george polya problem solving model

Polya's Problem Solving Process | Overview & Steps

Anita Dunn graduated from Saint Mary's College with a Bachelor's of Science in Mathematics, and graduated from Purdue University with a Master's of Science in Mathematics. She has been certified as a Developmental Education Specialist through the Kellogg Institute. She has more than 10 years of experience as a college professor.

Maria has taught University level psychology and mathematics courses for over 20 years. They have a Doctorate in Education from Nova Southeastern University, a Master of Arts in Human Factors Psychology from George Mason University and a Bachelor of Arts in Psychology from Flagler College.

Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.

Additional Example of Using Polya's Four-Step Problem-Solving Process

In the following example, use Polya's Four-Step Problem-Solving Process as outlined in the video lesson. Be sure to execute each step of the process and to state what that step involves.

Farmer Brown has many animals on his farm. He has 72 chickens, which make up 60% of his total animals, and the rest of his animals are sheep. How many legs in total do his animals have?

1) Step One of Polya's Process is to understand the problem. We are trying to count how many legs the animals have in total. The animals are chickens (which have 2 legs each) and sheep (which have 4 legs each).

2) Step Two of Polya's Process is to devise a plan. We will work with an equation. An example of an equation to use to solve the problem is (72 chickens * 2 legs) + (number of sheep * 4 legs) = total legs. However, we do not know the number of sheep. We know that 60% of the total number of animals is equal to 72, so if n is the total number of animals, we have 0.60n = 72 so the total number of animals is 72/0.6 = 120. Then the number of sheep is the remaining amount of animals. A revised equation to use to solve the problem is (72 chickens * 2 legs) + ((120-72) sheep * 4 legs) = total legs.

3) Step Three is to carry out the plan. We will solve our equation. 120 - 72 = 48 sheep, and so we have (72 * 2) + (48 * 4) = total legs. 72 * 2 = 144 and 48 * 4 = 192, so the total number of legs is 144+192 = 336 legs.

4) Step Four is to look back. Does this answer make sense? There should be more legs than animals and the number should be an even number (the animals each have an even number of legs) and 336 fits this. We can check that 0.6(120) = 72 chickens and that 0.4(120) = 48 to make sure the number of animals is correct. Our answer checks out.

How can Polya's Four-Step Problem-Solving Process help you solve problems?

Guide to Discussion

This is a pretty open-ended question - something that may help guide students on their discussion is to talk about word problems in math class. For many students, the hardest part of word problems is finding out what is even being asked and translating it into an equation - Polya's process helps with these things.

What is Polya's 4 steps in problem solving?

Polya's four step method for problem solving is

1) Understand the Problem-Make sure you understand what the question is asking and what information will be used to solve the problem.

2) Devise a Plan-Figure out what method you will use to solve the problem.

3) Carry out the Plan-Use that method to solve the problem

4) Look Back-Double check your answer and make sure it is reasonable.

What made George Polya famous?

George Polya's book: "How to Solve it" sold over a million copies and has been translated into at least 21 different languages. He is most famous for his "four step problem solving process" which helps students solve word problems.

What is Polya's first principle for solving problems?

Polya's first principle for solving problems is arguably one of the most important steps: Understand the Problem. You first need to make sure you understand any vocabulary words, understand what the problem is asking for, and understand what information is given in the problem which will help you solve it.

Table of Contents

Polya's problem solving process, how to solve using polya's method, lesson summary.

George Polya (1887-1985) was born in Hungary. He received his Ph.D. in mathematics at the University of Budapest. For many years he served as a professor at the Swiss Federal Institute of Technology in Zurich. Then, in 1940, Dr. Polya moved to the United States where he taught briefly at Brown University, and then he moved to Stanford University.

Dr. Polya maintained a lifelong interest in the thought processes we use when we solve math problems. Dr. Polya wrote many books, including How to Solve It (1945). This book sold over a million copies in at least 21 different languages. His methods are now commonly used amongst students when solving word problems. (Long et al., 2015)

His four step process can be summarized by

• Understand the Problem

Devise a Plan

• Carry out the Play

Understand the problem

Are there any vocabulary words you don't understand within the problem? If so, it'd be a good idea to look them up.

Figure out what you are being asked to find.

Can you restate the problem in your own words?

Would a diagram or a picture help you to solve the problem? If so, draw it.

Do you have enough information to solve the problem? Is there information included which you don't need?

Example: Isaac has 5 apples and he has 10 friends. He wants to give 2 apples to each of his friends. How many apples does he need to buy?

For the "understand the problem" step, we want to decide what we are being asked to find. Thankfully, the last sentence tells us. We want to find out "how many apples does he need to buy?" In other words, how many more apples does he need.

Do we have enough information? We know that he has 5 apples already. He has 10 friends. He wants to give two apples to each of his friends. That should give us the ability to solve this problem.

Decide how you are going to try to solve your problem. You could use any of the following methods:

• use algebra
• use basic arithmetic
• look for a pattern
• guess and check ...guess an answer, see if it works. If it doesn't work, try something else.
• use a model or a diagram...sometimes just by drawing a model we can figure out the answer.
• use a formula

Or, something else! There are many methods we could use here. Be creative!

For our earlier example, we can answer this question a few different ways. We could draw out 10 picture of people. Then we can draw two apples per friend, and count out how many Isaac currently has and count how many he needs.

If we had a classroom of children, we could have ten of them stand up and hand out 5 apples. Then we could figure out how many more apples we need to make sure each child has two apples. Hint: The kids could hold out their empty hands, so the other children could count how many more apples are needed.

We could use a bit of arithmetic, finding out how many apples he needs for all of his friends and subtracting to find out how many more he needs.

To solve my example problem, I'm going to use arithmetic.

Carry out the Plan

This is where we do the math and figure out our answer! Whatever you decided your plan was earlier.

Back to my example problem: we know that Isaac has 10 friends and each friend needs two apples. We can multiply {eq}2*10=20 {/eq}

So Isaac needs 20 apples. He currently has 5 apples. Therefore we can subtract {eq}20 - 5 = 15 {/eq}

Therefore Isaac needs 15 more apples. This is how many he will be buying.

This is the step everyone wants to skip. And yet it is also one of the most important steps. Basically, you will be checking your answer and thinking about whether or not your answer makes sense.

• Is there another way I could solve this problem?
• Does my answer make sense?
• Are there similar problems that I could use this strategy on? This will help you later on as you complete more problems.

Thinking about my example problem: Does the answer make sense? Yes, 15 apples is a reasonable number of apples, considering we started with 5 apples and we needed 2 apples per person for 10 people.

I can take a quick look at my arithmetic and make sure I didn't make a simple mistake. Sometimes it's even a good idea to completely redo the problem, it is sometimes easier to find a mistake this way.

Can I solve this problem some other way? I could also draw out a picture to see if the answer is correct. It doesn't have to be fancy, some stick figures for the friends and circles for the apples works just fine. We could color in 5 of the circles to show that Isaac currently has those apples, and then count the rest.

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• 0:32 Understanding the Problem
• 1:47 Devise a Plan
• 2:32 Carry out the Plan
• 3:33 Look Back
• 5:04 Example
• 6:56 Lesson Summary

Let's try this method with another example problem.

Two trains leave the train station at the same time. They are going in opposite directions. One train is going 60 miles an hour, while the other train is going 50 miles an hour. In two hours, how far apart are the trains?

Understand the problem:

What is the question asking us? It is asking us how far apart the trains are after 2 hours.

What information is important here? The trains leave the station at the same time. They are going in opposite directions. One has a rate of 60 miles an hour, the other has a rate of 50 miles an hour.

We are finding the distance given the rate and the time, so we should remember the distance formula: {eq}d=r*t {/eq}

A diagram may help us:

Make a plan:

How are we going to solve this problem. We could label the diagram and show how far each of the trains have traveled away from the train station after two hours. We could use arithmetic. We could make a formula and solve it for 2 hours.

Carry out the plan:

Let's make a formula!

First we need to define the variables that we will use.

We want to know the distance between the trains, so let d= distance between the trains.

We know that the distance is going to depend on the amount of time, so let's let t=time since the trains departed the station.

We also know that d=r*t, and that the trains are moving away from each other, so the total distance between the two trains will be found by adding the distance the first train has traveled and the distance the second train has traveled.

This gives us the following formula: {eq}d=60t+50t {/eq}

We want to know the distance between the trains after two hours, so let {eq}t=2 {/eq}

And we get {eq}d=60*2+50*2 = 120 + 100 = 220 {/eq}

So, our answer is that after 2 hours, the trains will be 220 miles apart.

Does the answer make sense? Yes, 220 miles is a reasonable distance between two trains moving farther apart after two hours.

Could we do it a different way? Yes. We could find out the distance the first train moves after two hours: 120 miles, and the distance the second train moves after 2 hours: 100 miles, and then add them together to get 220 miles.

Polya's problem-solving process is a systematic method to solve a mathematical problem. By following each step, students are more likely to be able to solve the problem correctly.

Video Transcript

Polya's 4-step process.

George Polya was a mathematician in the 1940s. He devised a systematic process for solving problems that is now referred to by his name: the Polya 4-Step Problem-Solving Process .

In this lesson, we will discuss each step of the Polya process while working through the solution to a problem. At the end of the lesson, you will have the opportunity to try more examples before taking your quiz.

Understanding the Problem

So, to start, let's think about a party. Sally was having a party. She invited 20 women and 15 men. She made 1 dozen blue cupcakes and 3 dozen red cupcakes. At the end of the party there were only 5 cupcakes left. How many cupcakes were eaten?

The first step of Polya's Process is to Understand the Problem . Some ways to tell if you really understand what is being asked is to:

• State the problem in your own words.
• Pinpoint exactly what is being asked.
• Identify the unknowns.
• Figure out what the problem tells you is important.
• Identify any information that is irrelevant to the problem.

In our example, we can understand the problem by realizing that we don't need the information about the gender of the guests or the color of the cupcakes - that is irrelevant. All we really need to know is that we are being asked, 'How many cupcakes are left of the total that were made?' So, we understand the problem.

Now that we understand the problem, we have to Devise a Plan to solve the problem. We could:

• Look for a pattern.
• Review similar problems.
• Make a table, diagram or chart.
• Write an equation.
• Use guessing and checking.
• Work backwards.
• Identify a sub-goal.

In our example, we need a sub-goal of figuring out the actual total number of cupcakes made before we can determine how many were left over.

We could write an equation to show what is unknown and how to find the solution: (1 dozen + 3 dozen) - 5 = number eaten

Carry Out the Plan

The third step in the process is the next logical step: Carry Out the Plan . When you carry out the plan, you should keep a record of your steps as you implement your strategy from step 2.

Our plan involved the sub-goal of finding out how many cupcakes were made total. After that, we needed to know how many were eaten if only 5 remained after the party. To find out, we wrote an equation that would resolve the sub-goal while working toward the main goal.

So, (1 dozen + 3 dozen) - 5 = number eaten. Obviously, we would need the prior knowledge that 1 dozen equals 12.

1 x 12 = 12, and 3 x 12 = 36, so what we really have is (12 + 36) - 5 = number eaten.

12 + 36 = 48 and 48 - 5 = 43

That means that the number of cupcakes eaten is 43.

The final step in the process is very important, but many students skip it, feeling like they have an answer so they can move on now. The final step is to Look Back , which really means to check your work.

• Does the answer make sense? Sometimes you can add wrong or multiply when you should have divided, then your answer comes out clearly wrong if you just stop and think about it. In our problem, we wanted to know how many cupcakes were eaten out of a total of 48. We got the answer 43. 43 is less than 48, so this answer does make sense. (It would not have made sense if we got an answer greater than 48 - how could they eat more than were made?)
• Check your result. Checking your result could mean solving the problem in another way to make sure you come out with the same answer. Basically, in mathematical terms, we are saying that 48 - 5 = 43. If we were to draw out a diagram of the 1 dozen blue cupcakes and 3 dozen red ones, then separate out the 5 that did not get eaten, we would see that we do, indeed, have 43 represented as the eaten cupcakes. Our answer checks out!

And that is all there is to Polya's 4-Step Process to Problem Solving:

So how about you try? Try using Polya's 4-Step Process to solve this riddle: There are 10 people at a party. Each person must say hello to each other person exactly once. How many times is the word 'Hello' said?

Step 1 - Understand the problem Okay, so we have 10 people saying hello, but they don't have to say hello to themselves, only to the 9 other people. I need to know how many times the word 'hello' is said. Got it.

Step 2 - Devise a plan A diagram might be a great to show me what is happening here. If I draw the diagram as a circle with 10 points (representing each of the 10 people), I can visualize each saying hello.

Step 3 - Carry out the plan Drawing the diagram of one person saying hello, we see that each person will have to say hello 9 times, thus there will be 10 people each saying hello 9 times. 10 x 9 = 90 hellos said.

Finally, Step 4 - Check your work 90 hellos might not make sense if there are 10 people; you might think the answer should have been 100. Well, to check our work with a problem like this, we could set up a different diagram. If we put the people in a straight line and then count them saying hello to each other one at a time, we will again see that the final tally is 90 hellos. 90 must be the correct answer. Remembering that they do not have to say hello to themselves may help you see why the answer can't be 100.

In this lesson, we reviewed Polya's 4-Step Process for Problem Solving , which is simply a systematic process used to reach a solution to a problem.

• Understand the Problem Restating the problem and identifying necessary information is a key to this step.
• Devise a Plan Use equations, diagrams, tables or any other tool needed to create a plan for solving the problem.
• Carry Out Your Plan Just do it!
• Look Back This means to review your work to double check your answer.

If you use these four steps when you approach any problem, be it math or otherwise, you will find your path to the solution much more direct and easy. Good luck!

Learning Outcomes

Following this lesson, you should have the ability to:

• Describe the steps in Polya's 4-Step Process for Problem Solving
• Explain the importance of having a plan to solve problems
• Apply Polya's process to problems

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Polya's Problem Solving

George Polya was a famous Hungarian mathematician who developed a framework for problem-solving in mathematics in 1957. His problem-solving approach is still used widely today and can be applied to any problem-solving discipline (i.e. chemistry, statistics, computer science). Below you will find a description of each step along with strategies to help you accomplish each step. Having a specific strategy like this one may help to reduce anxiety around math tests.

Understand the Problem  

Understanding the problem is a crucial first step as this will help you identify what the question is asking and what you need to calculate. Strategies to help include: 

• Identify (i.e. highlight or circle) the unknowns in the problem or question.
• Draw or visualize a picture that can help you understand the problem. 

Devise a Plan  

Devising a plan is a process in which you find the connection between the data/information you are given and the unknown. However, you may not have been given enough data/information to find a connection immediately, so this process may involve calculating/finding additional variables before the final unknown can be solved. Strategies to help you devise a plan include:  

• List the unknowns and knowns. 
• Identify if a theorem would help you calculate the unknown (i.e. a2 + b2 = c2). 
• Decide what variables you need to know the value of to solve for the unknown. 
• Select which variable you will solve for first.

Carry Out the Plan  

This step involves calculating the steps identified in the “Devise a Plan” stage. Strategies to help you carry out the plan include:  

• Focus on solving one part of the problem at a time.
• Clearly write out each step. 
• Double check each variable or step as you solve.
• Repeat this process until you solve for the final unknown. 

Look Back 

This step involves reviewing your answer and steps to confirm that your final calculation is correct. Strategies to help you review your work include:  

• Recalculate each step to see if you get the same answer.
• Check if your final calculation has the appropriate units (i.e. m/s, N/m2). 
• Repeat steps to correct any errors found.

Polya's problem-solving phases

Four Steps of Polya's Problem Solving Techniques

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In the world of mathematics and algorithms, problem-solving is an art which follows well-defined steps. Such steps do not follow some strict rules and each individual can come up with their steps of solving the problem. But there are some guidelines which can help to solve systematically.

In this direction, mathematician George Polya crafted a legacy that has guided countless individuals through the maze of problem-solving. In his book “ How To Solve It ,” Polya provided four fundamental steps that serve as a compass for handling mathematical challenges. 

• Understand the problem
• Devise a Plan
• Carry out the Plan
• Look Back and Reflect

Let’s look at each one of these steps in detail.

Polya’s First Principle: Understand the Problem

Before starting the journey of problem-solving, a critical step is to understand every critical detail in the problem. According to Polya, this initial phase serves as the foundation for successful solutions.

At first sight, understanding a problem may seem a trivial task for us, but it is often the root cause of failure in problem-solving. The reason is simple: We often understand the problem in a hurry and miss some important details or make some unnecessary assumptions. So, we need to clearly understand the problem by asking these essential questions:

• Do we understand all the words used in the problem statement? 
• What are we asked to find or show? What is the unknown? What is the information given? Is there enough information to enable you to find a solution?
• What is the condition or constraints given in the problem? Separate the various parts of the condition: Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
• Can you write down the problem in your own words? If required, use suitable notations, symbols, equations, or expressions to convey ideas and encapsulate critical details. This can work as our compass, which can guide us through calculations to reach the solution.
• After knowing relevant details, visualization becomes a powerful tool. Can you think of a diagram that might help you understand the problem? This can serve as a bridge between the abstract and tangible details and reveal patterns that might not be visible after looking at the problem description.

Just as a painter understands the canvas before using the brush, understanding the problem is the first step towards the correct solution.

Polya’s Second Principle: Devise a Plan

Polya mentions that there are many reasonable ways to solve problems. If we want to learn how to choose the best problem-solving strategy, the most effective way is to solve a variety of problems and observe different steps involved in the thought process and implementation techniques.

During this practice, we can try these strategies:

• Guess and check
• Identification of patterns
• Construction of orderly lists
• Creation of visual diagrams
• Elimination of possibilities
• Solving simplified versions of the problem
• Using symmetry and models
• Considering special cases
• Working backwards
• Using direct reasoning
• Using formulas and equations

Here are some critical questions at this stage:

• Can you solve a portion of the problem? Consider retaining only a segment of conditions and discarding the rest.
• Have you encountered this problem before? Have you encountered a similar problem in a slightly different form with the same or a similar unknown? Look closely at the unknown.
• If the proposed problem proves challenging, try to solve related problems first. Can you imagine a more approachable related problem? A more general or specialized version? Could you utilize their solutions, results, or methods?
• Can you derive useful insights from the data? Can you think of other data that would help determine the unknown? Did you utilize all the given data? Did you incorporate the entire set of conditions? Have you considered all essential concepts related to the problem?

Polya’s Third Principle: Carry out the Plan

This is the execution phase where we transform the blueprint of our devised strategy into a correct solution. As we proceed, our goal is to put each step into action and move towards the solution.

In general, after identifying the strategy, we need to move forward and persist with the chosen strategy. If it is not working, then we should not hesitate to discard it and try another strategy. All we need is care and patience. Don’t be misled, this is how mathematics is done, even by professionals. There is one important thing: We need to verify the correctness of each step or prove the correctness of the entire solution.

Polya’s Fourth Principle: Look Back and Reflect

In the rush to solve a problem, we often ignore learning from the completed solutions. So according to Polya, we can gain a lot of new insights by taking the time to reflect and look back at what we have done, what worked, and what didn’t. Doing this will enable us to predict what strategy to use to solve future problems.

• Can you check the result? 
• Can you check the concepts and theorems used? 
• Can you derive the solution differently?
• Can you use the result, or the method, for some other problem?

By consistently following the steps, you can observe a lot of interesting insights on your own.

George Polya's problem-solving methods give us a clear way of thinking to get better at math. These methods change the experience of dealing with math problems from something hard to something exciting. By following Polya's ideas, we not only learn how to approach math problems but also learn how to handle the difficult parts of math problems.

Shubham Gautam

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2.1: George Polya's Four Step Problem Solving Process

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Step 1: Understand the Problem

• Do you understand all the words?
• Can you restate the problem in your own words?
• Do you know what is given?
• Do you know what the goal is?
• Is there enough information?
• Is there extraneous information?
• Is this problem similar to another problem you have solved?

Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.)

George Pólya & problem solving ... An appreciation

• General / Article
• Published: 06 May 2014
• Volume 19 , pages 310–322, ( 2014 )

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• Shailesh A. Shirali 1  

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George Pólya belonged to a very rare breed: he was a front-rank mathematician who maintained an extremely deep interest in mathematics education all through his life and contributed significantly to that field. Over a period of several decades he returned over and over again to the question of how the culture of problem solving could be nurtured among students, and how mathematics could be experienced ‘live’. He wrote many books now regarded as masterpieces: Problems and Theorems in Analysis (with Gábor Szegö), How to Solve It, Mathematical Discovery , among others. This article is a tribute to Pólya and a celebration of his work.

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Sahyadri School, Tiwai Hill, Rajgurunagar, Pune, 410 513, India

Shailesh A. Shirali

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Shailesh Shirali is Director of Sahyadri School (KFI), Pune, and also Head of the Community Mathematics Centre in Rishi Valley School (AP). He has been in the field of mathematics education for three decades, and has been closely involved with the Math Olympiad movement in India. He is the author of many mathematics books addressed to high school students, and serves as an editor for Resonance and for At Right Angles . He is engaged in many outreach projects in teacher education.

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Shirali, S.A. George Pólya & problem solving ... An appreciation. Reson 19 , 310–322 (2014). https://doi.org/10.1007/s12045-014-0037-7

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Published : 06 May 2014

Issue Date : April 2014

DOI : https://doi.org/10.1007/s12045-014-0037-7

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