This is a guest post from Chelsey Cooley, GMAT and GRE instructor at Manhattan Prep. Manhattan Prep empowers students to accomplish their goals by providing an outstanding test prep curriculum and the highest-quality instructors in the industry.


In 1945, mathematician George Pólya published a small book entitled How to Solve It. At first glance, How to Solve It is a simple guidebook on solving math problems, addressed mainly to primary-school math teachers. However, it has far deeper implications. We all come across difficult problems from time to time, regardless of our field of interest, and Pólya's simple method -- understand, plan, solve, or UPS -- can help us all refine our problem-solving skills.

Can problem-solving really be learned?

Problem-solving ability, Pólya argues, isn't innate:

"Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming."

This aligns with modern research in learning science, which suggests that adopting a growth mindset can help you improve even those skills that are conventionally thought to be inherent. For instance, students who can be convinced that there's no such thing as 'bad at math' end up improving their mathematical skills more than those who believe that mathematical ability is inborn. So, can you really get better at the abstract skill of solving problems? You probably can -- as with swimming, painting, or math -- but, like those skills, better problem solving requires thoughtful practice.

Understanding a problem

'Understanding the problem' is often the first step to be thrown out when you need to find a solution quickly. However, you can solve problems more efficiently by making this the longest step of your problem-solving process.

Imagine getting a flat tire while riding a bicycle. If you chose to skip the understand step, all you'd need to do is replace the inner tube and ride away. But, ten minutes down the road, you'd find yourself with a second flat, because you missed the tiny piece of glass that was stuck in the tire. The understand step is the step where you check the tire and the wheel.

When you're trying to understand a problem, don't worry about possible solutions. Instead, focus on rephrasing the problem, making it more general or more specific, and understanding all of its facets. Pólya suggests a series of questions to ask yourself while working towards understanding. Although they come from a framework of mathematics, they can be used while solving any problem at all.  

What is the unknown? What do I not know or have?

What resources do I have? What do I know or have already?

Am I sure it's possible to solve the problem? Is it reasonable?

Have I ever seen a problem like this one before?

Creating a plan

The central example in How to Solve It, is the problem of finding the distance between two opposite corners of a large room:

missing_length

Understanding this problem means drawing the diagram above, including the known dimensions of the room. Next, you need a plan. One of the most useful questions, during this step, is: what information would help me solve the problem?

You could look at the unknown distance as the longest side of a right triangle. There's a formula that relates the length of the hypotenuse to the lengths of the two legs. You only know the length of one of those legs right now -- it's the height of the room, which you measured earlier. However, you now have a new, intermediate goal: find the length of that second leg. That's a refinement of your plan.

 

 

ups2

 

No two great plans are identical. However, many of them have something in common, and identifying those commonalities can also help you approach tough problems more effectively. Ask yourself: Have I ever solved a problem like this one before? How? Can I solve one small part of this problem? Can I think of a simpler or easier version of the problem, and solve that first? These three tools -- identifying information you need, finding commonalities with already-solved problems, and solving a related problem -- are powerful tools to use when creating a plan.

Solving the problem, and beyond

Solving the problem is the least interesting step in the process, when using this approach. One of the toughest habits to break, in changing your problem-solving style, is the desire to begin solving immediately. Experimentation does have its place -- it can help you both understand and plan -- but it should be accompanied by thoughtful reflection.

Reflection, by the way, is the last step of the process. The oft-cited 10,000 hour rule has faced recent criticism for glossing over the role of reflection and attention in improvement. If you really want to get better at solving problems, don't just solve dozens of problems in the same old way, putting them out of mind once the crisis is over -- rather, reflect deeply on the problems you've solved. Can I use this solution for another, related problem? Was there a quicker, cheaper, smarter, or more elegant solution?  

Consider some of the problems you regularly solve, and how you solve them. Then, commit to getting better at solving problems. A number of successful organizations have adopted variants on understand, plan, solve, and you can also apply it to your own endeavors. Problem-solving is a skill, so practice it!