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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

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Problem Solving

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problem solving

Problem Solving

Shortcuts through the Problem Space

problem solving2
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
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
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
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
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
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 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
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
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” 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
“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
set interfering with choosing an effective solution strategy
“Set” interfering with choosing an effective solution strategy
  • "Persistence of set" in the water jar problem (Luchins, 1942)