what is polya's problem solving strategies

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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!

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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2.3.1: 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.)

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.

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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