If I asked you to think of a really tough problem, what is the first thing that comes to your 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 are reading this topic 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.
(Tip: 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 post.
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 - George Polya
So, (Tip: 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 illustrated below. Let's assume you are given these two summation problems to solve.
Now, looking at the two problems at a glance, problem 2 looks more complicated and complex than problem 1
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, the summation problem 2 has a fractional exponent in it which makes it seem complicated, and that's what scares you away if you are not good in math. (Tip: Never get scared by the complexity of any question, they all have the same working method as the simpler one)
Now, maybe some of you math wizards out there could solve problem 1and 2 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. In this case, you will have to deal with the fractional exponets first before solving the summation. But what if you don't know how to treat exponents? Then, you will have to go back to the topic to learn about it. The truth is that (Tip: math builds upon itself) and so, if you are not good at beginning topics, you will definitely not do well in higher topics. You definitely need to go back to those lower and easier topics to learn and begin climbing from there.
If you can't solve a problem, then there's an easier problem you can solve: Find it. - George Polya
So, let's solve probelm 1
All we have done here is to evaluate that expression four times and then add up the answers which gets us to a final answer of 66.
For problem 2 that has exponent which is fraction, we would make it appear simple like the problem 1 by rewriting the fractional part as four to the power of three times the power of one half. And then we 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, (Tip: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. Do you know that 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, (Tip: 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 post which are the best ones I've been able to find are WolframAlpha and Symbolab. Just google the two tools and see for yourself the usefulness.
Both of these websites will allow you to type in an equation and get an answer. 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, (Tip: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.
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