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In this thread is a problem solving guide I have written for students in STEM. Bear in mind that these techniques specifically apply to problems in non-proof based mathematics and the physical sciences.
In addition to including a lot of my own ideas, many of the following methods were inspired by How to Solve It by George Polya and The Art of Insight in Science and Engineering by Sanjoy Mahajan, and Mathematical Problem Solving by Alan Shoenfeld.
Solving problems is the most tried-and-true method to learn in science and mathematics. It doesn’t matter whether you are taking a course or self-teaching; solving problems is the best way to test your understanding of the learned concepts. Now. let’s get to the actual guide.
1st Phase. Understanding.
The most crucial step to solving a problem is understanding the problem. What are you asked to find or show? Are you asked to solve for a quantity or property (integral, energy, some vector quantity, probability, etc.)?
The most crucial step to solving a problem is understanding the problem. What are you asked to find or show? Are you asked to solve for a quantity or property (integral, energy, some vector quantity, probability, etc.)?
Are you asked to derive an equation or, perhaps, manipulate a given equation to obtain another equation? Is it a conceptual question? Make sure to identify the elements or values that are given as well as potentially useful elements that are not given.
What principle does the problem require to be solved? To understand and solve the problem, you need to classify it. Is this a boundary value problem? Do you see any indications of Gauss’s law? Can you re-formulate the problem? See if you can attempt a change of perspective.
2nd Phase. Exploring.
This is the step where we explore the problem and plan our attack. Can you break the problem up and establish sub-goals? Try to identify not only your target, but also the intermediate steps/variables (which are not given) needed to obtain your target.
This is the step where we explore the problem and plan our attack. Can you break the problem up and establish sub-goals? Try to identify not only your target, but also the intermediate steps/variables (which are not given) needed to obtain your target.
What can you normally get from the given variables and can you use it to your advantage? Can you think of a picture or a diagram that might help you? More often than not, using visualizations can help you. It’s fine if the sketch is not entirely accurate.
For example, maybe you can envision a moving frame of reference as a train. Do you see a pattern? Often, it’s possible to exploit symmetry or invariance; doing so can greatly simplify the problem. Can you solve a modified version of the problem or consider a more special case?
What if you relax a condition (without loss of generality) and try to reimpose it? Try examining limiting cases or choosing special values to exemplify the problem. Now gather the information you acquired from your exploration, and see if you can devise a plan.
3rd Phase. Executing.
Before carrying out your plan, make sure you actually understand your approach. Does your approach really make sense? Try to clearly elucidate your plan. Have you used all of the pertinent information?
Before carrying out your plan, make sure you actually understand your approach. Does your approach really make sense? Try to clearly elucidate your plan. Have you used all of the pertinent information?
4th Phase. Verifying.
Before settling on your result, ask yourself, “Does my answer make sense?”. Does it withstand tests of symmetry, dimensional analysis, and limiting cases? If your solution does not withstand a special case, then you can be sure that it’s wrong.
Before settling on your result, ask yourself, “Does my answer make sense?”. Does it withstand tests of symmetry, dimensional analysis, and limiting cases? If your solution does not withstand a special case, then you can be sure that it’s wrong.
Does it comply with known rules and measurements? For example, speed greater than 299, 792, 458 m/s probably isn’t what you want. You either made a mistake in your calculation or your approach was wrong. If your calculation seems ok, then go back to the third phase.
5th Phase. Learning.
We’re not done yet. Now is your opportunity to learn from your solution. Can you solve the problem using another technique? You might find that there’s a more efficient method for solving the problem.
We’re not done yet. Now is your opportunity to learn from your solution. Can you solve the problem using another technique? You might find that there’s a more efficient method for solving the problem.
Can your answer be reduced to known results? See if you can manipulate your solution in a way that it can lead to a known formula or principle.
And there you have it. Here are some final notes:
DO NOT LOOK AT THE SOLUTIONS. At least until you are sure you are done with the problem. Too many times I see other students jump to the back of the book after struggling with the problem for only 20 to 30 minutes.
DO NOT LOOK AT THE SOLUTIONS. At least until you are sure you are done with the problem. Too many times I see other students jump to the back of the book after struggling with the problem for only 20 to 30 minutes.
If you still find yourself getting nowhere with a problem, there are a lot of people online willing to help. Two websites where you can find such help are http://physicsforums.com and http://stackexchange.com .
If you are solving a numerical problem, it’s often better to solve it symbolically and plug in the values at the end. This way, you are less likely to make a mistake.
If you have enough time (you most likely do),then solve additional problems (even if they were not assigned by the instructor). Tackle the challenging problems that use techniques or concepts which you have a weak grasp on.
