Problem Solving

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# Problem Solving - PowerPoint PPT Presentation

Problem Solving. Shortcuts through the Problem Space. Problem Solving. Problem = a situation in which one is trying to reach a goal Problem solving = finding a means for arriving at a goal. Stages of Problem Solving. Define the problem -- Problem identification and representation

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## Problem Solving

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### Problem Solving

Shortcuts through the Problem Space

Problem Solving
• Problem = a situation in which one is trying to reach a goal
• Problem solving = finding a means for arriving at a goal
Stages of Problem Solving
• Define the problem -- Problem identification and representation
• Select a strategy -- plan a solution
• Carry out the strategy -- execute the plan
• Evaluate the plan and the solution -- determine whether it worked
Defining the Problem
• Identify the initial state and the goal state
• Well-defined Problem: clear definition of problem and goal states (example: locked out of your house)
• Ill-defined Problem: the problem or goal state is not clearly defined (example: increase crop production in USSR -- Voss, Grene, Post, & Penner, 1983)
• Decide how to represent the initial and goal states
Problem Representation
• How difficult it is to solve a problem often depends on how you choose to represent it.
• Some examples:
• "number scrabble" (Newell & Simon, 1972)
• "Bird and Trains" math problem (from Posner 1973, Cognition: An introduction)
• The monk problem(Duncker, 1945)
Selecting a Strategy:Algorithms vs. Heuristics
• Algorithm
• a procedure that is guaranteed to produce a solution to the problem
• Examples:
• solving the anagram "xbo“ by enumerating all possible combinations: xbo, xob, oxb, obx, bxo, box
• What about "ntraoc"? There are 6! (or 72) possible combinations.
Heuristics in Problem Solving
• Heuristic = a rule of thumb, or "mental shortcut" for solving a problem
• not guaranteed to give the right answer
• usually much more efficient than an algorithm
• Heuristics for solving anagrams:
• “xbo”
• vowel in the middle
• assume “x” is not word-initial
• “ntraoc”
• start with likely groupings of letters: "ant, car, tan, tar, ton"
Problem Space and Computational Complexity
• Problem space = all the possible states of affairs that could be produced from transformations of the initial problem state.
• Problem solving consists of searching the problem space for a state that matches your goal state.
• Algorithms search the entire space; heuristics search only part
• If the problem space is too large, an algorithmic approach is impossible. Example: Chess.
Useful Problem-Solving Strategies (Heuristics)
• Simple Search (Hill Climbing)
• Means-end Analysis
• Break the problem into subgoals
• Used in the General Problem Solver (Newell & Simon, 1963; Newell, Simon, & Shaw, 1958)
• Working Backwards. Useful when:
• There is only one goal state and it is clearly specified
• There are a number of possible ways to represent the problem state
Try Out Your Problem Solving Skills
• The “Calvin & Hobbes” Game
• The Water Jar Problem (Luchins, 1942)
• The “9 Dots” Problem
• Can you connect all 9 of these dots by drawing 4 straight lines, without lifting your pencil from the page?
• Give up? Here is a solution.
“Set” in Problem Solving
• "set" = state of mind a person brings to a problem solving situation
• An inappropriate "set" can keep you from representing the problem in the most productive way, or from choosing the best solution strategy.
“Set” interfering with problem representation
• The nine dots problem -- including an unnecessary boundary in your representation of the problem
• Functional Fixedness: failing to see a new use for an object
• Duncker (1945) -- mount a candle on the wall
• The two string problem
• "Persistence of set" in the water jar problem (Luchins, 1942)