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# Content: Polya’s Problem-Solving Method

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## Purpose

The purpose of this tool for the field is to help paraprofessionals become more familiar with, and practice using, Polya’s four-step problem-solving method.

## Procedures

1. Read the section below entitled “Background Information,” and familiarize yourself with the chart of Polya’s four-step problem-solving method.

2. Read the example below about Mrs. Byer’s class, and then look over the example of how Polya’s method was used to solve the problem.
3.                                                                                 Choose one of the following problems and try to solve it yourself using Polya’s method:

Every person at a party of 12 people said hello to each of the other people at the party exactly once. How many “hellos” were said at the party?

A new burger restaurant offers two kinds of buns, three kinds of meats, and two types of condiments. How many different burger combinations are possible that have one type of bun, one type of meat, and one condiment type?

A family has five children. How many different gender combinations are possible, assuming that order matters? (For example, having four boys and then a girl is distinct from having a girl and then four boys.)

Hillary and Marco are both nurses at the city hospital. Hillary has every fifth day off, and Marco has off every Saturday (and only Saturdays). If both Hillary and Marco had today off, how many days will it be until the next day when they both have off?

### TAKE NOTES

1. In which types of situations do you think students would find Polya’s method helpful?
2. Are there types of problems for which students would find the method more cumbersome than it is helpful?
3. Can you think of any students who would particularly benefit from a structured problem-solving approach such as Polya’s?

## Background Information

Nearly 100 years ago, a man named George Polya designed a four-step method to solve all kinds of problems: Understand the problem, make a plan, execute the plan, and look back and reflect. Because the method is simple and generalizes well, it has become a classic method for solving problems. In fact, the method is applicable to all areas of our lives where we encounter problems—not just math. Although the method appears to be a straightforward method where you start at Step 1, and then go through Steps 2, 3, and 4, the reality is that you will often need to go back and forth through the four steps until you have solved and reflected on a problem.

## Polya’s Problem-Solving Chart: An Example

A version of Polya’s problem-solving chart can be found below, complete with descriptions of each step and an illustration of how the method can be used systematically to solve the following problem:

### Scenario

There are 22 students in Mrs. Byer’s third grade class. Every student is required to either play the recorder or sing in the choir, although students have the option of doing both. Eight of Mrs. Byer’s students chose to play the recorder, and 20 students sing in the choir. How many of Mrs. Byer’s students both play the recorder and sing in the choir?

#### Module: Helping Students Do Math

• Unit 1: What is Math?
• Unit 2: Experiences with and Attitudes Toward Math
• Unit 3: Math Instruction: The Big Picture
• Unit 4: Learning from a Math Textbook
• Unit 5: Math and Technology
• Unit 6: Asking Questions That Help Students Think
• Unit 7: Problem Solving
• Unit 8: Math and the Common Core
• Unit 9: Number Sense (Math Facts, Memorization, and Practice)
• Unit 10: Visual Math
• Unit 11: Putting it All Together for Paraprofessionals
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