Problem Solving Methods

By the end of this lesson, you should be able to:
1. Determine whether a problem and its proposed method of solution are well-matched.

2. Compare a variety of problem solving methods.

3. Use formula substitution for solving problems.

4. Discuss trial and error approaches and ways to maximize their usefulness.

5. Describe the scientific method and state its advantages.

6. Describe one creative problem solving method.

7. Discuss the importance of documenting work.

8. Explain how failed attempts to solve problems can be of use.

Matching the Method to the Problem

There are different approaches to problem solving. If you were attempting to alleviate repetitive stress injury by redesigning a keyboard, you would probably use a different strategy than if you were trying to determine the volume of a light bulb.

I teach a course where students design and build automated devices and remote manipulators to solve technical problems of manufacturing engineering. But the lab my students work in is used for general material processing and construction. As a result of the dominant equipment and materials, many of the students build their devices out of wood, even though metals or plastics are sometimes a much better choice. Attempting to build a robotic arm out of 2x4 spruce is not a wise decision, for a number of reasons (bulk, hardness, friction.)

Similarly, when you attempt to solve a problem, you should ask yourself if you're trying to build a robot out of a 2x4.

Sometimes a calculator can be an invaluable tool, but other times it can get in the way. Would you use a calculator to solve the following problem:
There is exactly one liter of pure water in one container, and exactly one liter of water-soluble blue ink in another. You take one (.1 ml) drop of ink from the second container and put it into the first, then you stir it in. You then take one (.1 ml) drop from the first container (which is nearly all water, with just a little bit of ink) and put it into the second container. Is there more ink in the water container, or more water in the ink container?
So, would you use a calculator to solve this? How would you set up the equation?

While certain assumptions must be made concerning how the fluids mix, evaporation, and fluid remaining on stirring devices, try to reconceptualize the problem as follows:

In the Acorn Bag there are 50 acorns, and in the Walnut Bag there are 50 walnuts. You combine these and then randomly divide the 100 nuts so that each bag contains 50 mixed nuts. Are there more acorns in the Walnut Bag, or more walnuts in the Acorn Bag? Well, if there are 50 nuts in each bag, then the number of acorns in the Walnut Bag must be equal to the number of walnuts in the Acorn Bag.
Would a calculator have helped? I suspect it would have interfered in solving this problem.

In another class I teach, my students learn to create trend lines by extrapolating current data to the future. For example, if I weigh 215 pounds, and I lose 5 pounds this week, 5 pounds the next week, and 5 pounds the third week, they could tell you exactly when I would vanish.

Once again, the mistake was made of using the wrong method for solving a problem.

Methods of Problem Solving

What are some different methods of solving problems? There are very many methods, and you have been successful at quite a few of them or you would not have gotten this far.

Please visit the following article by Dale Anderson, but I only ask that you examine (Table 1. A Comparison of Problem Solving Methods [Lumsdaine, 1995, p. 16]" from this article.

Formula Substitution Problem Solving

When using the "graphite method" to solve typical problems (given to you by others) in math and physics, one common approach is as follows:

  • 1. Study the problem statement.

  • 2. Write down the known values and variables.

  • 3. Write down the unknown variables you wish to solve for.

  • 4. Write down the equations that contain the variables in question.

  • 5. Manipulate the equations so that they solve for the unknown variables.

  • 6. Substitute the know values in these equations.

  • 7. Proceed step-by-step to solve the problem.

  • 8. Check your answer.
  • This is also a good approach when teaching, although you may wish to simplify a few steps. For example:

    Sample Shear Strength Problem:
    A cylindrical rod has a diameter of .4 inches. In a double shear test, the maximum load at failure is 10,000 Lbs. Determine the rod's radius, cross sectional area, double shear strength, and single shear strength.


  • Key:
  • DSS = double shear strength
  • SS = single shear strength
  • CSA = cross sectional area
  • R = radius
  • D = diameter
  • pi = circumference / diameter, approximately 3.1415926535
  • L = maximum load
  • Known:
  • L = 10,000 Lbs
  • D = .4 in.
  • Formulas:
  • R = D / 2
  • CSA (of a cylindrical rod) = pi * R ^ 2
  • DSS = L / CSA
  • SS = DSS / 2
  • Calculations and Answers
  • R = D / 2
  • R = .4 in / 2
  • R = .2 in
  • CSA = pi * R ^ 2
  • CSA = pi * .2 in ^ 2 [note: Remember that exponentation precedes multiplication.]
  • CSA = 3.1415926535 * .04 sq in
  • CSA = .1256637061 sq in
  • DSS = L / CSA
  • DSS = 10,000 Lbs / .1256637061 sq in
  • DSS = 79,577.47155 psi (or pounds per square inch.)
  • SS = DSS / 2
  • SS = 79,577.47155 psi / 2
  • SS = 39,788.73577 psi (Now let's round off.)

  • SS = 39,789 psi

    Notice that when the know values are substituted in the equations, their units are also used.

    But what type of answers does this method produce? Do we find out if something works or do we just get a numeric value?

    Trial and Error

    Trial and error works best for some people when they start with their intuitions. Those with less experience may find it more helpful to use a systematic trial and error approach.

    One important aspect of trial and error is that the trial accurately models reality. The vehicle proving grounds used to test US military vehicles contains many different terrains, with dirt, dust, and sand that behaves like the varieties of dirt, dust, and sand likely to be encountered throughout the world. This was an attempt to more closely mimic reality.

    But can trial and error accurately test the reasons or causes behind phenomena? Do we find out if our guesses and theories are correct?

    Scientific Method

    Another great problem solving method is the scientific method. This is especially useful in testing a theory. It usually includes the following steps:

  • 1. Gain background information (study, observation).

  • 2. Formulate a hypothesis or educated guess to test.

  • 3. Develop a methodology to test that theory.

  • 4. Objectively perform an experiment, complete with control groups, and accurately record observations.

  • 5. Analyze the results to determine whether they support the hypothesis.

  • 6. Draw conclusions.
  • There are a number of advantages to the scientific method, and the results are usually held in higher regard because of the rigor and objectivity involved. But what is the result? Is it an invention? What type of output do we get when we use the problem solving method? If a radio is broken and we want to fix it, should we employ the scientific method?

    Creative Problem Solving

    A different method that results in divergent, creative ideas has been called "creative problem solving." There are many different methods of creative problem solving; the following "OFPISA" method is one approach suggested by Sidney J. Parnes:

  • 1. Observation (or keeping one's mind, eyes and ears tuned for seeing possible problems and solutions)

  • 2. Fact-finding (or gathering data, especially through study)

  • 3. Problem-finding (wherein a well-formulated problem statement is developed)

  • 4. Idea-finding (or the generation of very many possible solutions, deferring judgment)

  • 5. Solution-finding (or the application of criteria to determine a single solution)

  • 6. Acceptance-finding (or finding ways for oneself and others to accommodate and live with the solution)
  • Some criticize this particular creative problem solving method because it does not include implementation or testing, others content that the process of solving technological problems tends to be cyclic rather than linear.

    One advantage of this approach is the attention it pays to developing the problem statement. Another advantage is the wealth of creative possibilities that are generated.

    Document Your Work

    Much of the time, work on solving problems by any of these methods can be improved by keeping clear and accurate documentation. It is very frustrating to test circuits if you find yourself re-testing the same area because you do not have an organized plan or good documentation.

    Embrace Failure

    Too often, we look at failure as failure. However, it is often the case that there is more learning resulting from failure than from success. Therefore, we should embrace failure and use it wisely. (The scientific method often fails to show a hypothesized effect, yet that failure represents a successful addition to our collective understanding.)

    Embracing failure can be particularly difficult in schools and in the workplace. In schools, teachers who emphasize the process rather than the product, and grade accordingly, help their students embrace failure. In industry, employees could be encouraged to freely discuss failure without fear of defensiveness.

    However, where the goal is not knowledge, a failure is a failure. But that's okay - sometimes we fail.


    There are many ways to solve a problem. Common approaches include formula substitution, trial and error, the scientific method, and creative problem solving. Some methods are more appropriate, depending on the problem to be solved.

    All information is subject to change without notification.
    © Jim Flowers
    Department of Technology, Ball State University