An article of faith for me is that good ideas come from everywhere. So, I am always curious about how accomplished people in various fields solve problems. You could adopt their approach in its entirety or, more practically, take aspects of it, modify them to suit your circumstances and apply them.
From a chance conversation recently with my colleague, math professor Steve Schiffman, while we were visiting a university in Mexico, I learned about Hungarian mathematician George Pólya and his book How to Solve It which has been used widely by mathematicians. Drawing upon his many years of doing research and teaching, Pólya developed a process for solving math problems. I have no mathematical talent but I found Pólya’s process surprisingly simple and useful, with some modifications, to solving problems in other settings. Below, I will draw upon Pólya’s ideas and modify them such that they can be applied to solving big problems in other fields.
George Pólya was a leading mathematician who was born in 1887 in Budapest, Hungary and died in 1985 in Palo Alto, California. He made important research contributions to several topics within mathematics. Beyond research, he had substantial influence on the practice and teaching of mathematics through his books on problem solving. He was a professor at ETH Zürich, Switzerland, and Stanford University, USA.
There are 4 broad steps to Pólya’s approach:
- Understanding the problem
- Devising a plan
- Carrying out the plan
- Looking back.
The broad steps may seem obvious but Pólya was spelling them out for two reasons. One, they are often not applied. Two, applying them works. He goes on to develop the details of each step in his book.
Understanding the Problem
Noting that you have to first understand the problem, Pólya elaborates the first step thus:
- 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.
- Separate the various parts of the condition. Can you write them down?
To solve them effectively, complex problems have to be understood in their details even if their broad existence is known (e.g., poverty, Alzheimer’s disease). In my June 29 blog-post (below) I explain how most of us fall into the plunging-in decision trap and begin to solve problems without first understanding them well. Pólya’s first step shows that this natural tendency is common among math problem solvers as well. He advised: “Do not rush… understand fully… try to see clearly what it means… convince yourself of the truth.”
Pólya’s suggestion of identifying the unknowns in a situation is useful in solving any complex problem. It is easy to make mistakes of omission and not recognize sources of uncertainty which later result in poor outcomes. What is it that is important to our situation but we do not know or understand? We often hear that complex situations have “unknown unknowns” and we have to move ahead with our actions despite them. There is some validity to it but it misses the point that everything that is unknown is not unknowable. With some data collection or generation, it is possible to learn at least something useful about what is unknown to make better decisions. Moreover, knowing what we do not know keeps us alert to the world and we may recognize useful information that emerges later about what is currently unknown. Dismissing what is unknown as also inherently unknowable is an easily prevented trap. What is unknown is partly a characteristic of the situation and partly of our current knowledge base. Even if we cannot alter the former, we can the latter.
“Visualize the problem as a whole as clearly and as vividly as you can,” wrote Pólya. His suggestion of drawing a figure is useful for solving any complex problem. Doing so helps us conceptualize and simplify complexity. A diagram helps us capture the parts that are important (drawing upon his last point above) and the relationships among them. He advised that we “consider the principal parts of the problem attentively, repeatedly, and from various sides.” Mathematicians are not alone in drawing diagrams. Engineers and architects draw them. So do ecologists. As do medical researchers. And many others. People in business talk of business models, which show how a company’s activities fit together to make a profit, but they rarely draw complementary diagrams of the phenomenon they are dealing with. Business analysis is full of standard frameworks (e.g., SWOT, 5-Forces) but they are not the same as drawing diagrams of particular situations, especially those that are complex, new, or unfamiliar. A diagram is a visual story of how things happen. You can even write this story briefly in words. Once we understand the story of how things happen, we can make better decisions and reduce the chances of failure. Finally, diagrams are effective communication tools. Complex problems are rarely solved alone. They require engaging others. Diagrams can help tell the story of what is being tackled and how.
In blog-posts that follow, I will write about Pólya’s remaining 3 steps for solving problems.
Pólya, G. (1957). How to Solve It: A New Aspect of Mathematical Method. 2nd edition, Princeton University Press, Princeton, New Jersey.
Copyright © 2013 Gaurab Bhardwaj