Slide 1: Hello, and welcome to this webinar on problem solving with Daniel Showalter.
Slide 2: Aside from basic arithmetic, problem solving is one of the most important skills that students can learn in math class. It will help them in future math classes and, more importantly, throughout their lives.
Slide 3: With technology available to assist with calculations, in fact, educators are increasingly focusing attention on problem-solving.
Slide 4: There are many styles and techniques for effective problem solving. Many of them can be traced back to a method designed by the “Father of Problem Solving,” George Polya.
Slide 5: Polya’s problem-solving method consists of four steps: understand the problem; make a plan; execute the plan; and look back and reflect. Many of the problems we all encounter in life benefit from this approach. For example, when we recognize that a car has mechanical problems or when we realize that we are unhappy with a particular job, we first try to understand the problem. Then we make a plan for addressing it, execute the plan, and reflect on how well the plan worked. Let’s take a look at each of these steps as they apply to math problems.
Slide 6: Step 1: Understand the problem. The first step in solving any math problem should be attempting to figure out what is being asked. If the problem is to solve for x in the equation 2 x = 14, we can rephrase the problem to state, “What number, when multiplied by 2, gives 14?” What else is needed in order to understand this problem? For example, we need to know that 2x means two times an unknown number. We also need to know that an equal sign mean that the expressions on either side of it represent the same value. In other words, that 2x is another name for 14.
Slide 7: Of course, many problems are more complicated than the one in this example. Here are some questions that can be used to help understand the problem better: “What do the terms mean?” “What information am I given?” “In what form should I express my answer?” “Have I solved any problems like this before?”
Slide 8: Step 2: Make a plan. Once the student understands what a problem is asking, it is time to come up with a strategy for solving the problem. This is often the most difficult step, and students are much more likely to succeed if they have learned a few of the main ways to approach a problem. The good news is that the more strategies that a student masters, the easier it is for the student to recognize where a particular strategy might be useful. Let’s look at a few of the most powerful strategies.
Slide 9: The first strategy is to make an organized list. In some cases, it is possible to find the solution by listing out all the possible results. If the problem involves a series of many results, calculate the first few and then see if there is a clear pattern emerging. If so, is there a way to figure out a future result without listing out every single result in between?
Slide 10: The second strategy is to draw a picture. Not only are pictures helpful in solving geometry problems, but they can also help with many other types of problems. Pictures are particularly helpful for visual learners or to help isolate individual parts of a larger problem.
Slide 11: The third strategy is to eliminate possibilities. On a multiple-choice test, eliminating one or two possible answers can drastically increase the chance of selecting the correct answer. In the case of a story problem, it is often possible to eliminate outcomes that don’t make sense, such as negative lengths or fractions of a person.
Slide 12: The fourth strategy is to solve a simpler, related problem. If the problem asks for a general answer, solve the problem in a couple specific cases and then look for a pattern. If possible, break the problem into smaller problems that are easier to solve.
Slide 13: The fifth strategy is to use a formula. This is a more advanced technique. Although formulas can be quite powerful and can help solve a problem quickly, it is also very possible to use a formula blindly without understanding what it means. When you see students using formulas, ask them why they chose that particular formula and have them explain what it means. One nice point about Polya’s method is that it will often catch an incorrect formula when the student checks how reasonable the answer is in Step 4.
Slide 14: The final strategy is to guess and check. Most math teachers are not fond of the guess and check strategy because students tend to use it when they don’t have a more efficient way of solving the problem. One way to help students use this approach in a way that will advance their learning is to let them guess and check first and, then, if they are correct, to work backwards and figure out why the answer was correct.
Slide 15: Step 3: Execute the plan. This step is often easier than coming up with the plan although it can require patience, especially when a planned strategy does not work. In this case, the problem-solver needs to go back to Step 2 and select a different strategy.
Slide 16: Step 4: Look back and reflect. Students have a tendency to move on to the next problem as soon as they find a solution. Even in the 1940s when Polya created his 4-step method, he claimed that this was the most overlooked step. After finding a solution, students should check whether or not it is reasonable or even use a different method from Step 2 to confirm their answer. They should then reflect on the overall process. Reflecting on the process makes it more likely that the student will be able to identify cases in future problems where a similar strategy would work well.
Slide 17: Here are some example questions that you can share with students to help them reflect on the problem-solving experience. “Is this answer reasonable? Why or why not?” “How could you improve the way you solved the problem?” “Are there other types of problems for which you could use this solution method?”
Slide 18: You probably use something similar to Polya’s four-step plan already, perhaps just not in math. Let’s try applying the method to a problem that we’re having. Let’s say that our car starts making a funny tapping sound when we drive it. First, we want to understand the problem. We notice that the sound gets faster when we accelerate. It goes away as soon as we stop the car.
Slide 19: Second, we need to make a plan. One solution would be to take it to a mechanic; but that’s quite expensive! Another solution would be to look under the hood and see if we can notice anything obviously wrong.
Slide 20: Even though we don’t have much hope in this plan of investigating the problem ourselves, it’s free, and so we move onto step three and execute the plan. Our inspection doesn’t turn up anything obvious, and so we return to step two and try a different method: We take the car to a mechanic. For $159.40, the mechanic discovers that our oil pressure is low. We have an old car that leaks some oil, and we hadn’t bothered checking the level recently.
Slide 21: Finally, we look back together on the problem. The tapping sound did stop, and so the problem is resolved, but it cost a fair bit of money for something we could have handled ourselves. Upon reflection, we realize that in Step One, we could have factored in the growing oil slick in our driveway as part of understanding the problem. In Step Two, we could have considered the few things we do know how to deal with a car—such as checking the oil level—before rushing to the mechanic for help.
Slide 22: Notice how important the fourth step was in our example. Because we reflected on the problem, we’ll be more likely to consider more low-cost options the next time something is wrong with our car. Even better, we might apply what we learned with our car problem to problems that occur with our computer, our phone, and other possessions. In the same way, when students reflect on how they solved a problem, they are more likely to generalize their method to similar types of problems.
Slide 23: In the car example, it was natural to apply a basic problem-solving method. In math, sometimes we have to be more intentional about applying it. Polya’s problem-solving method is an excellent tool to have ready when a student says, “I don’t even know where to start!” You can start with Step 1, asking something like, “Okay, what information does the problem give us? What are we being asked to find?”
Slide 24: The beauty of Polya’s problem-solving method is that you can use it to help students solve almost any type of math problem, even if it’s not a story problem, and even if you don’t quite know how to solve the problem yourself! In this unit, you will have some practice using Polya’s method to either solve a problem with a student or on your own.