Problem Solving pdf - Prof. Cavillo PDF

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Introduction ! • Mayer (1992) a problem consists of three ideas! • The problem is presently in some state (the initial state)! • It is desired that it be in another state (the goal state)! • There is no direct obvious way to accomplish this change! • problem solving is the process of moving from the initial state to the goal state! • Problems have obstacles! • Restrictions that make it difficult to proceed from the initial to the goal state! • Problem solving cycle! • Identifying the problem! • Defying the representing the problem! • Constructing a strategy! • Organizing information! • Monitoring problem solving! • Evaluating problem solving ! • Well- and ill-defined problems! • Well-defined problems:! • Have clear initial and goal states, and clear solution paths! • Ill defined problems:! • Do not have clear initial and goal states, or clear solution paths! • Reproductive and productive problem solving! • Reproductive problem! • The application of a known solution! • Productive problem solving ! • Creatively thinking of a new solution ! • Historical note: most early work on problem solving used simple, well-defined problems solved in a lab! • Problems had specified solutions! • Researchers could follow the steps toward solutions! • Problems could be done in the lab in little time! • Historical note: in the 1970s researchers began to study ‘real-world’ (often ill-defined) problem solving! • Problem solving may be different in different domains! • Research has been done in a variety of fields (physics, legal reasoning, writing, game playing, mathematics)!

Three types of problems! • Greeno (1988) claims that most fit into one of three categories! • Problems of transformation! • Problems of arrangement ! • Problems of inducing structure ! • Not all problems fit nearly into one of these categories ! • Problems of transformation! • Well defined problems in which the problem solver must find a sequence of operations that produces the goal state! • examples:! • Water jug problem! • You have 3 different sized water jugs that carry different amounts of water (simple math)! • Your biggest jug has 127 and your goal is to get to 100. ! • Fill up A, dump out A, dump out C twice. ! • Tower of hanoi. (REMEMBER NAME)! • Move the disks from the left peg (initial state) to the right peg (goal state) moving only one at a time and never allowing a larger disk to be on top of a smaller disk. (obstacles)! • Tower of london (remember name)! • Developed by Shallice (1982)! • Created the goal state from the initial state in as few moves as possible! • Usually several restrictions! • Only move one disk at a time! • Stack height! • Used to measure executive functioning ! • People with frontal lobe damage, or young children just grab one and start, are unable to complete this task correctly.! • Problems of arrangement ! • Given all the elements and a general description of the goal, the problem solver must arrange the events in a way that solves the problem! • examples:! • Anagrams! • Make as many words as you can out of this word! • Some insight problems! • ex: the matchstick problem! • Given 6 matches of equal size, create four equilateral triangles ! • answer: 3d pyramid !

• Problem of inducing structure! • Given several examples or instances, the problem solver must discover a general rule or pattern that is consistent with the information! • We will discuss inductive reasoning next lecture! • Examples! • Series completion problems! • Analogy problems! • Series completion problems! • A a a b b b c c c d d __! • B A C B D C E D F ___! • 1 10 3 9 5 8 7 7 9 6 __ __ ! • Analogy problems ! Problem Solving Strategies ! • Several widely used strategies (approaches) to problem solving have been identified! • Working forward! • Working backward! • Generate-and-test! • Means-ends analysis! • Solving by analogy ! • Working forward ! • Also referred to as hill climbing ! • Attempt to solve a problem by decreasing the difference between the initial and goal states. ! • Useful when you cannot discover sufficient information about your alternatives! • Problematic when the “best” alternative is not the one that appears to be the most direct! • Also problematic when problems are ill-defined (no clear initial or goal states)! • Ex: little kids playing soccer go straight to the goal and through the field. ! • Example of when working forward is problematic ! • Suppose a students goal is to earn a lot of money! • The most direct path toward this is to take the best paying job as possible! • however, earning a graduate degree may yield greater long-term benefits. !

• Working backward! • The number of possible alternatives can sometimes be reduced by working backward from the goal state toward the initial state ! • Start with the goal state and think about how to arrive at the initial state! • Example: tower of Hanoi Problem! • Example: mazes! • Problematic in ill-defined problems ! • Ex: find a person with the job you want, and work backwards to figure out which steps you need to take! • Generate-and-test! • Generate possible solutions for a problem and then test them! • Effective when there are few possible alternatives and the goal is well-defined ! • Grads completing the lab requirement in one year! • Works poorly when there are many alternatives or a poorly defined goal! • Matchstick problem (many possible configurations of matchsticks)! • Problems of inducing structure (many possible solutions)! • Means-ends analysis! • One of the most effective problem solving approaches! • Two components ! • Divide the problem into subproblems! • Try to reduce the difference between the initial state and the Goa estate for each of the subproblems! • example: completing the BA in psychology! • Divide goal state into sub-problems! • Take 9 units of lower div! • Take 16 units of required upper division courses! • Take developmental, personality/abormal, and applied psychology courses! • Take two labs! • Work toward each of the subgoals (by completing prerequisites, etc)! • example: getting into graduate school! • Divide goal state into sub-problems ! • Good GPA! • Good GREs! • Strong letters of rec! • Good essays, interviews, etc. ! • Work toward each of the subgoals ! • Application: the tower of Hanoi problem! • 3First goal is to get the large disk on the right peg! • Second goal is to get the medium disk on top of the large disk (on the right peg)! • Last goal is to get the small disk on the other two! •

• Limitations! • Like working forward, means-ends approach is sometimes ineffective when the correct solution to a problem depends on temporarily increasing the difference between the initial and goal state! • Means-end analysis can only be used in well-defined problems when the goal is known! • Can also only be used in problems that are divisible into component problems! Solving by analogy ! • Using a solution to an earlier problem to help solve a new problem! • Terminology ! • Surface features:! • • Structural features:! • • Problem isomorphs:! • • Target problem:! • • Source problem:! • ! Valid! Invalid?! Valid! Valid!...