easy but

I never look down on them if if I could

not solve a problem it was it was not

because I was too above them it's just

because that my mind had not

acclimatized to the theorems to the

concepts so you have to go through this

process I mean nothing no problem is

beneath you okay please do not ever

think like this this is absolutely a

wrong wrong way of going about things um

whether it is easy or whether it is hard

treat every problem with respect so the

next thing is you must be trying the

problems on your own nowadays there are

lots of books available lots of

resources available even on YouTube

where Solutions are there my Earnest

request to all of you is that you should

not be looking at the solutions directly

first of all try it on your own try your

best exhaust all possibilities of uh

that that you have at your disposal of

tackling the problem on your own only

and only then if you fail to make any of

sort of Headway try to then look up the

solution but then don't just get

satisfied by looking at the solution and

again this is something which is so very

important I had mentioned it at length

in my previous video on problem solving

so do watch it uh still I'm mentioning

it here because it is so very important

that you should be uh looking at the

solution through the lens of a

postmortem analysis meaning that since

you are not able to do it you must after

looking at the solution try to think for

yourself why was it that you were not

able to solve it so this kind of a

postmortem analysis is extremely

extremely important to make you realize

your shortcomings not just in for that

particular problem rather in your

overall thinking why you did not think

in that that particular direction and

it'll also be a reflection of your grasp

of the theorem that you had studied

earlier so as an application of that or

perhaps your grasp of the connections

that need to be made across different

kinds of theorems another important

point is that uh after you have

successfully solved a difficult problem

you must also make another attempt or

you must make a full analysis of why you

were able to solve it what was it that

made you proceed in the right direction

again this is something which I had

discussed in my earlier video on problem

solving so do make uh make it a point to

watch it so uh these are the some of the

broad points which I thought I'll

discuss in relation to mathematics

especially as it regards to making this

big jump from the school level that

means up to the 10th level from the 10th

level to the 10+2 level where the level

is really high now for the winner

so uh the two students who had

successfully solved the problem within a

few minutes of each other the community

post that had

uh had made are serves Krishna and

Krishna Prasad sures Krishna and Krishna

Prasad so congratulations to both of you

on solving the problems mind you there

were quite a few other students who came

very very close to the correct solution

but they were missing a key part of the

argument um uh here and there so uh

that's it I have to be fair in my

assessment uh but what made me really

happy was that quite a few students

actually figured out that uh this

problem had to be done by the method of

contradiction so congratulations to all

of you who thought in the proper

direction and came very very close to

the actual solution so all all right so

that's uh it uh

thank you very much for Patiently

listening to me and all the very best in

your preparation of studying mathematics

in in your in your preparation of

mathematics and your uh in your journey

of self-studying mathematics in a proper

comprehensive fashion as it should be

done thank you