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Polya’s Problem Solving Techniques

Teaching University students to carry out critical and independent science research is challenging, and they need to learn to flex new muscles and approaches in their brain, that are not always well stretched at the school stage. I have found the summary of George Polyas lessons that I reproduce below on a number of websites (e.g. here ) and I do not know the original source, but its great – have a read:

In 1945 George Polya published a 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 identifies four basic principles of problem solving.

Polya’s First Principle: Understand the Problem

This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don’t understand it fully, or even in part. Polya taught teachers to ask students questions such as:

• Do you understand all the words used in stating the problem?
• What are you asked to find or show?
• Can you restate the problem in your own words?
• Can you think of a picture or diagram that might help you understand the problem?
• Is there enough information to enable you to find a solution?

Polya’s Second Principle: Devise a Plan

Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

• Guess and check
• Look for a pattern
• Make an orderly list
• Draw a picture
• Eliminate the possibilities
• Solve a simpler problem
• Use symmetry
• Use a model
• Consider special cases
• Work backwards
• Use direct reasoning
• Use a formula
• Solve an equation
• Be ingenious

Polya’s Third Principle: Carry Out the Plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work, discard it and  choose another. Don’t be misled, this is how things are done, even by professionals.

Polya’s Fourth Principle: Look Back

Polya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn’t. Doing this will enable you to predict what strategy to use to solve future problems.

These principles and more details about strategies of carrying them out are summarized in this document: Polya’s Problem Solving Techniques

George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career. Even after his retirement from Stanford University in 1953, he continued to lead an active mathematical life. He taught his final course, on combinatorics, at the age of ninety.

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Polya’s Problem-Solving Process

Emma Moore, Teaching Excellence Program Master Teacher 

Problem-solving skills are crucial for students to navigate challenges, think critically, and find innovative solutions. In PISA, problem-solving competence is defined as “an individual’s capacity to engage in cognitive processing to understand and resolve problem situations where a method of solution is not immediately obvious” (OECD, 2014, p. 30). Returning to the classroom post-COVID, I found that students had lost their ‘grit’ for these deep-thinking tasks. They either struggled to start, gave up easily, or stopped at their first ‘answer’ without considering if it answered the problem or was the only possible solution.

To re-invigorate these skills, I investigated the impact of explicitly teaching Polya's problem-solving process in my Year Six class. This framework developed student agency and supported them to manage their feelings if they felt challenged by the work.

Here, I will share the impact of this initiative and how it empowered students to become effective and resilient problem solvers.  

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. (Polya, 1947)

In order to focus on the skills and knowledge of the problem-solving process, I began by using tasks where the mathematical processes were obvious. This allowed me to focus on the problem-solving process explicitly.

The question shown in Figure 2 is taken from Peter Sullivan and Pat Lilburn's Open-Ended Maths Activities book. This task was used to establish a baseline assessment for each stage of the process. I planned the prompts in dot points and revealed them one by one through the PowerPoint. After launching the task and giving the students time to think, they recorded all their possible answers in their workbook.

The student sample shown in Figure 3 demonstrates that the student followed a pattern and stuck to it but did not revisit their work. On line two, their response (1 half and 1 half is 2 quarters) is unreasonable.

Figure 3 is a sample gathered from a small group of students. This group required support to start. They used paper folding and paper strips to model their thinking.

Over half of the class could give at least one correct answer, but only four students showed signs of checking to see if their plans addressed the problem and yielded correct answers. Understanding the problem and revisiting the solutions became the focus of my inquiry.

The following series of lessons covering operations with fractions and decimals focused on the stages of Polya’s process.  

Step 1: Understanding the Problem

The first step of Polya's problem-solving process emphasises the importance of ensuring you thoroughly comprehend the problem. In this step, students learn to read and analyse the problem statement, identify the key information, and clarify any uncertainties. This process encourages critical thinking (Bicer et al., 2020) as students develop the ability to break down complex problems into manageable parts. I facilitated this process by engaging students in discussions and guiding them to identify the essential components of the problem. By fostering a collaborative learning environment, students shared their perspectives and learned to refine their questions when they were unsure. Figure 6 shares an example of a prompt I use for Step 1.

Figure 4: Example prompt for Step 1.

Initially, students who were stuck provided the classic ‘white flag’ responses.

Student: I just don’t get it.

Teacher: What part don’t you get?

Student: All of it!

As a starting point, the students and I co-created a classroom display of helpful questions the students could use to develop their understanding.

These questions supported me to develop a deeper understanding of what students didn’t understand when they expressed uncertainty. This could range from not understanding specific terminology (often easy to explain) to where numbers came from and why their classmates interpreted the problem differently. I found engaging in this step made triaging their misunderstandings easier.  

Step 2: Devising a Plan

Once students had grasped the problem, the next step was to formulate a plan of action. In this step, students explored different strategies and selected the most appropriate approach. I prompted students to brainstorm possible solutions, draw diagrams, make tables, and create algorithms, all the time fostering creativity and diverse thinking.

This step had been a strength during the baseline assessment data, and a wide range of strategies were explored. Polya’s strategies were displayed in the classroom as the mathematician’s strategy tool kit, so students were comfortable acknowledging the many ways to solve the problem.

Students developed critical thinking and decision-making skills by keeping this step in problem-solving. They become adept at evaluating multiple approaches and selecting the most effective strategy to solve a problem, thus promoting the development of mathematical reasoning abilities (Barnes, 2021). Figure 7 shows a slide used in Step 2.

Figure 5: Example prompt for Step 2.

Step 3: Try

The students implemented their selected strategy, performed calculations, made models, drew diagrams, created tables, and found patterns. This stage encouraged students to persevere and take ownership of their problem-solving process.

At Cowes Primary School, we have developed whole-school expectations around providing opportunities for hands-on learning, allowing students to engage in practical activities that support the development of ideas, expecting students to represent their work visually (pictures, materials and manipulatives), using language and numbers/symbols. This approach enhances students' problem-solving skills and fosters a sense of autonomy and confidence in their capabilities and ability to talk about their work (Roche et al., 2023). Figure 9 shows the slide used for Step 3.

Figure 6: Example prompt for Step 3.

Step 4. Re-visiting the solution

The last step in Polya's problem-solving process is re-visit. After finding a solution, students critically analyse and evaluate their approach after finding a solution. They consider the effectiveness of their chosen strategy, identify strengths and weaknesses, and reflect on how they could improve their problem-solving techniques. This step was missing from most students’ work during the baseline assessment.

As a class, we added to the display questions to facilitate better reflective practice and developed a more critical approach to looking at our work. This process encouraged students to refine their answers, not go too far down the wrong path, fostered resilience, embrace challenge and normalise uncertainty (Buckley & Sullivan, 2023).

Figure 7: Class display showing our questions.

  Figure 8: Student samples from the task.

Impact and Benefits:

Figure 9 shows four tasks, including the initial baseline assessment. The blue series shows the percentage of students who arrived at least one correct solution. The green series shows evidence that students were revisiting their initial solutions using other strategies to check they were correct or checking in with other groups and adjusting. There was a steady increase in both skills over the course of these four tasks.

By explicitly teaching Polya's problem-solving process, the students cultivated valuable skills that extend beyond maths problems. Some of the key benefits observed were:

Mathematical Reasoning: Polya's process promotes the development of mathematical reasoning skills. Students analysed problems, explored different strategies, and apply logical thinking to arrive at solutions. These skills can enhance their overall mathematical proficiency.

Self-efficacy: Through problem-solving, students gained confidence in their ability to tackle problems. They become more self-reliant, taking ownership of their learning, and seeking solutions proactively.

Collaboration and Communication: The process encouraged collaboration and communication among students. They discussed problems, shared ideas, and considered multiple perspectives, students developed effective teamwork and interpersonal skills.

Metacognition: The reflective aspect of Polya's process fostered metacognitive skills, enabling students to monitor and regulate their thinking processes. They learned to identify their strengths and weaknesses, supporting continuous improvement and growth.  

Overall using the 4 steps was a really effective and an explicit way to focus on developing the problem-solving skills of my Year 6 students.

This article was originally published for the Mathematical Association of Victoria's Prime Number.    

References:

Barnes, A. (2021). Enjoyment in learning mathematics: Its role as a potential barrier to children’s perseverance in mathematical reasoning. Educational Studies in Mathematics , 106(1), 45–63. https://doi.org/10.1007/s10649-020-09992-x

Bicer, Ali, Yujin Lee, Celal Perihan, Mary M. Capraro, and Robert M. Capraro. ‘Considering Mathematical Creative Self-Efficacy with Problem Posing as a Measure of Mathematical Creativity’. Educational Studies in Mathematics 105, no. 3 (November 2020): 457–85. https://doi.org/10.1007/s10649-020-09995-8

Buckley, S., & Sullivan, P. (2023). Reframing anxiety and uncertainty in the mathematics classroom. Mathematics Education Research Journal , 35(S1), 157–170. https://doi.org/10.1007/s13394-021-00393-8

OECD (Ed.). (2014). Creative problem solving: Students’ skills in tackling real-life problems. OECD.

Pólya, G. (1988). How to solve it: A new aspect of mathematical method (2nd ed). Princeton university press.

Roche, A., Gervasoni, A., & Kalogeropoulos, P. (2023). Factors that promote interest and engagement in learning mathematics for low-achieving primary students across three learning settings. Mathematics Education Research Journal , 35(3), 525–556. https://doi.org/10.1007/s13394-021-00402-w

George Pólya & problem solving ... An appreciation

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

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• Shailesh A. Shirali 1  

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

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Historical Events in the Background of Hilbert’s Seventh Paris Problem

Arnol’d’s 1974 Problems

Commentary on Part II of Mathematical Challenges For All: Making Mathematics Difficult? What Could Make a Mathematical Challenge Challenging?

Suggested reading.

http://www-history.mcs.st-and.ac.uk/Biographies/Polya.html

T Gowers, The Two Cultures of Mathematics , https://www.dpmms.cam.ac.uk/~wtg102cultures.pdf

http://en.wikipedia.org/wiki/George_Polya

T Passmore, Polya’s legacy: fully forgotten or getting a new perspective in theory and practice , http://eprints.usq.edu.au/3625/1/Passmore.pdf

G Pólya, Mathematics and Plausible Reasoning , Princetron University Press, Vols 1&2, 1954.

G Pólya, Mathematical Discovery , Vols 1&2, 1965.

G Pólya, How To Solve It , Princeton University Press, 1973.

Google Scholar  

G Pólya, Teaching us a Lesson (MAA), http://vimeo.com/48768091 (video recording of an actual lecture by Polya).

http://www.math.utah.edu/~pa/math/polya.html

Geoffrey Howson, Review of Mathematical Discovery, The Mathematical Gazette , Vol. 66, No. 436, pp.162–163, June 1982.

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

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George Pólya

1 george pólya.

American mathematician, Born: György Pólya in Budapest, Hungary in 1887, ( d. 1985 in Palo Alto, USA)

“ His first job was to tutor the young son, Gregor, of a Hungarian baron. Gregor struggled due to his lack of problem solving skills. ” Thus, according to Long ( [ 1 ] ), Polya insisted that the skill of “ solving problems was not an inborn quality but, something that could be taught ”.

In 1940, George Polya and his wife, Stella, (the only daughter of Swiss Dr. Weber, in Zurich) moved to the United States because of their justified fear of Nazism in Germany ( [ 1 ] ).

Understand the Problem

Devise a Plan on how to approach the Problem; such a plan may include one or several of the following:

Make a first guess to begin with, and then verify the answer

Solve a simpler problem

Consider special cases that are much easier to solve

Look for a pattern

Draw a picture

Use a model

Use direct reasoning but double-check your results

Eliminate possibilities

Carry out the Plan, as modified by partial solutions

If plan doesn’t work, make an improved plan but do not give up

Last-but-not-least, look back and examine critically your solution(s):

Does the solution make sense? Does it check out in particular cases?

Make sure there are no gaps and no steps missing.

He published also a two-volume book, “ Mathematics and Plausible Reasoning ” in 1954, and Mathematical Discovery in 1962.

• 1 Long, C. T., & DeTemple, D. W., Mathematical reasoning for elementary teachers . (1996). Reading MA: Addison-Wesley
• 2 Reimer, L., & Reimer, W. Mathematicians are people too . (Volume 2). (1995) Dale Seymour Publications
• 3 Polya, G. How to solve it . (1957) Garden City, NY: Doubleday and Co., Inc.
• 4 A. Motter,, http://www.math.wichita.edu/history/men/polya.html “A Biography of George Polya”

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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2.1: George Polya's Four Step Problem Solving Process

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Step 1: Understand the Problem

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

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