If I asked you to think of a really tough problem
what is the first thing that comes to mind?
Well, if you're anything like me
you probably thought of some big
complex math equation, right?
For most people, math seems like one of the most
difficult subjects out there.
It's abstract, it's complex, and unfortunately
for those reasons a lot of people
adopt the belief that they're just not math people.
Which is patently untrue because
math is a skill that can be learned
just like any other.
But since you clicked on this video
hopefully you are not one of those people.
Hopefully you have at least some degree
of belief that you can become better at math
and you have the motivation to do so.
And if you do, the obvious question is
how do you get better at math?
Well, fortunately, this is one of those
questions that has a pretty simple answer.
If you want to get better at math
you have to do lots and lots of math.
And the tougher the problems are, the better.
Because tough problems will stretch your understanding
and lead you to new breakthroughs.
But, in the course of studying math
and working through these tough problems
you are eventually going to come to problems
that just stump you, that you get completely stuck on.
And when you get to these points
it's important to know how to eventually
solve these problems, because these
are the ones that are really going to stretch
and build your skill set.
So that is what I want to focus on in this video.
I want to give you practical techniques
for working through, and eventually solving
those problems that seem insurmountable at first.
To start, I want to focus on a piece
of advice the Hungarian mathematician
George Polya shared in his 1945 book "How to Solve It."
It goes:
This is, in my opinion, the most important
technique to understand and put into practice
when you're trying to solve tough math problems.
Because math builds upon itself.
More complex concepts are built upon simpler concepts.
And if you don't have a strong grasp
on the fundamental principles, then a more
complex problem is going to likely stump you.
So, if you come across a problem
that you can't solve, first, identify the components
or the operations that it wants you to carry out.
A lot of times, complex problems will have multiple.
Now, what you can do in this case
is split the problem into multiple problems
that isolate just one of those
components or operations.
I want to show you this concept in action
so let's work through a quick example.
Now, I did have one example picked out
that would be pretty easy but it ended
up being a little bit too easy, so let's
do something a little bit more complicated.
So, this is a summation problem
which uses the Greek symbol, sigma.
And it essentially says that we're going
to add up a series of expressions
that use a variable starting at one and ending at four.
But, if you notice, this summation problem
also has a fractional exponent in it.
Now, maybe some of you math wizards out there
could do this kind of a problem in your sleep
but it's also possibly the case that
you don't have a really firm grasp
on either summation or fractional exponents.
So, when you're working a problem
that combines the two of them, you might get stuck.
So, assuming that's the case, let's break
this problem into two simpler problems
that each focus on just one of the underlying concepts.
First, let's create a simpler summation problem
that just gets rid of that fractional exponent altogether.
Now, all we have to do is evaluate that expression
four times and then add up the answers
which gets us to a final answer of 66.
And now let's move on to the fractional exponent.
Now, I'm going to go pretty quick here
because this is not a lesson on fractional exponents
but essentially you can rewrite this as
four to the power of three
times the power of one half.
And then you can rewrite that again
to the square root of four to the power of three.
And once you evaluate that, you get an answer of eight.
Now, the whole point of working these simpler
single concept problems is to
master the underlying concept
or operation that you're working on here.
So, if you solve a few and you still don't feel
really confident on that concept
keep working it until you do.
Remember, mastery means not being able to get it wrong.
Not just getting it right once.
Anyway, once you've mastered those underlying
components in an isolated setting
now you can come back to the more complicated
problem that combines them.
At this point, you should be able to work
those isolated concepts in your sleep
which means that all of your mental processing power
can go towards the new and novel problem
of how they work in tandem.
Now, there is one additional way of simplifying
tough problems that I want to talk about
and you might have already guessed it
if you paid really close attention to the examples.
I didn't use really complex numbers.
I didn't use long numbers.
I didn't use decimal points.
I didn't use big fractions.
And I stuck to a low limit on my summation problem.
Really complex, big numbers with lots of decimal points
can distract your attention away from the concepts
and the operations that you're supposed to be practicing.
So, if you're stuck on a tough problem
that has these kinds of numbers
go work a similar problem with really small
whole numbers that are easy to add
or operate in your head, that way
you can really zero in on the actual concepts.
Of course, sometimes you have too shaky
of an understanding of the concepts
and operations themselves for you to actually
work with them and solve that problem.
And in that case, it's time to go do some learning.
Go dig into your book, look through your notes,
or find example problems online
that you can follow along with step-by-step
so you can see how people are getting
to the solutions, using these concepts.
And, if you need to, you can actually get
a step-by-step solution to the exact problem
you're working on as well.
There are several tools out there
that you can use to do this.
The two that I want to focus on in this video
which are the best ones I've been able to find
are WolframAlpha and Symbolab.
Both of these websites will allow you
to type in an equation and get an answer
and also gypha and Symbolabs
that you can follow along with.
The difference between the two
is that WolframAlpha, while being much more
power and capable, does require you to be
part of their paid plan if you want
to get those step-by-step solutions.
By contrast, while I found that typing in
equations into Symbolab was a little bit slower
and less intuitive than it is with WolframAlpha
their step-by-step solutions are free.
Regardless of the tool that you choose to use here
the underlying point is that sometimes
it can be useful to see a step-by-step solution
for a problem you're stuck on.
But, there are two very important caveats here.
First and foremost, before you go running off
to find a solution, ask yourself
"Honestly, have I pushed my brain
to the limit trying to solve this problem first?"
Expending the mental effort required
to solve the problem yourself is going
to stretch your capabilities.
It's going to make you a better mathematician
in a way that just looking through solutions won't.
Now, if you do need to look up a solution, that's fine.
Look it up, follow the steps
and make sure that you understand
how the answer was arrived at.
But, once you've done that, challenge yourself
to go back and rework the problem
without looking at that reference.
It is really important to stay vigilant about this.
Because if you want to get better at math
the whole point is to master the concepts
that you're working with.
The danger that comes with looking up solutions
is that with math it's really easy
to follow along with a step-by-step solution
and comprehend what's going on.
But that is very different than
being able to do it on your own.
And that brings me to my final tip for you.
And this is especially important for anybody
in a math class working through assigned homework.
Don't rush when you work through math problems.
I know it's really tempting to try to work
through homework as fast as you can
and heck, I even made a video about it pretty recently.
But, with math and science and
any sort of really complex subject especially
rushing is only going to hurt you down the road.
Because when you rush, you don't master the concepts.
You just brute force your way to answers
or you look things up, or you otherwise
kind of cheaty-face your way to
a completed homework assignment.
And later on, when you're sitting in
a testing room, or you have to apply
what you've learned in the real world
you are going to get a harsh lesson
about exactly what it is you don't know.
So let's recap here.
If you want to get better at math
and you want to improve your ability
to solve those really tough problems
first, identify the combination of concepts
or operations being used in a problem
and then isolate them.
Work simpler problems that use just one
and then master each concept.
You can also simplify the problem
by leaving the combination of concepts intact
but swapping in smaller, easier to handle numbers.
If you need help with the concepts themselves
go to your book or an explainer article online
look up sample problems, or use a tool
like WolframAlpha or Symbolab to get step-by-step solutions
to the problem you're working on.
And finally, don't rush through your homework assignments.
Make sure that you're focusing intently
on mastering the concepts, not just finishing.
Hopefully these tips will give you the confidence
to tackle some really tough math problems
and to expand your math skill set.
And on that note, I want to leave you with a quote
from the great physicist, Richard Feynman, who said,
The bottom line is this:
Ultimately, your ability to get good at math
and anything else for that matter
starts with having the confidence to approach it.
And as you solve problems and make mental breakthroughs
your confidence is going to naturally increase.
It becomes a self-sustaining cycle.
If you're interested in starting the cycle
of learning now, a great place to begin
your journey is at Brilliant.
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I'm actually taking their computer science
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the quiz questions they gave me forced me
to learn new math I had never learned before.
I had to dig into wikis and example problems.
And eventually I had to get out a sheet
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