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Within just 4 years, I went from being a
very average student at a bad middle school
To becoming the youngest
contestant to ever win a medal
at the International Physics
Olympiad (IPhO) for my country
And a few years later, I landed one of the
highest paid internships in Wall Street
And all of this happened because I fundamentally
changed the way I think about mathematics!
In this video, I will teach you
how to understand math intuitively!
It doesn't matter if you're in high school,
in college, or even close to retirement
and it especially does not matter
if you think you're bad at math!
At the end of this video,
I'm going to give you the best resources
to get you started on this journey
depending on your age, and educational background
Brace yourself, I'm going to change
the way you see numbers forever!
As I said, I used to be rather average at math!
I used to do all of my homework problems
and then maybe solve a few
more problems before math exams
and then I would perform
"okay" in all of these exams
And sure, that approach kind of works
if your goal is just being
okay in all of your exams
But the sad reality is, that with this method
you will never become one of the people
who can truly understand mathematics intuitively
and who can utilize this
potential in their daily lives
And don't fool yourself by thinking that
only engineers and scientists
use math on a day-to-day basis
Almost every successful person on this planet
understands and uses math intuitively
Even without calculating anything explicitly
they can all make smart and intuitive decisions
based on very good estimates of numbers,
probabilities, risks, and other factors!
You probably think that I'm
some kind of child-genius
who just happened to become excellent in math
and therefore got offered
$15,000/month for an internship
Well, this couldn't be further from truth!
I would argue that understanding
mathematics is 90% dependent on:
First of all your mindset, which
I want to change in this video
and second of all lots and lots
of problem solving practice
using the right material, which I'm going
to give you at the end of this video
Math can be understood intuitively!
Contrary to history or geography
where you can only know the facts
that you have actually learned
you can develop the ability to
understand mathematics intuitively
so that you can solve new problems
that you have never encountered before
Let us now get to the main point of this video
How can you develop such mathematical intuition?
Obviously, this doesn't happen overnight!
If you have an exam in math tomorrow,
this video isn't going to save you
Instead, I wanted to use this video
as a start of a transformation
that will take at least several weeks or
even months to show significant impact
But, this may as well become THE most
valuable thing that you will ever learn!
The way to develop a great
mathematical intuition is the following
The first step is: make sure to
at least learn the basic concepts
Think about them as tools in your toolbox.
Additions, fractions, decimal
numbers – these are your basic tools!
They are the hammers, wrenches,
and screwdrivers of mathematics!
Every time you learn a new chapter in
school, you add one tool to your toolbox
By the time you graduate from
high school or university
you have accumulated an enormous amount of tools
that are most likely just rusting in your basement
Time to wipe off the dust and put
them to use in step number two!
The next step is an enormous
amount of problem solving practice
using the right materials,
that will provide you in a bit!
Instead of just studying for an exam, and
then moving on to learning the next tool
you should stick to your simple toolbox initially
Start with simple problems, and then
work your way up to very hard problems
using these exact same simple tools
You would be very surprised how
advanced problems can be solved
using just a hammer and a
screwdriver of mathematics
The third step, after you have really
exhausted all the tools in your toolbox
is going on and adding a new tool to your toolbox
The problem with most education
systems in the world is
that this step happens way too early!
Typically, you've just barely learned
how to solve like three or five
different types of problems using a given tool
And then you move on to learning the next one
And the word "learn" may be an overstatement here
because many students just memorize
the steps to solving a problem
and then just repeat the same thing
during an exam, but with different numbers
Most school systems are set
up to value good memorizers
who can reproduce what teachers show them
and push away those creative deep thinkers
As a rule of thumb, for every 1 hour
that you spend learning a new tool
You should spend at least 20 to 100
hours practicing using these tools
first and simple, and then very
advanced mathematical problems
All right! You've made it this far in the video!
Now you have to make a big decision:
Do you want to forever change your
academic and professional life
by learning to understand math intuitively?
If you decided yes, I will not give you
the right resources to get you started
depending on your age and educational background
The most comprehensive set
of resources is available
on a website called artofproblemsolving.com
When you open their website, you
will find a complete set of resources
ranging from elementary school, middle school,
all the way up to high school, and early college
don't be too focused on the year number,
because these are other advanced problems
So, for example, if you're in
the last year of high school
you don't have to start necessarily with grade 12.
You may want to start with something a
bit earlier, and then work your way up
Remember, it's all about the toolbox!
And you want to start with simple problems
and then use that same toolbox to
learn how to solve advanced problems
Speaking of simple toolboxes, another
great book to get you started is called
A Concise Introduction to Pure
Mathematics, by Martin Liebeck
Also, if you happen to speak German
there's a great book called Mathe ist
Cool, which is for everyone 12 and over
If your toolbox already contains somewhat
more advanced high school level tools
then you might wanna have a look at a book called
Problem-Solving Strategies, by Arthur Engel
This book is a collection of problems for many
national and international math competitions
and it contains very good resources and guides
on how to tackle these problems in the first place
Also, I know a guy who solved all
of these problems in the whole book
and he ended up winning 4 gold medals
at the International Math Olympiad (IMO)
For more mature audiences, I
would also recommend the book
Love and Math: The Heart of
Hidden Reality, by Edward Frenkel
It's not just a math textbook, but it
also talks about the beauty of mathematics
and how to understand it and
see it in a different way
Becoming good at math is, in some
way like becoming a good car mechanic
For sure, at some point you will have to
acquire a lot of tools to do your job well
But a professional mechanic
would be able to fix more cars
with a $20 toolbox from Amazon
Then I would be able to do
with a $100,000 mechanic shop
Let me give you an open-ended question
that captures the essence of
mathematical intuition really well
What is the total length, of all the tunnels
of London's public transport system – the Tube?
This is a slightly simplified
version of a question
that I was asked during the first interview
round for my aforementioned internship
that later paid me $15,000/month
as an 18 year old student
The worst way of solving such problems
would probably be like preparing for the TV
show called "Who Wants to be a Millionaire"
That is, just memorizing
random facts about everything
and hoping that one day some of these
facts will actually be useful for something
Memorizing some things may be really
good for your general knowledge
but it won't really get you far
Because, honestly, what were the chances
that you would have actually thought about
memorizing this specific fact,
unless you actually live in London?
Instead, I want you to develop a great
problem-solving strategy
for attacking such problems
with whatever weapons you have
available to you at that moment
Instead of just guessing,
I want you to explain to me
How you would calculate
the length of these tunnels
by using only information that is available
to you and without using a calculator
There are many ways to solve this problem
so I encourage you to describe your
solution in the comment section down below
and once the video hits 100,000 views,
I'm going to pick my favorite solution
and send the lucky winner a $50 Amazon gift card
If you're curious about how I solved
the problem during the interview process
I made a video about it and put it to
my Instagram account (@samuel.bosch)
so feel free to follow me on
instagram, and have a look at the video
and while you're there, feel
free to send me a nice message
and let me know if you enjoyed this video
Special shoutout to my friend and colleague at
MIT, Emily Mu, who helped me prepare this video
If you want to hear more from both of us, about
math competitions and interning on Wall Street
you can check out our podcast episode over here!
Or if you want to hear a video specifically about
this internship, you can check it out over here!
Thank you guys so much for watching!
If you enjoyed this video
please give it a thumbs up
and you can subscribe to my channel over here
I'll see you next time!
Please play the YouTube video first