 So, YouTube is a great educational resource. You can find out how to do just about anything by finding a YouTube video on it. But there's an important lack, and I haven't been able to find any good videos that explain how to fail a math course, so I've decided to make one. So, well, there's an easy way to fail a math course, so you could be not prepared for class. Maybe you're taking calculus, but you haven't yet passed algebra. Not going to class. Registering for a course and not showing up for class is sort of like buying lunch and leaving it at the counter. You've paid for it, but you're going to still be hungry an hour later. So, you could also not do the work. In general, your grade is going to depend on the work that you've done, so not doing the work is an easy way to fail the course. And so, these are the easy ways to fail a math course. On the other hand, we might also take a look at the hard way to fail a math course. So, what's a hard way to fail a math course? Well, you could be prepared for the course, have taken all the prerequisite courses, you can go to every class, you can do all the work, and then fail the class. And the difference is, if you're going to fail the class, you might as well do it the easy way rather than doing it the hard way. So, how can you do this the hard way? Well, I've been teaching math for years, but I'm still a math student and I'm still learning new things in mathematics every day. And I know a great deal about how to fail math, because I've done it quite a bit. So, let's take a look at this. Well, one of the things that we can do that will make sure that we fail a math course is we can memorize procedures. And I'll point out, procedures are great time savers and there are some things that you should memorize. But if you memorize a procedure without understanding why you're doing it, why it works, it's a fast way to failing a math course. So, for example, if I have the quadratic equation x squared minus x minus 6 equals 10, well, many of us have learned the procedure that when you have a quadratic equation, you can factor it. So, we'll factor it, and let's see that factors is x minus 3 times x plus 2, and we're really good at factoring, so we do this factorization perfectly and we arrive at the answer, x equals 3, x equals negative 2, which is the wrong answer. If you don't know why these answers are wrong, if you don't know why factoring doesn't work in this case, you probably shouldn't be using the procedure of factoring to solve a quadratic equation. Another good way to make sure that you fail a math course is to follow the examples, and many math teachers, myself included, I'll admit that I've done this myself, give examples of how to solve problems, and then we assign problems based on those examples. And unfortunately, this gives us the impression that problems can be solved by following the examples. The problem is that it's not possible to give examples for every situation that can occur. So you will always encounter problems for which you've never seen an example of that problem being solved. No matter how many examples you are given, you'll always encounter something new at some point. So what does this mean? Well, it means when you look at the examples, they are not directions of how to solve a problem. What you should understand is the reasoning behind the example, the logic of what and why we're doing something. Well, here's another common easy hard way to fail a math course is to work problems in your head. And many math students, and again, I am a math student, so I include myself in this category, we practice mathematics by thinking about what we would do to solve a particular problem. And it's a great workout. This is a great thing to do. You should be able to solve problems in your head. But the risk here is that it's very easy to convince yourself that you know how to solve a problem without actually being able to solve it. And this is because when you go to solve actual problems, you often run into unexpected complications that aren't apparent until you get to them. Well, maybe you don't want to fail math. Maybe your goal is not to fail a math course. In that case, here's three bumper sticker sayings that you might want to take to heart. You might want to memorize. Typically, if I can reduce something to a bumper sticker, it's going to be something that's either very trivial or maybe it's really profound. I'm hoping these are profound. So paper is cheap. Understanding is priceless. So what does this mean? Well, write stuff down. Even if you can do several steps in your head, one right after the other, even if you can do them perfectly, write stuff down. Write things down one step at a time. Paper is cheap. Understanding is priceless. Definitions are the whole of mathematics. All else is commentary. You have to know the definitions. You cannot do mathematics unless you know the definitions. And importantly, to understand the difference between a definition, which you have to know, a theorem and a rule. It's useful to know things besides the definitions. The theorems and rules are actually very helpful to know because they make things work a little bit more efficiently, a little bit faster. But if you don't know the definitions, you cannot do mathematics. It is impossible for you to succeed in mathematics without knowing the definitions. It doesn't matter if you get it right at first, as long as you get it right at last. Once you get past very, very, very, very, very, very simple arithmetic, there's a lot of trial and error in mathematics. There's a lot of things that we try something out, and if that doesn't work, well, the most important thing to remember about trial and error is that it involves making a lot of errors and trying again. And we'll try something. It doesn't work. We go on to the next thing. We'll try. And a little analysis goes a long way. Wait, did I say three? I didn't mean three. I don't mean three bumper stickers. I meant four bumper stickers. Let's change that. A little analysis goes a long way. We want to analyze the problem before we start to write things down. And this is in particular, this is particularly a problem if we have been trained to applying certain procedures. We might not want to automatically start applying a procedure. We do want to analyze the problem a little bit before we start working.