Problem Solving Notes

What is a Problem?

Engineers are problem solvers, hired to solve problems. The only way to learn problem-solving is by doing. Engineering education requires solving numerous homework problems, utilizing tools such as computers.

A problem is defined as a situation faced by an individual or group with no obvious solution.

Types of Problems

  • Research problems: These involve proving or disproving a hypothesis.
    • Example: Smoking causes lung cancer.
  • Knowledge problems: These occur when someone encounters an unfamiliar situation.
    • Example: A computer engineer finds their computer's performance sharply declines after 30 minutes and initially cannot determine why. Further analysis reveals that the fan placement obstructs its function, leading to increased temperature and processor slowdown.
  • Troubleshooting problems: These arise when equipment malfunctions or behaves unexpectedly.
    • Example: A computer displays a message indicating a limited wireless connection.
  • Mathematical problems: These are frequently encountered by engineers, who use mathematical models to describe physical phenomena. Other problem types are often converted into mathematical problems.
  • Resources problems: These involve accomplishing tasks with limited resources such as time, money, labor, equipment, and capacity. Engineers who can overcome these limitations are highly valued.
  • Social problems: These relate to people in a workplace or community.
    • Example: A factory experiences a shortage of skilled labor due to poor quality local schools. Engineers may need to implement a training program to accommodate the trainees' low reading abilities.
  • Design problems: These necessitate creativity, teamwork, and broad knowledge.

Good Problem Solver vs. Poor Problem Solver

Good Problem Solver:

  • Motivated, persistent
  • Logical, careful
  • Thoroughly understands the problem before starting
  • Divides and conquers the problem
  • Uses basic logical principles
  • Organized in analysis
  • Has a good feel for the correct answer

Poor Problem Solver:

  • Easily discouraged
  • Not logical, careless
  • Starts before thoroughly understanding the problem
  • Directly tries to solve the whole problem
  • Uses intuition and guesses
  • Disorganized
  • Believes that the answer produced by the computer is correct

Techniques for Error-Free Problem Solving

Engineers can never be absolutely certain an answer is correct. Example: A civil engineer designing a bridge cannot be sure of their calculations until a heavy load is placed on the bridge and the deflection agrees with those calculations.

To increase the probability of calculating a correct answer, use the following procedure:

  1. Given:
    • Draw a picture.
    • State any assumptions.
    • Indicate all given properties on the diagram with their units.
  2. Find:
    • Label unknown quantities with a question mark.
  3. Relationships:
    • Write the main equation.
    • Isolate the desired quantity in the equation.
    • Write subordinate equations for the unknown quantities.
  4. Solution:
    • Insert numerical values with units.
    • Ensure units cancel appropriately.
    • Compute the answer.
    • Mark the final answer (indicate units).
  5. Check:
    • Ensure the final answer makes physical sense.
    • Ensure all questions are answered.

Estimating

Estimating is an important tool to quickly check answers, ensuring the answer "feels" correct.

How to improve estimating capabilities:

  • Practice
  • Broaden interests by reading, watching, and being involved
  • Get comfortable with numbers; do not solely rely on calculators
  • Learn the units, especially those related to your major

Strategies for Solving Problems

Researchers like Polya and Schoenfeld have described various problem-solving strategies to help develop or refine problem-solving skills. Individuals may develop their own unique problem-solving approaches.

George Polya’s How to Solve Method

This method is suitable for mathematical problems, but can be applied to other types.

  1. Understand the problem:
    • What is the unknown?
    • What are the data?
    • What is the condition?
    • Is the condition sufficient to determine the unknown? Or insufficient? Or redundant? Or contradictory?
    • Draw a figure; introduce suitable notations.
  2. Find a connection between data and the unknown:
    • Have you seen it before? Or have you seen the problem in a slightly different form?
    • Do you know a related problem? Do you know a theorem that could be useful?
    • Look at the unknown and try to think of a familiar problem having the same or a similar unknown.
    • Here is a problem related to yours; could you use its results? Its solving method? Etc.
  3. Carry out your plan:
    • Carry out your plan of solving the problem; check each step.
    • Can you see clearly that the steps are correct?
    • Can you prove they are correct?
  4. Examine the solution obtained:
    • Can you check the results? Can you check the argument?
    • Can you derive the results differently?

Bransford and Stein Method

  • Identify the problem
  • Define and represent it
  • Explore possible strategies
  • Act on the strategies
  • Look back and evaluate the effects of your activities

Schoenfeld Method

  • Analyze the problem
  • Explore it
  • Plan
  • Implement
  • Verify

Krulik and Rudnick

  • Read the problem
  • Explore it
  • Select a strategy
  • Solve
  • Look back