What is the proper way to study Mathematics? | IIT prof's tips
Summary
TLDRThe 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
- đ Mathematics requires self-study for true understanding beyond classroom learning.
- đ A significant number of students overly rely on teachers for theory and examples.
- đ Achieving mathematical maturity is essential for mastering mathematics.
- đ Effective self-study involves understanding theorems deeply and reproducing them from memory.
- đ Select a comprehensive textbook for self-study rather than relying on books filled with solved problems.
- đ Treat solved examples as opportunities to apply knowledge, not just pattern matching exercises.
- đĄ Approach problems independently before looking at solutions to reinforce learning.
- 𧩠Solving easy problems is crucial for building a foundation before tackling more complex ones.
- đ§ Postmortem analysis of why a problem was solved or not is essential for improvement.
- đ 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
đ 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.
đ§ 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.
đ 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.
đ 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.
Mindmap
Keywords
đĄMathematical maturity
đĄSelf-study
đĄTheorems
đĄSolved examples
đĄProblem-solving
đĄPattern matching
đĄCommercial textbooks
đĄ
đĄPostmortem analysis
đĄCompetitive examinations
đĄNCRTL books
đĄLogical and geometrical problems
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
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