polya's stages of problem solving

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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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

George Pólya & problem solving ... An appreciation

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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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Analysis of problem-solving skills with Polya's steps in solving numeracy problems in class VIII junior high school in terms of gender differences

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Shafira Ramadhani , Adi Nurcahyo , Nuraini Kasman , Hardianti , Jamaluddin Ahmad; Analysis of problem-solving skills with Polya's steps in solving numeracy problems in class VIII junior high school in terms of gender differences. AIP Conf. Proc. 17 January 2024; 2926 (1): 020045. https://doi.org/10.1063/5.0183389

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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 on gender. Types of research using qualitative. Data collection techniques using written tests, interviews, and documentation. The subjects of this study were 30 class VIII students at SMP Negeri 2 Banyudono. Indicators of problem-solving skills based on Polya's four steps. The results of the study showed that female students were superior with an average score of 62.91 while male students with an average of 55.67. Problem-solving skills at the step of understanding the problem female students can write down information that is known and asked on the question even though it is not complete, male students mostly do not write down important information on the questions. In the second step, developing a plan, students can use important information on the questions to help solve problems, but there are still shortcomings. In the third step, implementing the plan students are able and able to answer the questions asked even though there are still shortcomings. The last step, re-examining students, there are still many who do not confirm whether the answer has answered the question on the question or not, but students can make conclusions about each question.

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