How to Solve It: George Pólya's Four Steps, Still Useful Today

George Pólya’s How to Solve It, first published by Princeton University Press in 1945, treats problem solving as four phases, each with a short list of questions to ask yourself. The Pólya How to Solve It method fits on the two-page checklist that opens the book.¹

1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Look back

The phases are easy to remember. The book’s lasting value is in the questions under each one and in its heuristic dictionary, dozens of entries on making a problem more manageable.²

What are Pólya’s four steps of problem solving?

Pólya’s four steps are understand the problem, devise a plan, carry out the plan and look back, and he stressed that the borders between them are not rigid: your view of a problem shifts as you work on it.¹ Each step comes with self-questions. A 2014 OECD report on teaching mathematics calls his four stages and self-directed questioning the foundation of later metacognitive teaching models, which train learners to think about their own thinking.³

Ask the questions out loud or on paper: a named question gives you something to do when the problem itself offers no way in. For the wider work of defining a problem, testing a fix and reviewing it with a team, there is a six-step process for everyday problem solving; what follows stays with Pólya’s own four phases.

1. Understand the problem

Pólya held that students often get stuck simply because they have not fully understood the problem. His three opening questions are “What is the unknown?”, “What are the data?” and “What is the condition?”, meaning the link between the two. He adds others: do you understand every word, can you restate the problem in your own words, can you draw a figure, and is the condition enough to settle the unknown?¹

Restating is the quickest test. A manager told that “sales are down” has no unknown yet; “which region fell, by how much, since when” does. If you cannot say the problem in a sentence of your own, the test behind the Feynman technique for learning, you are not ready to plan.

2. Devise a plan

Pólya considered this the most important and hardest phase. The idea for a plan may build up slowly, or arrive suddenly after a period of hesitation as a “bright idea”, the experience psychologists study as the aha moment. His central question here is “Do you know a related problem?” Most of the book is suggestions for this phase.¹

A related problem helps because its method already exists; you only have to adapt it. Before inventing an approach, spend two minutes listing problems you have solved that share the same unknown, such as an earlier budget, a past hiring round or last term’s timetable.

3. Carry out the plan

Pólya saw carrying out the plan as easier than devising it; what it needs is patience and a check of every detail as you go.¹ A University of Kentucky mathematics course page adds a rule many readers miss: give the plan a reasonable amount of time, return to step 2 if it is not working, and return to step 1 if you run out of strategies.⁴

4. Look back

After solving, Pólya asks you to check the result, confirm it uses all the data, look for another route to the same answer and ask where else the method could be used.¹ It is easy to skip, because the problem feels finished, yet it is the phase that turns one solved problem into a method you can reuse. A team that traces one late invoice to a missing approval step can ask where else that step is missing and fix several delays at once.

1. Understand the problem: What is the unknown? What are the data? What is the condition?
2. Devise a plan: Have you seen it before? Do you know a related problem?
3. Carry out the plan: Is each step correct? Can you prove that it is?
4. Look back: Can you check the result? Can you use the result, or the method, for another problem?

What are examples of Pólya’s heuristics?

Pólya’s heuristicsheuristic: A mental shortcut or rule of thumb that gives a quick, usually good-enough answer without weighing all the information. Heuristics often serve people well; they produce biases when used where they do not fit.Full entry in the glossary are rules of thumb for finding a plan, and the largest part of How to Solve It is a dictionary of them, 67 entries according to Alan Schoenfeld’s 1987 count reported on Wikipedia.¹ In his sense a heuristic is a suggestion for making progress when you do not know a method. Many of the entries do the same job in different ways: they swap the problem you cannot solve for a related one you can.

Each heuristic gives you a foothold when the original problem offers none. In Pólya’s words, as quoted on Wikipedia, you should “try to solve first some related problem”, which may give you the courage to return to the original.¹ When you are blocked, run down the left-hand column and try the first question that produces a smaller problem.

A worked maths example: the diagonal of a box

The book’s running example asks for the diagonal of a rectangular box, given its length, width and height. In Pólya’s classroom dialogue, the teacher’s key move is asking “Do you know a related problem?”, which leads students to the right triangle.⁵ The numbers below are ours. Will a thin curtain rod 125 cm long fit inside a moving box that measures 30 by 40 by 120 cm?

The look-back row does two jobs. The special case catches errors, because a formula that fails a case you can check by eye is wrong. The general rule is what you take away: the next time you pack a tripod or a ski pole, you skip straight to the last line.

Can Pólya’s method be used outside maths?

Pólya’s method can work outside mathematics wherever a problem has a clear unknown and a goal you can check. An early reviewer in the Chicago Tribune said in 1946 that the method applies in principle to science, engineering and social work, and programmers adopted the book as a debugging guide.¹ Alan Schoenfeld, a mathematics education researcher who spent years working out how to teach Pólya’s strategies, conjectured that his framework of knowledge, strategies, self-monitoring and beliefs applies to problem solving in other fields, writing included.⁶

A traveler has a 07:10 flight from an airport 40 minutes’ drive away and wants the latest safe time to leave home. The unknown is a departure time; the data are the flight, the drive, the airline’s rules and the queues. Working backwards from the goal is the natural plan. Each step back subtracts one allowance from the time before it.

1. 05:15Leave home (40-minute drive)
2. 05:55Arrive and drop bags (allow 15 minutes)
3. 06:10Join the security queue (allow 30 minutes)
4. 06:40Leave security for the gate (allow 10 minutes)
5. 06:50Gate closes (assume 20 minutes before departure)

Looking back is where the plan improves. Does the answer use all the data? A traveler who forgot parking or a fuel stop finds the gap now rather than at the gate. The same backwards plan handles a report due to a client on Friday at noon: work back through review, drafting and the data you need, and the real start date appears.

What later research says about teaching heuristics

Alan Schoenfeld’s research on teaching problem solving concluded that Pólya’s heuristics fall short by themselves: learners also have to be taught when to use each one and how to check whether it is working. Mathematicians who read Pólya felt his strategies were right, yet in the mid-1970s nobody had managed to teach students to use them effectively.⁶ His 1985 book Mathematical Problem Solving gives separate chapters to resources, heuristics, control and belief systems.⁷

He identified two problems. Pólya’s labels are too broad: “solve an easier related problem” covers at least a dozen different techniques, each of which has to be taught on its own. And knowing strategies does not help a solver who never stops to check progress; research he drew on found that unsuccessful solvers tended to pick a path quickly and stick with it despite making little or no progress.⁶

The study

Limited evidence

What changed after Schoenfeld's problem-solving course (Schoenfeld, 1992)

In Alan Schoenfeld’s collection of more than a hundred videotapes of college and high-school students working on unfamiliar problems, roughly 60 percent of attempts followed one pattern: read the problem, decide quickly on an approach and pursue it no matter what. After his course, which coached students to monitor their own progress, fewer than 20 percent of his students’ attempts were of that type, and their problem-solving success rose.⁸

The result is encouraging but thin. It comes from the researcher’s own course, reported by the researcher, and the authors of a 2014 OECD report on these methods note that the gains could partly reflect the attention students received and the extra time spent on the problems. The same report summarizes meta-analyses in which training in planning, monitoring and evaluating one’s own thinking was linked to better results at school, with larger effects in primary school mathematics than in secondary school mathematics and science.³

Myth

Learning Pólya's list of heuristics is enough to make you a good problem solver.

Fact

Students also need to learn which heuristic fits a problem and to check, while working, whether their approach is making progress.

Schoenfeld turned self-monitoring into three questions he could ask students at any moment during a problem:

What (exactly) are you doing? (Can you describe it precisely?) Why are you doing it? (How does it fit into the solution?) How does it help you?

Ask them every few minutes. Someone who has spent twenty minutes rewriting one spreadsheet formula and cannot say how the next rewrite will help has found a dead end, and the useful move is back to the plan, the move his students learned to make. If the plan itself has stalled, taking a deliberate break from an unsolved problem is the other option worth knowing.

Who Pólya was, and why a 1945 maths book is still read

George Pólya was a Hungarian-American mathematician who taught at ETH Zürich from 1914 to 1940 and at Stanford University from 1940 to 1953.⁹ He wrote How to Solve It mainly for teachers and students of mathematics, with examples from school algebra and geometry, yet other fields took it up for its plain account of problem-solving strategies. Wikipedia, citing his 1985 obituary in the Los Angeles Times, reports that it has been translated into more than 15 languages and has sold over a million copies.¹

The book lasted because it teaches questions instead of formulas. A formula belongs to one kind of problem; a question such as “what exactly am I trying to find?” works on a geometry exercise, a bug in a spreadsheet and a supplier who keeps missing deadlines. That portability is one plausible reason the book reached readers far from the maths classroom. In a 1980s interview, the Microsoft programmer Charles Simonyi said new programmers there received it on their first day, and he singled out the two-page checklist at the front as the part that matters.¹

Keep the questions general when you copy them out, since generality is what lets them travel. Write “what is the unknown?” on the card and it serves a timetable clash as well as a triangle; “what is the missing side?” serves only the triangle.

Is the Pólya How to Solve It book worth reading?

How to Solve It is worth reading for teachers, students, programmers and anyone who helps other people think through problems, provided they accept that its examples are school mathematics. Princeton’s current paperback runs to 288 pages.²

Reviews from its first years still describe it fairly. The Mathematical Gazette wrote in 1946 that it held little a good teacher would not discover alone, but might speed up that discovery.¹ Schoenfeld’s later caution matters too. He wrote that the book’s descriptions lack the detail a newcomer needs to apply the strategies,³ and he argued that knowledge of the subject is essential as well.¹

Read the opening checklist and the worked box example, then dip into the dictionary entries for the heuristics in the table above. A reader who wants practice can rework one problem from their own job each week with the questions beside them.

A printable Pólya question list

The list below gathers Pólya’s main questions by phase, as summarized by Wikipedia and the OECD, plus Schoenfeld’s monitoring questions. Ticks are saved in your browser, so you can work through it on your next problem.

Pólya's questions, phase by phase

• Understand: What is the unknown? What are the data? What is the condition?
• Understand: Can I restate the problem in my own words, and draw a figure of it?
• Understand: Is the condition enough to settle the unknown?
• Plan: Have I seen this problem, or a related one, before?
• Plan: Can I work backwards, try a special case or solve part of it first?
• Carry out: Is each step correct, and can I show that it is?
• While working: What exactly am I doing, why, and how will it help?
• Look back: Can I check the result, and did I use all the data?
• Look back: Could I reach the result another way, or use the method elsewhere?

The bottom line

Pólya’s four phases are simple; the skill lies in asking his questions at the right moment and noticing when an approach has stopped working. Print the question list, keep it beside your next stubborn problem, and stop every few minutes to ask what you are doing and why.