Have you ever had the experience of driving somewhere, but then forgetting how you got there or even when you arrived? Most of us know that disorienting feeling. In fact, the more senior among us sometimes forget how–or why–we walked into a room!
If you spend more time on the road than in a classroom these days, you may not recall that this phenomenon plays out every single day in schools and on tests around the world. Students diligently toil on math problems to arrive at solutions they suddenly forget how they found. George Pólya, the influential mathematician who authored How to Solve It: A New Aspect of Mathematical Method, knew all too well how quickly people abandon and encapsulate their efforts:
Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. By looking back at a completed solution, by reconsidering and reexamining the result and the path that led to it, they could consolidate their knowledge and develop their ability to solve problems.
According to Polya, failure to look back strips experiential value from effort, slowing growth and ensuring that similar problems will be just as difficult to solve. Even worse, by not looking back, students may not recognize mistakes made during problem solving:
The student has now carried through his plan. He has written down the solution, checking each step Thus, he should have good reasons to believe that his solution is correct. Nevertheless, errors are always possible, especially if the argument is long and involved. Hence, verifications are desirable.
We see time and again how students lose points on the SAT and ACT simply because they forget to look back. These exams test the highest levels of math ability not only through advanced content but also very wordy problems. Many math problems require so many steps that students typically forget what they’re doing mid-question. This is exactly what the test makers want.
Effective problem solving should always involve that final step of looking back. In the classroom, looking back often entails checking work and taking additional steps to ensure accuracy and quality. On timed exams, checking work is a luxury few test takers can afford. Looking back, then, requires attention to the earlier steps in the problem solving process. Smart problem solvers always begin by understanding the problem, then determining their quickest, easiest, and most accurate path to the solution before actually doing the work. Once the work is done, take a moment to confirm that your solution answers the question that was asked. Looking back this way protects you from the common trap of answering the wrong question or presenting the result of an intermediate step as the ultimate solution.
Life throws all sorts of problems our way, which means that problem solving is a core life skill. If you don’t want to be buried under an avalanche of unsolved issues, always take the time to look back at how you arrived at your solutions and whether you actually solved the problem at hand. I’m not saying that consistently following this system will lead to a problem-free existence, but at least you won’t be struggling with the same ones over and over!
Mike Bergin