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.
数学是抽象的 复杂的 不幸的是
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,
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 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 his1945 book”How to Solve It.”
他说：“如果你不能解决一个问题 那么你一定能够解决一个比较简单的问题 找到它”
It goes: “If you can’t solve a problem, then there is an easier problem you can solve:find it”
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 arebuilt upon simpler concepts.
And if you don’t have a strong grasp
掌握得不牢 那么 复杂的
on the fundamental principles, then a more complex
problem is going to likely stump you. So,
if you come across a problem
解不了的问题时 首先 找到你要解决问题时
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 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 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 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 notbeing 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 limiton my summation problem.
复杂的数字 大数字 有许多小数点的数
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 similar problem with really small
whole numbers that are easy to add or operate
相加或运算 这样 你才能真正
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 timeto 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
WolframAlpha 和 Symbolab
are WolframAlpha and Symbolab.
Both of these websites will allow you to type
in an equation and get an answer and also Alpha 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 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 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,
Look it up, follow the steps and make sure
that you understand how the answer was arrived at. But,
但是 一旦你看完了 考考自己
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 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 tomy final tip for you.
And this is especially important
for anybody in a math class working through 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 video about it pretty recently. But,
但是 对于数学 科学
with math and science
and any sort of really complex 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 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,
或使用工具 WolframAlpha 或Symbolab找
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 throughyour 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 problemsand make mental breakthroughs
your confidence is goingto 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.
A learning platform that uses hands-on problem solving as
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I’m actually taking their computer science fundamentals course right now,
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new math I had never learned before.
I had to dig into wikisand example problems.
And eventually I had to get out a sheet
of notebook paper and literally draw out algorithms step-by-step
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This process provided me
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sitting through mostof my college lectures.
And these types of challenges that really force you to dig in,
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包括概率 逻辑 微积分 天文
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I want to give a huge thanks to Brilliant
for sponsoring this video and helping to support this channel. And,
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