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Understanding Problems and Generating Solutions

Introduction​

The foundation of effective problem solving lies in our ability to accurately understand the nature of a problem and generate viable solutions. This process involves constructing mental representations, identifying relevant information, and applying strategic thinking to develop potential pathways toward resolution.

Understanding the Nature of the Problem​

Internal Representation​

According to Greeno (1977), understanding a problem involves constructing an internal representation - a mental pattern or model that mirrors the structure and relationships within the problem itself. This cognitive process is similar to how we understand language.

Example: When you understand a sentence, you create an internal representation in your mind where concepts are related to each other in the same way they appear in the original sentence. This mental model becomes the framework through which you process and work with the problem.

The Role of Background Knowledge​

Creating an effective internal representation requires:

  1. Background Knowledge: Understanding the meaning of various elements in the problem
  2. Contextual Understanding: Recognizing how different components relate to each other
  3. Domain Familiarity: Previous experience with similar problems or concepts

Practical Example: If solving a mathematical word problem, your background knowledge of mathematical operations, units of measurement, and logical relationships helps you construct an accurate mental model of what the problem is asking.

Key Steps in Understanding Problems​

1. Attention to Important Information​

The problem solver must focus attention on essential information while filtering out irrelevant details. This selective attention is critical because:

  • Cognitive Load: Our working memory has limited capacity
  • Efficiency: Processing irrelevant information wastes mental resources
  • Accuracy: Focusing on key elements improves solution quality

2. Identifying Essential vs. Irrelevant Information​

Decision Process:

  • Which information is essential for reaching the solution?
  • Which details are distracting or non-contributory?
  • What assumptions can be safely made?

Example: In the problem "A train traveling at 60 mph leaves Station A at 2 PM heading toward Station B, 180 miles away. The train is painted blue with red stripes," the color of the train is irrelevant information that should be disregarded.

3. Representation Methods​

Once essential information is identified, the next step is to represent the problem effectively. Various representation methods can be employed:

Symbolic Representation​

Using letters, numbers, or mathematical symbols to represent problem elements:

  • Variables: x, y, z for unknown quantities
  • Equations: Mathematical relationships between elements
  • Logical Operators: AND, OR, NOT for logical problems

Lists and Matrices​

Organizing information in structured formats:

  • Simple Lists: Sequential ordering of elements
  • Two-dimensional Matrices: Showing relationships between two variables
  • Tables: Organizing multiple attributes of problem elements

Graphs and Diagrams​

Visual representations that show relationships:

  • Tree Diagrams: Hierarchical relationships and decision paths
  • Network Graphs: Connections between multiple elements
  • Flow Charts: Sequential processes and decision points

Visual Images​

Mental imagery and sketches:

  • Spatial Visualization: Mental rotation and manipulation
  • Diagrams: Physical sketches of problem scenarios
  • Mind Maps: Radial organization of concepts

Generating Solutions​

After understanding the problem, the next critical stage is generating potential solutions. This creative and analytical process involves several approaches:

1. Drawing on Past Experience​

Analogical Reasoning: Looking for similarities between the current problem and previously solved problems:

  • What similar problems have you encountered?
  • How were those problems solved?
  • Can those solution strategies be adapted to this situation?

Pattern Recognition: Identifying familiar patterns or structures:

  • Does this problem follow a known template?
  • Are there recurring elements that suggest a solution approach?

2. Utilizing Present Resources​

Available Tools and Information:

  • What resources are currently at your disposal?
  • What information can be gathered or accessed?
  • What tools or techniques are available for this problem domain?

Environmental Constraints:

  • Time limitations
  • Material resources
  • Access to expertise or assistance

3. Critical Thinking About the Problem​

Analytical Approach:

  • Decomposition: Breaking the problem into smaller, manageable parts
  • Hypothesis Generation: Proposing potential explanations or approaches
  • Logical Reasoning: Applying deductive and inductive reasoning

Creative Thinking:

  • Brainstorming: Generating multiple possible solutions without initial judgment
  • Lateral Thinking: Approaching the problem from unconventional angles
  • Innovation: Combining elements in novel ways

4. Strategic Considerations​

Different solution strategies vary in:

Time Investment​

  • Exhaustive Methods: Time-consuming but thorough
  • Heuristic Methods: Faster but may not guarantee solutions
  • Balanced Approaches: Combining efficiency with thoroughness

Success Guarantee​

  • Algorithmic Approaches: Guaranteed success if properly executed
  • Heuristic Approaches: Probable but not certain success
  • Hybrid Strategies: Combining both approaches strategically

The Iterative Nature of Solution Generation​

Generating solutions is rarely a linear process. It often involves:

  1. Initial Generation: Creating first-pass solutions
  2. Evaluation: Assessing feasibility and effectiveness
  3. Refinement: Improving upon initial ideas
  4. Alternative Generation: Creating backup solutions
  5. Selection: Choosing the most promising approach

Practical Applications​

Academic Problem Solving​

Mathematical Problems:

  • Identify known values and unknowns
  • Represent relationships algebraically
  • Apply relevant formulas and principles

Scientific Problems:

  • Define variables and controls
  • Formulate hypotheses
  • Design experiments to test solutions

Real-World Problem Solving​

Personal Decisions:

  • List goals and constraints
  • Generate multiple options
  • Evaluate pros and cons

Professional Challenges:

  • Analyze stakeholder needs
  • Consider resource limitations
  • Develop feasible action plans

Creative Problem Solving​

Design Challenges:

  • Understand user needs (problem representation)
  • Sketch multiple concepts (solution generation)
  • Prototype and test (evaluation)

Innovation Projects:

  • Identify pain points
  • Brainstorm novel approaches
  • Develop and refine ideas

Common Obstacles in Problem Understanding​

1. Functional Fixedness​

Difficulty seeing alternative uses or representations of familiar objects or concepts.

Example: Struggling to use a coin as a screwdriver because we're fixed on its monetary function.

2. Mental Set​

Tendency to approach new problems using familiar methods, even when they're not optimal.

Example: Continuing to use long division when a calculator would be more efficient.

3. Confirmation Bias​

Focusing only on information that confirms our initial understanding while ignoring contradictory evidence.

4. Information Overload​

Becoming overwhelmed by too much information, making it difficult to identify what's essential.

Strategies for Effective Problem Understanding​

1. Slow Down and Reflect​

  • Take time to thoroughly read and understand the problem
  • Don't rush to solutions before fully grasping the issue
  • Ask clarifying questions

2. Restate the Problem​

  • Put the problem in your own words
  • Explain it to someone else
  • Verify your understanding

3. Draw or Diagram​

  • Create visual representations
  • Use sketches to externalize your mental model
  • Identify relationships visually

4. Identify Knowns and Unknowns​

  • List what information you have
  • List what you need to find out
  • Note any assumptions you're making

5. Consider Multiple Perspectives​

  • How might others view this problem?
  • Are there alternative ways to frame this issue?
  • What assumptions am I making?

Integration with Problem Solving Stages​

Understanding problems and generating solutions are the first two critical stages in the broader problem-solving process:

  1. Understanding the Problem (Current Focus)
  2. Generating Solutions (Current Focus)
  3. Judging Solutions (Next Stage)
  4. Implementing Solutions (Subsequent Stage)
  5. Evaluating Outcomes (Final Stage)

Each stage builds upon the previous one, with the quality of your problem understanding directly impacting the quality of solutions you can generate.

Conclusion​

Effective problem solving begins with accurate problem understanding and creative solution generation. By constructing clear internal representations, identifying essential information, and using appropriate representation methods, problem solvers create a solid foundation for finding effective solutions. The combination of past experience, present resources, and critical thinking enables the generation of multiple potential solutions, setting the stage for evaluation and implementation.

Source PDFs:

Self-Assessment Questions​

  1. What is meant by "internal representation" in problem solving, and why is it important?

  2. Explain the role of background knowledge in understanding problems.

  3. How do you distinguish between essential and irrelevant information in a problem?

  4. List and describe three different methods of representing problems.

  5. What factors should you consider when generating potential solutions?

  6. How does past experience contribute to solution generation?

  7. Describe the difference between time-consuming methods that guarantee solutions versus faster methods that may not.

  8. What is the relationship between problem understanding and solution quality?

Memory Aids​

RISE Framework for Problem Understanding:

  • Represent: Create an internal mental model
  • Identify: Recognize essential information
  • Simplify: Remove irrelevant details
  • Evaluate: Check understanding accuracy

PEG for Solution Generation:

  • Past Experience: What worked before?
  • Existing Resources: What's available now?
  • Generate Alternatives: Create multiple options

Further Reading​

  • Greeno, J. G. (1977). Process of understanding in problem solving. In N. J. Castellan, D. B. Pisoni, & G. R. Potts (Eds.), Cognitive theory (Vol. 2). Hillsdale, NJ: Erlbaum.
  • Newell, A., & Simon, H. A. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.
  • Anderson, J. R. (2010). Cognitive psychology and its implications (7th ed.). New York: Worth Publishers.