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polya's problem solving approach

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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 .

polyas-problem-solving-steps-to-solve-unknowns-in-elementary-and-middle-school-classes

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.

This table lists “unknown situations” from modern math standards and suggests a problem-solving challenge for each grade level. Use this list to apply Polya’s Four-Step Problem-Solving Process in the lower grades!

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!

polya's problem solving approach

Physics Network

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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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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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.)

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  4. POLYA's 4 Step Problem Solving

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  6. POLYA'S PROBLEM-SOLVING STRATEGY (PART 1)

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  3. Polya's Problem Solving

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  5. CHP 3 Lesson 2, Polya's 4 steps in problem solving

  6. MATH 1350

COMMENTS

  1. PDF Polya's Problem Solving Techniques

    Polya's Problem Solving Techniques In 1945 George Polya published the book How To Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this book he identi es four basic principles of problem solving. Polya's First Principle: Understand the problem

  2. 10.1: George Polya's Four Step Problem Solving Process

    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. Guess and test.

  3. Polya's Problem Solving Process

    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...

  4. PDF 1. Understand Polya's problem-solving method. 2. State and apply

    1.1 Objectives Understand Polya's problem-solving method. State and apply fundamental problem-solving strategies. Apply basic mathematical principles to problem solving. Use the Three-Way Principle to learn mathematical ideas. Problem Solving KEY POINT George Polya developed a four-step problem-solving method.

  5. Polya's Problem-Solving Process: Finding Unknowns ...

    April 19, 2023 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.

  6. PDF Polya's Four Phases of Problem Solving

    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? Or is it insufficient? Or redundant? Or contradictory? Draw a figure. Introduce suitable notation.

  7. The Impact of Applying the First Two Steps of Polya's Four

    The Impact of Applying the First Two Steps of Polya's Four ProblemSolving Steps in an Advanced Mathematics Classroom. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/5037 This Open Access Dissertation is brought to you by Scholar Commons.

  8. PDF Polya's four-step approach to problem solving

    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. 2. Devise a plan— You must start somewhere.

  9. Polya, Problem Solving, and Education

    Solve It was hardly Polya's first foray into the world of problem solving. It was, however, an absolutely critical one. How to Solve It marked a turning point both for its author and for problem solving.

  10. Mastering Problem-Solving: A Guide to Polya's Four-Step Approach

    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 ...

  11. Polya's four steps to solving a problem

    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. ... Design a plan for solving the problem: In essence, decide how you are going to work on the problem. This involves making some choices about what strategies to use.

  12. What is Polya's method of problem solving?

    What is Polya's method of problem solving? October 9, 2022 by George Jackson Spread the love 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.

  13. Problem Solving Strategies & Polya's 4-step Process

    Math in the Modern World Playlist: https://www.youtube.com/watch?v=vb-hDOm3GzU&list=PLbZl6MGLeYnsoaxa2L-xouDPHcoe9z23x&index=3A more logical answer to 13:36 ...

  14. The Problem-Solving Process in a Mathematics Classroom

    Polya's (1957) four-step process has provided a model for the teaching and assessing. problem solving in mathematics classrooms: understanding the problem, devising a plan, carrying out the plan, and looking back. Other educators have adapted these steps, but the. essence of these adaptations is very similar to what Polya initially developed.

  15. Revisiting Polya's approach to foster problem solving skill development

    To build this approach we examined mathematician Gyorgy Polya's heuristic of a generalized problem solving model in the context of software engineering (SE) and propose the necessary augmentation to support skill development.

  16. (PDF) George Pólya & problem solving ... An appreciation

    Problem solving is a systematic series using a certain process to solve a problem (G Polya, 1945;Schoenfeld, 2017; Shirali, 2014). Problem solving is part of a complex thinking process with ...

  17. Overview

    This generality is the primary strength of Polya's method. Given any problem, you can try to understand it (Step 1), make a plan to solve it (Step 2), execute the plan (Step 3), and then reflect back on how you solved the problem (Step 4). Figure 1 details the flow of Polya's problem-solving method. A weakness of Polya's method is that it ...

  18. Analysis of problem-solving skills with Polya's steps in solving

    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. ... Profile of students' problem-solving skills viewed from Polya's four-steps approach and elementary school students. Eur J Educ Res ...

  19. 2.3.1: George Polya's Four Step Problem Solving Process

    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. Guess and test. 11. Solve an equivalent problem. 2.

  20. PDF A rubric for problem-solving strategies based upon Polya's stages

    Devising a plan or strategy to solve the problem. Making a general plan and selecting relevant methods, "heuristics" that might be useful for solving the problem based on the understanding of the problem. At level 4 a student can recognize or classify the structure of the problem. They can consider one or more strategies, coordinate several ...

  21. Polya theory to improve problem-solving skills

    K R Daulay I Ruhaimah Abstract and Figures This research is to improve problem-solving skills in SMP Muhammadiyah 8 Medan students through Polya learning theory in the material of linear...

  22. (PDF) Revisiting Polya's Approach to Foster Problem Solving Skill

    To build this approach we examined mathematician Gyorgy Polya's heuristic of a generalized problem solving model in the context of software engineering (SE) and propose the necessary augmentation ...

  23. Profile of Students' Problem-Solving Skills Viewed from Polya's Four

    In general, the learners' problem-solving performance decreased from the first step of the Polya Problem-Solving Steps to the third step, as was found in other studies (Akyüz, 2020;Lubis et al ...