 So let's talk about a term that's gaining more and more cachet among those who are looking to change how we teach mathematics, and that's the term problem-solving. And many mathematics education standards now emphasize problem-solving as an important goal in mathematics education. And this often raises the question, wait a minute, what are all those things that we do at the end of each section of a math book? Aren't those problems, and aren't we problem-solving when we do these things? The answer to that question is probably not. Those things that we do at the end of each section of a math book, even though they might be called problems, are not problems within the context of the term problem-solving. And to illustrate the difference, let's consider a lesson on the rules of exponents as presented in a traditional math course, well, actually, as I've actually taught these in the past. So let's consider that. So what we might do, we might start out with by saying, okay, so this thing, this exponent a to power n, well, that's the product of n factors of a. And so, for example, 2 to power 3, that's the same as 2 times 2 times 2. And so now I might ask, well, what happens when I multiply two exponential expressions? And so I'll say, okay, well, here's what happens. 2 to the third is 2 times 2 times 2, 2 to the second is 2 times 2. And altogether, we have 1, 2, 3, 4, 5 factors of 2. And this looks like 2 to power 5. And well, this leads us to the following generalization. If I multiply two exponential expressions, then what I do is add the exponents. And then I might say, okay, well, here's an example, 5 to power 10 times 5 to the third. And that's going to give us that. And we'll assign this homework problems a bunch of finding products of exponential expression. Now, this is not problem solving. It actually doesn't have a specific name for it, but maybe we'll call it example following. The reason it's not problem solving is that once you've been shown how to find the value a to power m times a to power n, finding these products is not a problem. It's a matter of following an example. And one of the problems here is that when mathematics is presented this way, we train students to look for examples where similar questions have been solved. And then to follow those examples to get an answer. However, there's some practical difficulties with this. The main one is that it's impossible to give examples of every type of question that could possibly appear. So sooner or later, students are going to encounter questions for which they've never seen an example. This also a second problem, which is that judging similarity requires some level of experience and sophistication. So if I've just introduced, here's how you multiply two exponential expressions, it's fairly easy for students to identify that two to the fifth, two to the sixth is equivalent is the same problem as five to the third, five to the tenth. And it's easy to see that the two problems are the same. But at some point they're going to not have all their problems labeled and they'll have to make a decision. And so for example, 3x plus 5 equals 2x, 3x plus 5 equals x squared. Well, these problems appear similar, but they're not solved in the same way. They are radically different problems from a perspective of what you do with them next. And that development of similarity, that development of this ability to judge similarity is something that takes quite a bit of time and practice. So this is why the shift has gone from following examples to solving problems. And how might that show up? Well, so instead of being given examples of how to multiply two exponential expressions, students can solve problems involving the multiplication of two exponential expressions. And so this might occur in the following way. We might begin the same way. We will define a to the power n as the product of n factors of a. And again, we might say, okay, well, here's an example of what we mean. 2 to the power 3 is 2 times 2 times 2, so there's three factors of 2. At that point, we'll introduce the problem. What is 2 to the power 3 times 2 to the power 2? And at this point, rather than having us as the teachers solve this problem, we want to shift the focus to the student. And so the student hasn't been shown how to solve this. This is an actual problem. They have to integrate their understanding of this exponential concept to be able to solve it. And if they understand the concepts of exponents, they can solve this with some guidance and then go on to, again, with some guidance, more sophisticated problems. Find 5 to the 8th, 5 to the 12th. Find x, y cubed to the second and x plus 3 to the second. And so on. And they can solve other types of problems, all based on this understanding of exponents as a repeated product problem. And so all of these become real problems because they're not taught how to multiply two exponential expressions. They're not taught how to expand or divide two exponential expressions. They're real problems whose solution is based on understanding the fundamental concept of powers as products. So how do we incorporate problem solving into the classroom? Well, there's three key ideas to... There's three key ways of making this work. Practice, patience, and preparation. The key is practice. Problem solving is a skill. You get better at it the more often you do that. The first time your students are exposed to a real problem and ask to solve it, they'll have some difficulty because if they've never done it before, it's difficult to get an idea of how to start. But the more often they solve problems, the easier it becomes for them to approach the problems. Now, the difficulty here is problem solving is like first impressions. You never, never, never, never, never get a second chance to solve a problem for the first time. And what this means is that the instant someone shows you how to solve a problem, you have forever lost an opportunity to solve that problem. You never get another chance to solve the problem. So going back to this problem, find two to the third times two to the second, the instant someone shows you that this is really found by two to the third looks like this, two to the second is two times two, the instant somebody shows you that problem, then you never get another chance of solving this problem ever again and you have lost forever that educational opportunity. And this means a couple of important things. First of all, you do want to avoid giving examples of solved problems because, again, as soon as somebody shows you how to solve a problem, you never have a chance of solving that problem ever again. So what do you do when they ask questions of how do you solve it? Well, again, we want to go back and emphasize the underlying concepts. So from the understanding of the underlying concepts, the solution to the problem should emerge. Now, this is the 21st century, so it's easy enough for a student to look up the answer. This is the age of Google, Math BFF, and a whole bunch of others. And if you don't show the student how to solve a problem, they're going to find somebody who will. They'll look up the answer online. And this needs to be discouraged. This really needs to be discouraged as much as possible. And I don't have any great way of doing this. I don't know how to discourage students from doing this. I will say that you should emphasize the once-in-a-lifetime opportunity to solve problems and that the instant that you look up an answer, you are never going to get another chance to solve the problem. This is a once-in-a-lifetime chance. So I don't know how well that works, but it does seem to be something that should be emphasized. Now, if you want to incorporate real problem solving, this does require patience. Problem solving requires these students to solve problems. They have to create their solutions, but they will need some guidance. And what this often translates into in practice is group work. And this is actually good preparation for the real world because real world problems usually require collaboration by hundreds, if not thousands of people. Time management is also important. In general, you don't have enough class time to give a lot of examples and then do problem solving, but that's okay because presenting a lot of examples defeats the purpose of trying to problem solve. Incidentally, one of the key tools that we have for making problem solving work is to use some sort of flipped or inverted class structure. And so students can read about or watch videos on basic concepts out of class and then go to class to work on problems. And finally, the last important component here is preparation. Let's be honest. Given five minutes, most of us, college professors, self-mathematics, most of us could prepare an hour-long lecture on some introductory math topics, solving linear equations, differentiation, Gaussian reduction, whatever. Most of us can prepare a lecture without really having to work too hard at it. If you want to build your class around problem solving, this requires significantly more preparation. You have to know your students. Can they go from the basic definition of exponents to solving something like this in one set of problems? Or is it going to take a little bit more guidance and will it be something they have to work through with several problems? This is one of the reasons why if you're going to go the problem solving approach, you can use standard books on the topic as resources and inspiration, but ultimately, you're going to have to make up your own sets of problems because a standardized book is good for standardized students, and the problem of teaching is that students are not identical. One other important thing, you have to be able to block the shortcuts. Some of the students are already going to say, oh, well, I know how to do this, so I'm going to follow the rule that I was taught. So many of them will already know this rule. You have to block the shortcuts. If you want this to be a problem-solving experience, the challenge is making it a problem-solving experience even for students who already know the rule. Is it worth it? Well, mathematics is a collection of rules and algorithms to follow, said no mathematician ever. Okay, technically I just said it, but we'll ignore the paradox of ethnicities. The idea is that nobody, no mathematician, is ever going to honestly claim that mathematics is a collection of rules and algorithms to follow. And so this is why problem-based learning is so very important. With problem-based learning, with a class centered around problem-solving, we have the chance to reinforce conceptual understanding. If you don't know what a to power n means, if you don't understand the concept that's embedded in this exponentiation, then it's impossible for you to find the product. And more importantly, you shouldn't be able to find the product to give somebody the power rule for exponents who doesn't understand what exponents mean is like giving a chainsaw to a toddler. The other reason, another reason, is that this chain students to think about mathematics the way that mathematicians think about mathematics. So Benjamin Perse, that's not a typo, his name is actually spelled P-E-I-R-C-E. Benjamin Perse is one of the founding figures of American mathematics, and he has this comment. Mathematics is the science of necessary consequences. What mathematicians do is they begin with some sort of concept, a to power n, and they see what follows from that concept. And finally, the other advantage to learning through problem-solving is this humanizes mathematics. Anything that can be solved by following an example can be done faster, more accurately, and less expensively by a computer. So we have to ask ourselves a question. Why learn how to follow examples when that can be done faster, more accurately, and less expensively by an inanimate device? The real lesson of John Hendry, folk hero who tried to compete with a steam shovel, is don't try to beat the machine. You can't do it. You cannot beat the machine. But what you can do, what you should do, is you should try to transcend the machine. Things that can be done by following an example can be done by a machine. So this is not really a human task. It's not worth trying to make yourself good at this because a computer will always be faster, more accurate, and less expensive. What a computer can't do, what a machine can't do, is problem-solving. And that is one of the main reasons we want to start focusing our efforts on learning how to solve problems instead of follow examples.