8.4 Overview of Problem Solving Processes
Problem Solving in Thought
Introduction to Problem Solving
Definition of problem solving: The process of finding solutions to challenges that we face in everyday life.
Importance: Problem solving is a necessary skill to overcome obstacles and challenges.
Approaches to Problem Solving
Algorithm Approach
Definition: A structured, step-by-step method for solving a problem.
Characteristics:
Follows a rigid set of predefined rules.
Similar to a computer's method of problem-solving, where programmed steps are executed systematically.
Can explore all possible solutions through exhaustive methods (brute force).
Example: Using the algorithmic approach to unscramble an anagram by trying every combination of letters until the correct word is found.
Efficiency: While guaranteed to produce a solution eventually, this method is often inefficient due to the vast number of combinations to check.
Heuristic Approach
Definition: A mental shortcut or rule of thumb based on previous experiences that aids problem solving.
Advantages:
More efficient than algorithms for many real-world problems.
Can lead to quicker solutions based on educated guesses rather than exhaustive search.
Example: In the case of the anagram, utilizing knowledge of common letter combinations to quickly identify the word instead of trying every possible combination.
Example in Hangman: Rather than guessing letters sequentially (algorithm), a player might start with common vowels due to their frequency in the English language.
Limitations: Heuristics do not guarantee correctness; they can mislead or overlook correct answers.
Pitfalls of Heuristics
Example of Misleading Heuristic: The bat and ball problem:
Statement: A bat costs $1 more than a ball, and together they cost $1.10.
Common intuitive answer: Ball costs 10¢ (incorrect).
Actual solution:
Let the ball = x.
Bat = x + $1.
Equation: x + (x + 1) = 1.10
Resulting in: 2x + 1 = 1.10, thus x = 0.05 (Ball = 5¢, Bat = $1.05).
Mental Sets in Problem Solving
Definition of Mental Sets
Definition: Established methods of problem-solving that may hinder finding solutions when faced with new or altered problems.
Cause: Relying on memorized steps or familiar patterns instead of adapting to the constraints of the new problem.
Examples of Mental Sets
Mathematical or Physics problem-solving where familiar equations must be adapted to slightly altered situations.
Nine Dot Problem: An exercise illustrating mental set limitations. The goal is to connect nine dots with four straight lines without lifting the pencil.
Students often struggle to find the solution due to self-imposed limitations (thinking within the bounds of the dot arrangement).
Correct solution involves extending lines beyond the confines of the dots.
Functional Fixedness in Problem Solving
Definition of Functional Fixedness
Definition: A cognitive bias that limits a person’s belief to the usual functioning of an object, preventing creative problem-solving.
Examples of Functional Fixedness
Two Strings Problem:
Challenge: Tie two strings together while unable to reach both at the same time.
Common instinct: Attempting to use pliers as an extension of reach.
Correct solution: Use the pliers as a weight to swing and grab the other string for tying.
Candle Problem:
Task: Attach a candle to a wall so that it doesn't drip wax onto the surface below.
Misguided approaches often include using tacks directly with the candle, which won't work.
Elegant solution: Use the box of tacks as a shelf to hold the candle when tacked to the wall.
Importance of framing the problem: Changing the description of the box’s utility increases the likelihood of recognizing it as a usable object.
Conclusion
Importance of recognizing different approaches to problem-solving: Knowing when to apply an algorithm, heuristic, or adapt to overcome mental sets and biases.
Creativity and flexibility in thinking are crucial for effective problem-solving. Additional problem-solving skills can lead to insightful and efficient solutions.