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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(1887-1985), a Hungarian mathematician, wrote "How to solve it." for high school students in 1957. Here is his four step method. Read the problem over carefully and ask yourself: Do I know the meaning of all the words? What is being asked for? What is given in the problem? Is the given information sufficient (for the solution to be unique)? Is there some inconsistent or superfluous information which is given? By way of checking your understanding, try restating the problem in a different way. In essence, decide how you are going to work on the problem. This involves making some choices about what strategies to use. Some possible strategies are: -- making a picture which relates the information given to what is asked for can often lead to a solution. -- this is a strategy which is especially useful in problems where you need to count the members of a set. -- almost any problem can be made simpler in some way. By working out simpler versions, you can often see patterns which help solve the original problem. -- Many problems can be broken into a series of smaller problems. This strategy can turn a problem which on first glance seems intractable into something more doable. -- the method of algebra. Very useful in a lot of problems. : Spend a reasonable amount of time trying to solve the problem using your plan. If you are not successful, go back to step 2. If you run out of strategies, go back to step 1. If you still don't have any luck, talk the problem over with a classmate. After you have a proposed solution, check your solution out. Is it reasonable? Is it unique? Can you see an easier way to solve the problem? Can you generalize the problem?
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Polya's problem-solving phases
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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?
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| 1. Understand the problem. | Figure out what is being asked. What is known? What is not known? What type of answer is required? Is the problem similar to other problems you’ve seen? Are there any important terms for which you should look up definitions? | There are 22 total students. There are three groups of students: Students who only play recorder, students who only sing in choir, and students who do both. Initially, we do not know how many students are in any of these groups, but we know the total of the three groups adds up to 22. We also know that a total of 8 students play the recorder, and a total of 20 students sing in the choir. We must find the number of students who do both. |
| 2. Make a plan. | Come up with some strategies for solving the problem. Common strategies include making a list, drawing a picture, eliminating possibilities, using a formula, guessing and checking, and solving a simpler, related problem. | We could list out the 22 students and then assign to each either recorder, choir, or both until we got the right totals. We could draw a Venn Diagram that separates out the three types of groups. We could try solving a similar problem with a class of fewer students. |
| 3. Execute the plan. | Use the strategy chosen in Step 2 to solve the problem. If you encounter difficulties using the strategy, you may want to use resources such as the textbook to help. If the strategy itself appears not to be working, return to Step 2 and select a different strategy. | Let’s try solving a similar problem with a class of 6 students, 5 of whom play recorder and 3 of whom are in the choir. In this case, we know that there is only one student who doesn’t play recorder, and so this student must sing in the choir. That means the other two choir singers must play the recorder, so there are 2 students who do both. Now, let’s try that same method with the original problem. Since only 8 of the 22 students play recorder, the other 14 must sing in the choir and not play recorder. But there are 20 students in the choir, so 6 of these choir students also play the recorder. So the answer is 6. |
| 4. Look back and reflect. | Part of Step 4 is to find a way to check your answer, preferably using a different method than what you used to solve the problem. Another part of Step 4 is to evaluate the method you used to solve the problem. Was it effective? Are there ways you could have made it more effective? Are there other types of problems with which you might be able to use this type of solution method? |
Let’s check our answer with a Venn Diagram, which was one of the other strategies we considered in Step 2. We first fill in each region based on the results we found in Step 3. Now we check to see if the numbers match the original problem. Notice that 2 + 6 + 14 = 22 total students, 2 + 6 = 8 students playing the recorder, and 6 + 14 = 20 students in choir. So our answer checks out! Looking back on our answer, we now see that our process of subtracting from the total can be used in any similar situation, as long as all students must be in at least one of the two groups. In the future, we wouldn’t even have to use the simpler related problem since we’ve found a more general pattern! |
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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.
- 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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What is Polya’s method of problem solving?
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.
Table of Contents
What are the 4 problem solving methods?
- Rubber duck problem solving.
- Lateral thinking.
- Trial and error.
- The 5 Whys.
What is Polya’s third step in the problem solving process?
Third. Carry out your plan. Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct?
What is the part of Polya’s four step strategy is often overlooked?
Understand the Problem. This part of Polya’s four-step strategy is often overlooked. You must have a clear understanding of the problem. To help you focus on understanding the problem, consider the following questions: • • • • • Can you restate the problem in your own words?
What are the 5 problem-solving methods?
- Step 1: Identify the Problem.
- Step 2: Generate potential solutions.
- Step 3: Choose one solution.
- Step 4: Implement the solution you’ve chosen.
- Step 5: Evaluate results.
- Next Steps.
What is the best problem-solving method Why?
One of the most effective ways to solve any problem is a brainstorming session. The gist of it is to generate as many ideas as you can and in the process, come up with a way to remove a problem.
What are the 7 steps of problem-solving?
- 7 Steps for Effective Problem Solving.
- Step 1: Identifying the Problem.
- Step 2: Defining Goals.
- Step 3: Brainstorming.
- Step 4: Assessing Alternatives.
- Step 5: Choosing the Solution.
- Step 6: Active Execution of the Chosen Solution.
- Step 7: Evaluation.
What are the 3 types of problem-solving?
- Social sensitive thinking.
- Logical thinking.
- Intuitive thinking.
- Practical thinking.
What are the 3 stages of problem-solving?
A few months ago, I produced a video describing this the three stages of the problem-solving cycle: Understand, Strategize, and Implement. That is, we must first understand the problem, then we think of strategies that might help solve the problem, and finally we implement those strategies and see where they lead us.
What are the three problem-solving techniques?
- Trial and Error.
- Difference Reduction.
- Means-End Analysis.
- Working Backwards.
Who is the father of problem-solving method?
George Polya, known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.
What are the examples of problem-solving strategies?
- Guess (includes guess and check, guess and improve)
- Act It Out (act it out and use equipment)
- Draw (this includes drawing pictures and diagrams)
- Make a List (includes making a table)
- Think (includes using skills you know already)
Which step of Polya’s problem-solving strategy where you can freely state the problems in your own word?
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.
Which method is also known as problem-solving method?
Brainstorming and team problem-solving techniques are both useful tools in this stage of problem solving. Many alternative solutions to the problem should be generated before final evaluation.
What is the 5 step approach?
Step 1: Identify the problem. Step 2: Review the evidence. Step 3: Draw a logic model. Step 4: Monitor your logic model. Step 5: Evaluate the logic model.
What is the problem-solving approach?
A problem-solving approach is a technique people use to better understand the problems they face and to develop optimal solutions. They empower people to devise more innovative solutions by helping them overcome old or binary ways of thinking.
What is another term for problem solving?
synonyms for problem-solving Compare Synonyms. analytical. investigative. inquiring. rational.
How many tools are used for problem solving?
The problem solving tools include three unique categories: problem solving diagrams, problem solving mind maps, and problem solving software solutions. They include: Fishbone diagrams. Flowcharts.
What are the stages of problem solving?
- Step 1: Define the Problem. What is the problem?
- Step 2: Clarify the Problem.
- Step 3: Define the Goals.
- Step 4: Identify Root Cause of the Problem.
- Step 5: Develop Action Plan.
- Step 6: Execute Action Plan.
- Step 7: Evaluate the Results.
- Step 8: Continuously Improve.
How do you teach problem solving?
- Model a useful problem-solving method. Problem solving can be difficult and sometimes tedious.
- Teach within a specific context.
- Help students understand the problem.
- Take enough time.
- Ask questions and make suggestions.
- Link errors to misconceptions.
What are the 4 common barriers to problem-solving?
Some barriers do not prevent us from finding a solution, but do prevent us from finding the most efficient solution. Four of the most common processes and factors are mental set, functional fixedness, unnecessary constraints and irrelevant information.
Why is Polya the father of problem-solving?
Pólya is considered the father of mathematical problem-solving in the 20th century. It was his constant refrain that problem-solving was not some innate special ability but can actually be taught to anyone.
What is George Polya known for?
He was regarded as the father of the modern emphasis in math education on problem solving. A leading research mathematician of his time, Dr. Polya made seminal contributions to probability, combinatorial theory and conflict analysis. His work on random walk and his famous enumeration theorem have been widely applied.
What is the most difficult part of solving a problem?
Contrary to what many people think, the hardest step in problem solving is not coming up with a solution, or even sustaining the gains that are made. It is identifying the problem in the first place.
What are 10 problem-solving strategies?
- Guess and check.
- Make a table or chart.
- Draw a picture or diagram.
- Act out the problem.
- Find a pattern or use a rule.
- Check for relevant or irrelevant information.
- Find smaller parts of a large problem.
- Make an organized list.
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Polya's Four Phases of Problem Solving The following comes from the famous book by George Polya called How to Solve It. 1. Understanding the Problem. You have to understand the problem. What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown?
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.) ... 10.1: George Polya's Four Step Problem Solving Process
Understanding Polya's Problem-Solving Process Polya's problem-solving process, developed by mathematician George Polya, provides a structured approach to problem-solving that can be applied across various domains. This four-step process consists of understanding the problem, devising a plan, trying the plan, and revisiting the solution.
Learn about Polya's problem-solving process and the steps involved. See what is the final step in the problem-solving process and how it provides a...
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.
The Polya problem-solving approach, also known as the Polya method or Polya's four-step approach, is a widely used framework for solving mathematical problems. This method, developed by ...
Polya's four steps to solving a problem George Polya (1887-1985), a Hungarian mathematician, wrote "How to solve it." for high school students in 1957. Here is his four step method. Understand the problem: Read the problem over carefully and ask yourself: Do I know the meaning of all the words? What is being asked for? What is given in the problem?
Polya's Four Step Problem Solving Process . ... Usually, a problem is stated in words, either orally or written.Then, to solve the problem, one translates the words into an equivalent problem using mathematical symbols, solves this equivalent problem, and then interprets the answer.This process is summarized in Figure 1.1. ...
Polya's problem-solving phases. Understand the problem. Devise a plan for solving the problem. Carry out the plan. Evaluate the solution for accuracy and for its potential as a tool in solving other problems. These shouldn't be understood as steps to be followed. They are phases that will be completed some time during the problem-solving process.
Much of the advice presented in this section is based on a problem-solving process developed by the eminent Hungarian mathematician George Polya (see the historical high-light at the end of this section). We will now outline Polya's method.
Back to: Helping Students Do Math 0 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 Read the section below entitled "Background Information," and familiarize yourself with the chart of Polya's four-step problem-solving method. Read the example below […]
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).
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.
George Polya's tips for problem solving: (from How to Solve It)
8/24/15 Section 1-1 Polya's four-step approach to problem solving 1. Understand the problem— It is impossible to solve a problem, if you do not know what the problem is. What is known or unknown? Is there enough information or is more information needed? What do the terms mean? Etc.
If you misunderstand either the problem or the underlying situation you may make mistakes or incorrect assumptions. One of our main goals for this semester is to become better problem solvers. To begin this task, we now discuss a framework for thinking about problem solving: Polya's four-step approach to problem solving.
Polya's Problem Solving Strategy is one of the topic included in the Mathematics in the Modern World. It is basically the steps on how to deal with various m...
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.
For mathematics education and for the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Polya. Since then Polya's influence both on the study of mathematical thinking and on the study of productive thinking in general has been enormous.
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roblem-solving steps of Polya. Data collection techniques in this s. udy were tests and interviews. The test instrument used. onsisted of two word problems. Problem number (1) is an arithmetic series problem, while number (2) is a geometric serie. tic series and is expressed aswhere P is the sum of production (in ton.
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.) 1.
The purpose of the study is to describe students' problem-solving skills in solving numeracy problems in relation and function materials using Polya steps based