A-2: PROBLEM SOLVING
Problem solving is an art best learned by practice. It requires understanding, systematic procedure, ingenuity, common sense, and creativity. Problem solving is learned easily by some students, for whom the process may seem "obvious" or "trivial." Most students will need quite a bit of practice before they attain confidence in their abilities.
The student who has little or no previous practice in problem solving should start by following a systematic procedure to ensure the avoidance of trivial mistakes and develop good habits.
Such a systematic procedure is given below. You might call it a "cook-book" method for problem solving! Some steps may seem obvious, but even good students would benefit from reading and using them. This procedure is one that the best and most successful students actually use.
As an example of what even good students can learn from this procedure, note that it suggests solving the problem algebraically before inserting numerical values. Many students, left to themselves, take a long time to discover that the algebraic solution often eliminates some quantities from the calculation entirely, and considerably simplifies the arithmetic that must be performed.
Most rules have exceptions. You will encounter problems, especially in advanced courses, where the algebra becomes so messy that it is simpler to numerically evaluate certain key quantities along the way. Knowing when to do this comes from having a good grasp of the complete solution strategy before actually doing the gory details.
Although the procedure summarized below is a good one, used by many successful students, it is not presented as the only correct one! Use it while you are learning. When you reach the point where you understand physics well, you will no longer need these "cook book" rules.
2. Terms: Do you know and understand the definitions of all words and terms used in the problem? If not, look them up with the aid of the index of your textbook. If you still don't quite understand them, try another book. Attempting a problem without knowing precisely what it means is futile; your time would be better spent learning basic definitions and principles.
3. List all given facts. Some important facts may not be explicitly stated, but are understood by the context of the problem. For example:
4. Draw a diagram to help you visualize the physical situation. Label it well, and insert any given values.
5. Decide what sort of answer is required. List and label (in the diagram) all relevant unknowns.
6. Consider the situation carefully. [This requires thinking about it.] List the principles that apply to this problem. Double-check: are you sure they apply? Write down the principles in formula form.
7. Stop to think about what you now have. Decide which combinations of facts, formulae and principles will most efficiently lead to the desired result. If necessary, split the problem into smaller parts that are easier to handle. Make approximations if high precision isn't necessary. Some given facts may not be needed, so ignore them if you are certain they don't apply.
8. You may need some formulae or theorems from pure mathematics. List those you think might help. Look them up if you have the slightest doubt about the reliability of your memory.
9. Solve the problem mathematically for the desired unknown(s), without yet inserting numerical values.
10. Decide which unit system you will use. If most or all of the data is expressed in one unit system, that's probably the best system to use. Convert all of the data to the units of the chosen unit system.
11. Insert the given numerical values into the mathematical result, and perform the operations to obtain the answer(s).
12. Check each answer. Think about it critically. Is it reasonable? Make independent checks on the answer. Compare each answer with something with which you are familiar.
13. If the "answer" is an equation, test it with simple values, or for simple cases, where you are sure of the results.
14. Never submit your first draft. Reorganize and rewrite it, explaining the reasons for all steps (except for the obvious elementary operations). This process may seem like extra work, but it is part of the important process of communicating your understanding to others, and doing it will strengthen your own understanding. It might even clarify some points in your mind, or even expose a blunder or error you might have made.
Some students become so good at this that their first drafts are nearly good enough for final submission. I once had a student who worked all his homework and exam problems using a fountain pen, and they were as organized and clear as anyone could want. The few occasional errors were "lined out" neatly. But note, in my 38 years of teaching, I only had one student who was that organized and confident in understanding. It's rare.
GENERAL HINTS FOR WORKING PROBLEMS:
Don't be reluctant to use several pages for a problem. Crowded work is hard to read and hard to check in case of a blunder. Write on only one side of each sheet and leave margins around all sides.
Be neat and orderly. Never submit your original "scratch" solutions. Recopy them into organized form. This step is not merely to make the work "look good." Doing this helps you to organize your thoughts and reinforce your understanding of the problem.
Include labeled diagrams as appropriate, the labels consistent with the notation used in the problem.
Include explanatory words stating what principle is being used, what assumptions have been made, which case is being considered, etc. The problem solution should never be a disjointed collection of equations. It should read as a narrative of words, equations, and diagrams. The worked examples of textbooks serve as a good model for your problem solutions.
Carry all units through the numerical calculations as a check on the work. Label final answers and give their units.
Do problems as a way to strengthen your understanding and improve your skills. Do not be content with just "getting the answer." Be sure that you know why the method works. Ask yourself if there are other ways the problem might be done. If there are, you can check your results and perhaps find an easier method, or a clearer one.
Physics textbooks today have many examples of worked problems. If you treat them merely as recipes or patterns, hoping that exam questions will be "just like" them, only with different numbers, they will do you no good at all. You won't have learned a thing from them.
Someone once said that you should never do a problem without first knowing the answer! This means that you should first think the problem through, using your physical intuition and common sense, to get a good idea of what a reasonable answer would be. This is good advice. If your advance "intelligent guess" turns out to be right, you will gain confidence in your understanding. If it turns out to be wrong, you may discover a flaw in your analysis that will help you avoid such mistakes in the future.
"Getting the right answer" isn't the only important goal. You want to learn something about physics as you do the problem, and you want to polish your skills at doing physics.
In my undergraduate days at the University of Iowa I knew a physics stockroom clerk, Mr. Grescher, who hadn't a degree to his name, but could make any physics apparatus work. He was responsible for all the equipment professors used for demonstrations in their classes; he helped them design it, and he'd set it up and have it working before the professor came to lecture. His genuine interest, cleverness and willingness to "tinker" had given him a really good intuitive understanding of physics. Sometimes we student assistants would need to borrow a piece of physics equipment from his stockroom. Before letting it out of his care, he'd question us about what we intended to do with it, and make sure we knew how to use it properly. When students tried to show off their knowledge by high-powered mathematical analyses. Mr. Grescher would tolerate this just so long, then haul them up short with the question "Do you understand all you know about it?" Students ought to ask themselves that question, often. There's a great difference between knowing facts and information, and genuine understanding.
Another lesson I learned as an undergrade was from the director of the radiology lab at the University Hospitals. I had been given the task of taking some metal Geiger-Müler tubes there to be x-rayed, so we'd have a picture of their internal geometry for a published paper. We wanted the X-ray source to be far from the tubes, for least geometric distortion, but that put the x-ray tube well off its calibrated scale. The lab technician assigned to help me could not calculate the proper exposure, since she was used to simply reading that value from the scale. I suggested calculating it using the inverse square law, but she was unsure whether that was appropriate, and asked her boss. He shrugged, and scribbled some calculations on a prescription pad (using the inverse square law) and looked at the result. "Set the current at the same value you'd use for a Baby's arm." That she understood! Later, when she was out of the room, he said to me "It's so hard to get good help. They are trained, not educated. They know only what they've been taught."
A philosophy professor at that school challenged me by asking why I was taking his philosophy of science course, since it wasn't required of physics majors, and I was the only physics major taking it that year. I responded with something naive like, "I think it might give me broader perspective to enhance my understanding of what physics is all about." "Nonsense," he replied. "To be good at physics, you have to `have physics in your bones'. If you don't, no course will do you any good, not even mine."
Galileo Galilei observed, "You can't teach anyone anything; you can only help them find it within themselves." The education that lasts the longest and does you the most good in the long run is that you achieve by your own hard work, sweat, brainstorming and perseverance. Instructors can give you hints, encouragement, point out your errors, show where you went wrong, prod you to try harder; but if they have to finally give up and show you how to do it they know that the process has failed.
© 1996, 2004 by Donald E. Simanek.