If I asked you to thinkof a really tough problem what is the first thingthat comes to mind? Well, if you're anything like me you probably thought of some big complex math equation, right? For most people, mathseems like one of the most difficult subjects out there.
It's abstract, it'scomplex, and unfortunately for those reasons a lot of people adopt the belief thatthey'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 canbecome 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 theproblems are, the better.
Because tough problems willstretch your understanding and lead you to new breakthroughs.
But, in the course of studying math and working through these tough problems you are eventually goingto come to problems that just stump you, thatyou 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 arereally going to stretch and build your skill set.
So that is what I want tofocus on in this video.
I want to give you practical techniques for working through, and eventually solving those problems that seeminsurmountable at first.
To start, I want to focus on a piece of advice the Hungarian mathematician George Polya shared in his1945 book “How to Solve It.
” It goes: This is, in my opinion, the most important technique to understandand put into practice when you're trying tosolve tough math problems.
Because math builds upon itself.
More complex concepts arebuilt upon simpler concepts.
And if you don't have a strong grasp on the fundamental principles, then a more complex problem is goingto likely stump you.
So, if you come across a problem that you can't solve, first, identify the components or the operations that itwants you to carry out.
A lot of times, complexproblems will have multiple.
Now, what you can do in this case is split the probleminto 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 littlebit 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 startingat one and ending at four.
But, if you notice, this summation problem also has a fractional exponent in it.
Now, maybe some of youmath wizards out there could do this kind ofa problem in your sleep but it's also possibly the case that you don't have a really firm grasp on either summation orfractional exponents.
So, when you're working a problem that combines the two ofthem, you might get stuck.
So, assuming that's the case, let's break this problem into two simpler problems that each focus on just oneof the underlying concepts.
First, let's create asimpler summation problem that just gets rid of thatfractional exponent altogether.
Now, all we have to do isevaluate 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 tothe fractional exponent.
Now, I'm going to go pretty quick here because this is not a lessonon 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 fourto the power of three.
And once you evaluate that, you get an answer of eight.
Now, the whole point ofworking these simpler single concept problems is to master the underlying concept or operation that you're working on here.
So, if you solve a fewand you still don't feel really confident on that concept keep working it until you do.
Remember, mastery means notbeing able to get it wrong.
Not just getting it right once.
Anyway, once you'vemastered those underlying components in an isolated setting now you can come backto 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 ofyour mental processing power can go towards the new and novel problem of how they work in tandem.
Now, there is one additionalway of simplifying tough problems that I want to talk about and you might have already guessed it if you paid really closeattention 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 limiton my summation problem.
Really complex, big numberswith lots of decimal points can distract your attentionaway from the concepts and the operations that you'resupposed to be practicing.
So, if you're stuck on a tough problem that has these kinds of numbers go work a similarproblem with really small whole numbers that are easy to add or operate in your head, that way you can really zero inon the actual concepts.
Of course, sometimes you have too shaky of an understanding of the concepts and operations themselvesfor you to actually work with them and solve that problem.
And in that case, it's timeto go do some learning.
Go dig into your book, look through your notes, or find example problems online that you can followalong 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 solutionto 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 tofocus on in this video which are the best onesI'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 Symbolabwas a little bit slower and less intuitive thanit is with WolframAlpha their step-by-step solutions are free.
Regardless of the toolthat you choose to use here the underlying point is that sometimes it can be useful to seea step-by-step solution for a problem you're stuck on.
But, there are two veryimportant 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 tosolve this problem first?” Expending the mental effort required to solve the problem yourself is going to stretch your capabilities.
It's going to make youa better mathematician in a way that just lookingthrough solutions won't.
Now, if you do need to lookup 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 donethat, challenge yourself to go back and rework the problem without looking at that reference.
It is really important tostay 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 comeswith looking up solutions is that with math it's really easy to follow along with astep-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 tomy final tip for you.
And this is especiallyimportant for anybody in a math class workingthrough assigned homework.
Don't rush when you workthrough 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 videoabout it pretty recently.
But, with math and science and any sort of reallycomplex subject especially rushing is only going tohurt you down the road.
Because when you rush, youdon'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 thecombination 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 combinationof concepts intact but swapping in smaller, easier to handle numbers.
If you need help withthe concepts themselves go to your book or anexplainer article online look up sample problems, or use a tool like WolframAlpha or Symbolabto get step-by-step solutions to the problem you're working on.
And finally, don't rush throughyour homework assignments.
Make sure that you're focusing intently on mastering the concepts, not just finishing.
Hopefully these tips willgive you the confidence to tackle some really tough math problems and to expand your math skill set.
And on that note, I wantto leave you with a quote from the great physicist, Richard Feynman, who said, The bottom line is this: Ultimately, your abilityto get good at math and anything else for that matter starts with having theconfidence to approach it.
And as you solve problemsand make mental breakthroughs your confidence is goingto naturally increase.
It becomes a self-sustaining cycle.
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I want to give a huge thanks to Brilliant for sponsoring this video and helping to support this channel.
And, as always guys, thankyou so much for watching.
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Thanks for watching and I willsee you in the next video.
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