Better In Math

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

of notebook paper and literally draw out

algorithms step-by-step so I could

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for sponsoring this video and helping

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