How To Solve Any Problem
Summary
TLDRIn this video, the narrator introduces 'How to Solve It,' a book by renowned mathematician George Pólya, which teaches a systematic approach to problem-solving. The narrator, a math book collector, highlights the book's significance and shares insights from Pólya's method, emphasizing understanding the problem, finding connections, devising a plan, and examining the solution. The video also discusses the applicability of these strategies to real-world issues, contrasting the predictability of mathematics with the unpredictability of human interactions. The narrator encourages viewers to engage with the content, appreciate the genius of Pólya, and continue learning about problem-solving.
Takeaways
- 📚 The book 'How to Solve It' by George Pólya is a classic guide on problem-solving techniques, particularly in mathematics.
- 🧠 The first step in problem-solving is to understand the problem thoroughly, which can be a significant challenge in itself.
- 🔗 The second step involves finding connections between the given data and the unknown, and considering auxiliary problems if necessary.
- 📝 The third step is to devise a plan for the solution, which is crucial in mathematics but can be more complex in real-life scenarios due to unpredictable human behavior.
- 🏃♂️ Carrying out the plan is essential; many people plan but fail to act, which is a common issue in self-study and learning.
- 🔍 The fourth step is to examine the solution obtained, which is vital for understanding and verifying the correctness of mathematical solutions.
- 🤔 The book emphasizes the importance of asking questions like 'What is the unknown?', 'What are the data?', and 'What is the condition?' to gain a deeper understanding of the problem.
- 🔄 It suggests looking back and checking the result and argument, which is a critical part of the problem-solving process in mathematics.
- 🕵️♂️ The script mentions the value of breaking down complex problems into smaller, more manageable auxiliary problems to simplify the process.
- 🎓 George Pólya was a renowned mathematician known for his clever solutions and innovative approach to mathematical methods.
- 📖 'How to Solve It' was first published in 1945 and has had multiple printings, indicating its enduring relevance and popularity.
Q & A
What is the main topic of the book 'How to Solve It'?
-The main topic of the book 'How to Solve It' is a systematic approach to problem-solving, particularly in mathematics, and it is written by George Pólya.
Who is George Pólya?
-George Pólya was a renowned mathematician known for his clever solutions to problems and his contributions to mathematical methods.
What is the first step in solving a problem according to George Pólya?
-The first step in solving a problem, according to George Pólya, is to understand the problem thoroughly.
Why is understanding the problem considered a big step?
-Understanding the problem is a big step because it lays the foundation for the entire problem-solving process and helps in formulating a plan of action.
What does the second step in Pólya's problem-solving method involve?
-The second step involves finding the connection between the given data and the unknown, and considering auxiliary problems if an immediate connection cannot be found.
How does the problem-solving process in mathematics differ from real-life situations?
-In mathematics, the process is more linear and structured, whereas in real life, the process can be more complicated due to the unpredictable reactions of people and other variables.
What is the third step in Pólya's method?
-The third step is to carry out the plan of the solution that has been devised.
Why is it important to actually carry out the plan?
-It is important to carry out the plan because planning without action leads to indecision and inaction, which hinders progress and learning.
What does the fourth step in Pólya's method entail?
-The fourth step is to examine the solution obtained, ensuring that it is correct and that the argument used to reach it is valid.
What is the significance of examining the solution in mathematics?
-Examining the solution is crucial in mathematics to ensure accuracy and to learn from the process, which can be applied to solving other problems.
How does the book 'How to Solve It' relate to critical thinking and problem-solving in general?
-The book 'How to Solve It' provides a framework for critical thinking and problem-solving that can be applied to various fields beyond mathematics, emphasizing the importance of understanding, planning, and examining solutions.
What is the significance of the book 'How to Solve It' in the context of its publication time?
-The book 'How to Solve It', published in 1945, was a pioneering work in the field of problem-solving methods and had a significant impact on the way people approach problem-solving in mathematics and beyond.
What advice does the speaker give regarding taking action on learning mathematics?
-The speaker advises to take action by picking a book and starting to study or solve a math problem, as doing so can provide clarity and stimulate the mind.
What does the speaker suggest about the value of the book for its time?
-The speaker suggests that the book was very original and valuable for its time, as it was one of the first to focus on problem-solving methods in such a systematic way.
What is the speaker's opinion on George Pólya's contributions to mathematics?
-The speaker holds a high opinion of George Pólya, describing him as a genius and emphasizing the importance and uniqueness of his contributions to mathematics.
Outlines
📚 Introduction to Problem Solving with 'How to Solve It'
The speaker introduces a book titled 'How to Solve It' by George Pólya, a renowned mathematician known for his innovative problem-solving methods. The book, which is signed by Pólya, is a prized possession of the speaker, who is a collector of math books. The speaker emphasizes the importance of understanding the problem as the first step in solving it, using the example of a mathematical problem where clarity on a simple statement took hours to achieve. The speaker also discusses the process of finding connections between data and the unknown, considering auxiliary problems, and planning a solution. They note the difference between problem-solving in mathematics, which is more structured and predictable, and in real life, where human reactions and unpredictable elements can complicate the process. The importance of executing the plan and examining the solution obtained is highlighted, with the speaker sharing personal anecdotes about the challenges of taking action and the clarity that comes from doing so.
🎓 Deep Dive into Pólya's Problem-Solving Techniques
This paragraph delves deeper into the problem-solving techniques outlined by George Pólya in his book. The speaker discusses the process of questioning to understand what is unknown, the conditions that need to be met, and whether these conditions are sufficient to solve the problem. They mention the importance of recognizing related problems and leveraging previous knowledge or related solutions. The speaker also touches on the originality of Pólya's approach, especially for the time when the book was published (1945, with the speaker holding a fifth printing from 1948). The speaker admires Pólya's humility in stating that even modest problems can lead to a sense of discovery and triumph when solved through one's own means. The paragraph concludes with a brief mention of the book's contents, which include guidance for students, questions, recommendations, and general comments on problem-solving. The speaker encourages the audience to subscribe if they find value in the content and promotes their own math courses on their website.
Mindmap
Keywords
💡Problem Solving
💡George Pólya
💡Mathematical Method
💡Auxiliary Problems
💡Understand the Problem
💡Plan of the Solution
💡Carry Out the Plan
💡Examine the Solution
💡Discovery
💡Critical Thinking
Highlights
Introduction of the book 'How to Solve It' by George Pólya, a renowned mathematician known for his clever problem-solving methods.
Emphasis on the importance of understanding the problem before attempting to solve it, a crucial first step in problem-solving.
The book's focus on a systematic approach to problem-solving, applicable not just in mathematics but also in real-world scenarios.
The anecdote about spending hours understanding a simple statement in mathematics, highlighting the complexity and depth of the subject.
The concept of finding connections between data and the unknown, and considering auxiliary problems when direct connections are not evident.
The challenge of applying mathematical problem-solving strategies to real-life situations where human reactions and unpredictability are involved.
The third step in Pólya's method: carrying out the plan, emphasizing the importance of action over endless planning.
The advice to just start with one book or problem when feeling overwhelmed by choices, to initiate the learning process.
The fourth step: examining the solution obtained, which is crucial for understanding and solidifying mathematical knowledge.
The difficulty of applying the examination of solutions to real-life problems due to the variability of human reactions.
Questions to ask when understanding a problem, such as identifying the unknown, data, and conditions, and assessing the feasibility of meeting those conditions.
The process of devising a plan by relating the problem to previously seen problems and considering if related problems have been solved before.
The originality of 'How to Solve It' when it was published, and its significance in the field of mathematics and problem-solving.
The book's publication year of 1945, indicating its historical context and the innovative nature of its content at the time.
The preface's message that every problem, no matter how modest, can lead to a discovery and the joy of solving it through one's own means.
The humility in Pólya's approach, acknowledging that even geniuses like him understand the value of modest problems.
The contents of the book, which include helping the student, questions, recommendations, mental operations, and generality comments.
The recommendation of the book for its unique approach to problem-solving and its applicability beyond mathematics.
Transcripts
how to solve any problem that's right
this book will actually teach you how to
solve any problem how to solve it a
system of thinking which can help you
solve any problem and I believe this is
an older
addition I think this might be signed
this is written by George Pia so George
Paia was a super famous mathematician he
has very clever solutions to problems he
was very smart very brilliant
mathematician very very famous Super
Famous and it is signed by George Pia
really cool right so I'm a collector of
math books this is uh a version of the
book I have I'll leave a link in the
description to this book in case you
want to check it out how to solve it a
new aspect of mathematical method by
George Paia Princeton University
Princeton University
press cool right and so here he talks
about
it how to solve it first you have to
understand the problem that's right so
think about it so no matter what you're
trying to do your first step is to
understand the problem if you are doing
a mathematics problem let's just take
mathematics right because George Pao was
a
mathematician that is a huge deal I
remember spending one time maybe like
four hours on maybe okay maybe not four
hours maybe two hours on the simple
statement fix snot and S and I didn't
know what that meant I was like what
does that mean and I was on some IRC
Channel talking to this guy and I I
won't mention his name I remember who he
was great guy really good at mathematics
he would help me all the time and he
knew how to write proofs and stuff so he
you know so that really really helped me
and two hours right so just to
understand the problem that is a big
step second find the connection between
the data and the unknown you may be
obliged to consider auxiliary problems
if an immediate connection cannot be
found you should a eventually a plan of
the solution right so this is good for
mathematics it's a little bit harder in
the real world let me explain why so in
mathematics the way you're trained when
you learn mathematics is you have a lot
of areas of math like linear algebra for
example or abstract algebra everything
builds it's very very linear you have
some places where you can Branch off but
like before you learn XYZ you have to
know about ABC a lot of times and
there's a lot of that so it's pretty
deep some of the proofs have some tricky
things in them um it's a lot of work
right so you get used to building on
things in the real life you're not
trained like you are in mathematics
right in the real world if if you have a
problem in life maybe it's with a
significant other maybe it's a job May a
problem at work you have a problem with
your teacher um problems with friends
any type of problem you have right if
you can actually break it down into
auxiliary problems then you know you
that can really help sometimes if you
really think it out but it's harder to
plan the solution because human
beings are not like mathematics right
they their reactions aren't the same so
if your plan depends on the reaction of
other people or things like that it
becomes more complicated right in math
everything is is black and white so it's
a little bit easier step three carry out
your plan yeah so a lot of people are
planners and they're not doers this is a
real thing and this is something that I
uh I I I know I'm aware of this and I
think you should be too whenever you're
doing self-study for example if you're
trying to learn mathematics and you have
a bunch of books one common issue that
people have is that they have a plan for
study but they don't take action and the
reason people don't take action is
because they can't make a decision so if
you have 30 books which one do you pick
to study with well you just grab one and
start studying just just pick one and
just do it right after this video just
do a math problem and what's going to
happen is once you do one problem it's
going to clear your head you're going to
feel like wow I have some clarity here
I've used my mind you know it's it's
really good for you I think that that is
a good one fourth examine the solution
obtained right mathematics this is very
very important especially the further
you go I first actually you know I
didn't learn this until I was I mean I'm
sure my teachers said it but it didn't
really stick to me until I was in
graduate school for mathematics you know
examine the solution obtained and the
same thing with other problems right
like you know was the outcome um you
know what you wanted
um but yeah this is mainly meant for
mathematics but you could you could
apply this to things in life it's a
little bit harder though right because
um a lot of things in life depend on
people and people's reactions aren't
perfect in mathematics you know
something is true or false you know you
you can you have statements and so you
can work through them let's see what
this says here understanding the problem
so here it says what is the unknown
right what is the unknown what are the
data what is the condition is it
possible to satisfy the condition is the
condition sufficient to determine the
unknown so just a lot of questions there
devising a plan have you seen it before
do you know a related
problem look at the
unknown yeah here is a problem related
to yours and solv before could you use
it so it's really going deep here right
and that's really good these are things
that come up in mathematics and that's
why Paia does this right so it's just
really brilliant carrying out the
plan looking back can you check the
result can you check the argument just
this is a very very original book
especially for its time right this was
published wow signed by the legendary
Paia this was published in
1948 1948 well here it's actually
1945 so this must not be this is the
fifth printing so this is no this is not
the first I probably couldn't afford the
first that's probably what it was um you
know books are expensive I have a lot of
books people say well you have you spend
a lot of money on books I do but I've
been collecting books for you know like
15 years so I have a lot of
books yeah what's this say look at the
preface here see what this
says a great discovery solves a great
problem but there is a grain of
Discovery in the solution of any problem
your problem may be modest but if it
challenges your curiosity and brings
into play your inventive faculties and
if you solve it by your own means you
may experience the tension and enjoy the
Triumph of Discovery that's that's right
I because your problem may be modest
he's he's being humble here because Paia
was a freaking genius right I mean the
man was a genius that's what I don't
think people realize you should read
about him on Wikipedia it's very I'll
try to remember to leave a link in the
description uh to his Wikipedia page in
case you don't want like just go to the
description you can click
it and uh you can read about Paia
because the dude was amazing yeah just
really really good mathematician
right and then here we have um
contents you can look at the
contents so helping the student
questions recommendations mental
operations generality comments it's it's
a book so it's not not just we spent a
lot of time there at the beginning but
it's not just that but I mean that's
pretty much the idea um it gives you
examples and stuff it's a really pretty
interesting
book you know people always say problem
solving is really important you know
critical thinking all those things this
is a book on problem solving right
that's what this is and it's not a math
book but at the same time it is you know
it's written by a mathematician a very
famous genius mathematician I mean this
was written by a genius right so I think
that's uh and I mean he was a genius for
writing this book this is a great book
and again for the time no one else was
really writing books like this I mean I
can think of other unique books from
from this era before um there's like a
calculus Made Easy by Sylvanas Thompson
that's another like you know wow what is
that book you know that's another like
Game Changer um pretty pretty incredible
move but yeah great book um I think it's
great if you found any value in this
content feel free to hit subscribe if
you want to if not that's okay too I
have courses they're on my website math
sourcerer docomo vids.com uh check them
out I have calculus differential
equations algebra all that stuff trig
but yeah great book I recommend it and
um yeah hope it's been helpful keep
learning math
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