*Although the four processes of problem solving are listed sequentially, they may occur in many different orderings and with many iterations in the course of solving a problem.*

" In the problem translation phase, the problem solver may mentally represent the first sentence as "Lucky = 0.65," the second sentence as "Lucky = Vons - 0.02," and the third sentence as "4 × Vons = ___." In problem integration, the problem solver may construct a mental number line with Lucky at 0.65 and Vons to the right of Lucky (at 0.67); or the problem solver may mentally integrate the equations as "4 × (Lucky 0.02) = ____." A key insight in problem integration is to recognize the proper relation between the cost of butter at Lucky and the cost of butter at Vons, namely that butter costs more at Vons (even though the keyword in the problem is "less").

In solution planning, the problem solver may break the problem into parts, such as: "First add 0.02 to 0.65, then multiply the result by 4." In solution executing, the problem solver carries out the plan: 0.02 0.65 =0.67, 0.67 × 4 = 2.68.

Problem representation can be broken down further into problem translation, in which the problem solver translates each sentence (or picture) into an internal mental representation, and problem integration, in which the problem solver integrates the information into a coherent mental representation of the problem (i.e., a mental model of the situation described in the problem).

Problem solution can be broken down further into solution planning, in which the problem solver devises a plan for how to solve the problem, and solution execution, in which the problem solver carries out the plan by engaging in solution behaviors.

Cognitive processing aimed at figuring out how to achieve a goal is called problem solving.

In problem solving, the problem solver seeks to devise a method for transforming a problem from its current state into a desired state when a solution is not immediately obvious to the problem solver.For example, the following problem (reported by Robert Sternberg and Janet Davidson) is nonroutine for most people: "Water lilies double in area every twenty-four hours.At the beginning of the summer, there is one water lily on the lake.Rote instructional methods promote retention (the ability to solve problems that are identical or highly similar to those presented in instruction), but not problem solving transfer (the ability to apply what was learned to novel problems).For example, in 1929, Alfred Whitehead used the term inert knowledge to refer to learning that cannot be used to solve novel problems.It takes sixty days for the lake to be completely covered with water lilies. " In this problem, the problem solver must invent a solution method based on working backwards from the last day.Based on this method, the problem solver can ask what the lake would look like on the day before the last day, and conclude that the lake is half covered on the fifty-ninth day. It is also customary to distinguish between well-defined and ill-defined problems.Given water jars of size 21, 127, and 3 units and an unlimited supply of water, how can you obtain exactly 100 units of water?" This is a well-defined problem because the given state is clearly specified (you have empty jars of size 21, 127, and 3), the goal state is clearly specified (you want to get 100 units of water in one of the jars), and the allowable operators are clearly specified (you can fill and pour according to specific procedures).This definition has three parts: (1) the current state–the problem begins in a given state; (2) the goal state–the problem solver wants the problem to be in a different state, and problem solving is required to transform the problem from the current (or given) state into the goal state, and (3) obstacles–the problem solver does not know the correct solution and an effective solution method is not obvious to the problem solver.According to this definition a problem is personal, so that a situation that is a problem for one person might not be a problem for another person.

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