what is polya's four steps in problem solving

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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.

• 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.

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?

Reflect on your experience.

• In which types of situations do you think students would find Polya’s method helpful?
• Are there types of problems for which students would find the method more cumbersome than it is helpful?
• 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?

Helping Students Do Math

• Introductory Scenario and Pre-Test
• Content: Does Anyone Know What Math Is?
• Introductory Scenario
• Content: The Fennema-Sherman Attitude Scales
• Content: Past Experience with Math
• Content: Learning About Math
• Content: What is it like to teach math?
• Content: Using a Frayer Model
• Content: Helping a Child Learn from a Textbook
• Content: Using Online Math Resources
• Content: Helping a Student Learn to use a Calculator
• Links for More Information
• Content: Better Questions
• Content: Practice Asking Good Questions
• Content: Applying Poly’s Method to a Life Decision
• Content: Learning Progression Activities
• Content: Connecting Concepts and Procedures
• Content: Resources
• Activity: The Old Guy’s No-Math Test
• Take Notes and Post-Test
• Open All · Close All

Contact: [email protected] 

The Ohio Partnership for Excellence in Paraprofessional Preparation is primarily supported through a grant with the Ohio Department of Education and Workforce, Office for Exceptional Children. Opinions expressed herein do not necessarily reflect those of the Ohio Department of Education or Offices within it, and you should not assume endorsement by the Ohio Department of Education and Workforce.

• Jul 5, 2021

Problem-Solving Steps that Actually Work

Updated: Mar 6, 2023

Whenever our students encounter problems, it can be a tricky situation. On one hand, I get super excited about the idea of my students THINKING about everything they know to solve the problem. I love watching their brains work while they access that filing cabinet in their brain of math information and pull out the information to solve a challenging problem.

On the other hand, that same process can become a brick wall when it becomes too overwhelming. Students can shut down and refuse to move. They can cry and become frustrated. These same students can then begin believing they are just not good at math from this point forward.

That's a lot of pressure from a simple math problem.

If you haven't read Jo Boaler's Mathematical Mindsets, I strongly suggest it as a way to begin helping our students see math learning with a growth mindset. It's a helpful guide in teaching our students and ourselves that knowledge is something that grows and is not fixed. It is based on Carol Dweck's work with Growth Mindsets from Mindset: The New Psychology of Success . (Another great read in helping children as a parent, teacher or coach.)

So what do we do? Instead of bombarding our students with several strategies to make problem-solving easier, I think it's important to boil it down to the basics. What strategies can I give my students that help them with all problems? What's something that's easy for them to remember and recall? What's something that would give them confidence moving forward?

Enter in Polya's Problem-Solving Method by George Polya who was known as the father of problem solving. These four steps sum up everything our students need to solve problems successfully. They are easy to remember and easy to implement.

(This post does contain affiliate links.)

Understand the Problem: This is the focus on comprehension. What is the problem asking me to do? What do I know from reading the problem? What can I comprehend?

Plan: This is the time where students think about how they want to move forward. Before solving with mathematics, we want our students to determine what steps they should take.

Solve : This is where students do the math. They follow the steps in their plan and work out the problem.

Look Back: Now we want students to look back and see that their answer makes sense. We want them to check the answer using estimation or even by trying to solve it in another way.

Four steps...that's totally manageable right? I love the simplicity of it all and even find that it carries over to all aspects of our life when solving real-life problems.

Now that students have a way to solve problems, it's time to give them the tools to make a plan that will work. I've been talking about Singapore's heuristics in my Member's Facebook group, and I wanted to share some of those with you. Stay tuned in the next few weeks to learn about the heuristics and how these strategies help students determine a meaningful plan to solve problems.

In the meantime, be sure to grab your problem-solving poster by clicking below!

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April 19, 2023 3-5-operations-and-algebraic-thinking , k-2-operations-and-algebraic-thinking , 6-8-expressions-and-equations

Polya’s problem-solving process: finding unknowns elementary & middle school, by: jeff todd.

In this article, we'll explore how a focus on finding “unknowns” in math will lead to active problem-solving strategies for Kindergarten to Grade 8 classrooms. Through the lens of George Polya and his four-step problem-solving heuristic, I will discuss how you can apply the concept of finding unknowns to your classroom. Plus, download my Finding Unknowns in Elementary and Middle School Math Classes Tip Sheet .

It is unfortunate that in the United States mathematics has a reputation for being dry and uninteresting. I hear this more from adults than I do from children—in fact, I find that children are naturally curious about how math works and how it relates to the world around them. It is from adults that they get the idea that math is dry, boring, and unrelated to their lives. Despite what children may or may not hear about math, I focus on making instruction exciting and showing my students that math applicable to their lives.

Problem solving is a fundamental means of developing students' mathematical knowledge and it also shows them that math concepts apply to real-world concepts.

Problem solving is one way I show my students that math relates to their lives! Problem solving is a fundamental means of developing students' mathematical knowledge and it also shows them that math concepts apply to real-world concepts.

Who Is George Polya?

George Polya was a European-born scholar and mathematician who moved to the U.S in 1940, to work at Stanford University. When considering the his classroom experience of teaching mathematics, he noticed that students were not presented with a view of mathematics that excited and energized them. I know that I have felt this way many times in my teaching career and have often asked: How can I make this more engaging and yet still maintain rigor?

Polya suggested that math should be presented in the light of being able to solve problems. His 1944 book,  How to Solve It  contains his famous four-step problem solving heuristic. Polya suggests that by presenting mathematical thinking as a way to find “unknowns,” it becomes more engaging for students.

He even goes as far as to say that his general four-step problem-solving heuristic can be applied to any field of human endeavor—to any opportunity where a problem exists.

Polya suggested that math should be presented in the light of being able to solve problems...that by presenting mathematical thinking as a way to find “unknowns,” it becomes more engaging for students.

Polya specifically wrote about problem-solving at the high school mathematics level. For those of us teaching students in the elementary and middle school levels, finding ways to apply Polya’s problem-solving process as he intended forces us to rethink the way we teach.

Particularly in the lower grade levels, finding “unknowns” can be relegated to prealgebra and algebra courses in the later grades. Nonetheless, today’s standards call for algebra and algebraic thinking at early grade levels. The  download  for today’s post presents one way you can find unknowns at each grade level.

Presenting Mathematics  As A Way To Find "Unknowns" In Real-Life Situations

I would like to share a conversation I had recently with my friend Stu. I have been spending my summers volunteering for a charitable organization in Central America that provides medical services for the poor, runs ESL classes, and operates a Pre-K to Grade 6 school. We were talking about the kind of professional development that I might provide the teachers, and he was intrigued by the thought that we could connect mathematical topics to real life. We specifically talked about the fact that he remembers little or nothing about how to find the area of a figure and never learned in school why it might be important to know about area. Math was presented to him as a set of rules and procedures rather than as a way to find unknowns in real-life situations.

That’s what I am talking about here, and it’s what I believe Polya was talking about. How can we create classrooms where students are able to use their mathematical knowledge to solve problems, whether real-life or purely mathematical?

As Polya noted, there are two ways that mathematics can be presented, either as deductive system of rules and procedures or as an inductive method of making mathematics. Both ways of thinking about mathematics have endured through the centuries, but at least in American education, there has been an emphasis on a procedural approach to math. Polya noticed this in the 1940s, and I think that although we have made progress, there is still an over-emphasis on skill and procedure at the expense of problem-solving and application.

I recently reread Polya’s book. I can’t say that it is an “easy” read, but I would say that it was valuable for me to revisit his own words in order to be sure I understood what he was advocating. As a result, I made the following outline of his problem-solving process and the questions he suggests we use with students.

Polya's Problem-Solving Process

1. understand the problem, and desiring the solution .

• Restate the problem
• Identify the principal parts of the problem
• Essential questions
• What is unknown?
• What data are available?
• What is the condition?

2. Devising a Problem-Solving Plan 

• Look at the unknown and try to think of a familiar problem having the same or similar unknown
• Here is a problem related to yours and solved before. Can you use it?
• Can you restate the problem?
• Did you use all the data?
• Did you use the whole condition?

3. Carrying Out the Problem-Solving Plan 

• Can you see that each step is correct?
• Can you prove that each step is correct?

4. Looking Back

• Can you check the result?
• Can you check the argument?
• Can you derive the result differently?
• Can you see the result in a glance?
• Can you use the result, or the method, for some other problem?

Polya's Suggestions For Helping Students Solve Problems

I also found four suggestions from Polya about what teachers can do to help students solve problems:

Suggestion One In order for students to understand the problem, the teacher must focus on fostering in students the desire to find a solution. Absent this motivation, it will always be a fight to get students to solve problems when they are not sure what to do.

Suggestion Two A second key feature of this first phase of problem-solving is giving students strategies forgetting acquainted with problems.

Suggestion Three Another suggestion is that teachers should help students learn strategies to be able to work toward a better understanding of any problem through experimentation.

Suggestion Four Finally, when students are not sure how to solve a problem, they need strategies to “hunt for the helpful idea.”

Whether you are thinking of problem-solving in a traditional sense (solving computational problems and geometric proofs, as illustrated in Polya’s book) or you are thinking of the kind of problem-solving students can do through STEAM activities, I can’t help but hear echoes of Polya in Standard for Math Practice 1: Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.

In Conclusion

We all know we should be fostering students’ problem-solving ability in our math classes. Polya’s focus on “finding unknowns” in math has wide applicability to problems whether they are purely mathematical or more general.

Grab my  download  and start  applying Polya’s Four-Step Problem-Solving Process in the lower grades!

Beginning Algebra Tutorial 15

• Use Polya's four step process to solve word problems involving numbers, rectangles, supplementary angles, and complementary angles.

Whether you like it or not, whether you are going to be a mother, father, teacher, computer programmer, scientist, researcher, business owner, coach, mathematician, manager, doctor, lawyer, banker (the list can go on and on).  Some people think that you either can do it or you can't.  Contrary to that belief, it can be a learned trade.  Even the best athletes and musicians had some coaching along the way and lots of practice.  That's what it also takes to be good at problem solving.

George Polya , known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.  I'm going to show you his method of problem solving to help step you through these problems.

If you follow these steps, it will help you become more successful in the world of problem solving.

Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:

Step 1: Understand the problem.  

Step 2:   Devise a plan (translate).  

Step 3:   Carry out the plan (solve).  

Step 4:   Look back (check and interpret).  

Just read and translate it left to right to set up your equation .

Since we are looking for a number, we will let 

x = a number

*Get all the x terms on one side

*Inv. of sub. 2 is add 2  

FINAL ANSWER: 

We are looking for two numbers, and since we can write the one number in terms of another number, we will let

x = another number 

one number is 3 less than another number:

x - 3 = one number

*Inv. of sub 3 is add 3

*Inv. of mult. 2 is div. 2  

Another number is 87.

Perimeter of a rectangle = 2(length) + 2(width)

We are looking for the length and width of the rectangle.  Since length can be written in terms of width, we will let

length is 1 inch more than 3 times the width:

1 + 3 w = length

*Inv. of add. 2 is sub. 2

*Inv. of mult. by 8 is div. by 8  

FINAL ANSWER:

Length is 10 inches.

Complimentary angles sum up to be 90 degrees.

We are already given in the figure that

x = 1 angle

5 x = other angle

*Inv. of mult. by 6 is div. by 6

The two angles are 30 degrees and 150 degrees.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem .  At the link you will find the answer as well as any steps that went into finding that answer.

  Practice Problems 1a - 1c: Solve the word problem.  

(answer/discussion to 1c)

http://www.purplemath.com/modules/ageprobs.htm This webpage goes through examples of age problems, which are like the numeric problems found on this page.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on July 26, 2011 by Kim Seward. All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.