When a new problem is encountered knowledge relevant to that problem must be applied along with an appropriate problem soling strategy for attacking it. This process has been said to involve "critical thinking" by some, "reflective thinking" by others, or simply "problem solving." Classically, the cognitive domain consists of a hierarchy of six cognitive skill levels (upper of the two accompanying figures). The lowest level in this hierarchy is Knowledge. The next higher level, the lowest level of understanding, is Comprehension. This denotes the understanding of what is being communicated without necessarily relating this to other material. Third is Application involving the use of general ideas and methods in particular situations. Next there is Analysis in which things are broken down into their constituent parts and the relationships among these parts understood. Level five is Synthesis where elements and parts are put together to form a whole. Finally there is the level of Evaluation in which judgments are made about the value of methods and materials and the extent to which they satisfy certain criteria. All of these same cognitive levels appear in problem solving situations with only a slight change in labeling and sequencing (lower of the two accompanying figures). Here, for the sake of making things more explicit, the lowest problem solving level is divided into two parts, Knowledge and Tools. The label "Tools" simply indicates a background knowledge of various approaches to solving problems. Next comes Comprehension, as before, but following that in a typical problem solving sequence of operations comes Analysis in which all the elements of a problem, both explicit and implicit, are identified and understood. Once the pieces of a problem are understood these, together with new elements, are put together in some new arrangement to form one or more Strategies for solving a problem. This is clearly the operation of Synthesis. Next comes the process of Application to the problem of selected elements of both knowledge and tools together with an identified problem solving strategy. Referring to the hierarchy of problem solving skills, it is seen that application generally requires a lower level of skill than either the process of Analysis or that of devising a Strategy. It is at this point, however, that problem solvers may realize that they have failed to produce a satisfactory solution to a problem. For this reason a section entitled Help may be inserted following the Application section. The skill level required here may range over the full extent of the cognitive hierarchy depending upon the nature of the help required. Finally, the problem together with its solution is subjected to a critical Evaluation. Whether Strategies (synthesis) or Evaluation stands higher in the hierarchy of problem solving skills depends on the particular problem situation. Some would insist that the term "critical thinking" be reserved only for these highest two levels. Devising or inventing a new combination or rearrangement of elements (synthesis) may require creativity of the highest order. On the other hand, evaluation may involve making critical judgments about the extent to which certain criteria are satisfied. In such cases this might involve the highest level of cognitive skill. Problem solving COMMANDS (heuristics) are those things problem solvers tell themselves to do in seeking solutions to problems before they actually do them. They are self-instructions, i.e., "commands" that are given to oneself in order to solve a problem. These are then followed by the execution of these commands. At every level in the cognitive hierarchy there are commands appropriate to that level of cognitive activity. Commands consist of key words and phrases that suggest what might be done at a particular stage in solving a problem. Command examples include IDENT RELEV KNOWLEDGE, CLARIFY PROB, and EVAL PROB SOLUTION. These are the tips, hints, hues, and suggestions problems solvers often give themselves that help them solve problems. A list of such commands, or heuristics, is given in Appendix C. It is the author's list, full of redundancy, and quite extensive. This list serves only to suggest the kinds of items that might be included in one's personal repertoire. It is actually best for problem solvers to develop their own collection of such commands. The development of skills in the use of such heuristics is characterized by the following: the transferability of problem solving skills from one academic discipline to another, greater efficiency in learning achieved by problem analysis that is as applicable to sociology as it is to medicine as it is to anthropology, greater permanence in learning because of the generalized nature of the problem solving commands, the applicability of the command structure to unforeseen future problems. It is the thesis of this problem solving program that, particularly to novice problem solvers, problem solving commands should be made explicit. This is best done by writing these down and including them as a part of problem solutions. This procedure provides problem solvers with a constant reminder as to what instruction they should now be executing; permits the modification of a command at any time and even its rejection and the subsequent substitution of an alternative command; properly separates the plan of what problem solving steps are to be taken from the process of actually taking those steps; and results in a written problem solution that contains the commands together with the work done in performance of those instructions, thereby resulting in a more complete narrative of the problem solver's path to the problem solution. It is strongly recommended that beginning problem solvers keep a journal of problem solving heuristics (commands) that they, and others, have found useful. As this journal grows in size, so grows one's ability to grapple with new problems. The journal entries provide a reminder of past experience upon which future problem solving will rest. The precise words that are used to describe individual problem solving commands are not important. What is important is that commands be made explicit while remaining sufficiently general to have wide application. Chapter I is an introduction to the skill, and the art, of problem solving. It contains twenty-six "warmup" problems designed to get one' collective feet wet in the subject. With no particular background available at this point, individuals are made aware that good problem solving is often a matter of simply using one's head. In Chapter II probabilistic problems are encountered. Notions of probability are found almost everywhere and have applications in the fields of medicine, law, economics, management, and biology, to name a few. Problems presented in Chapter III involve decision making strategies together with the ever present decision trees. Decision making is found in most disciplinary areas, for example in history, social work, writing, sports, geology, medicine, and law. Next, in Chapter IV, dilemma "games" are treated, games in Von Neuman's sense of competition with one or more adversaries. Although Von Neuman's games were in the field of economics, application is now made to such fields as international politics, ethics, sociology, history, law, and biology. Graphs are the subject of Chapter V. Such graphs are not the kind we learned about in our first algebra class, but are simply networks of lines, called edges, drawn between points, called vertices. These graphs are important in that they denote relationships and impose a structure that is helpful in many problem situations. Graph theory is important in anthropology, sociology, systems engineering, political science, and management. Chapter VI deals with problems involving patterns and visual thinking. Patterns appear in linguistics, music, dance, art, architecture, theater, geography, meteorology, and elsewhere. Many of the characteristics of ordinary reasoning are presented in Chapter VII, entitled Plausible Reasoning. There is of course the problem of illogical thinking. Such considerations are particularly important in rhetoric, advertising, marketing, philosophy, and writing. Problem solving disabilities, both real and imagined, and problem solutions, both rational and irrational, are the subject of Chapter VIII. There are a far greater number of ways to flounder in attempting to solve problems than there are useful ways to proceed. In Chapter IX ten problems in ethics in the professions are presented. Important problems such as these are so often extremely difficult ones. Perhaps no "solution" as such is possible in many cases, and one may have to be satisfied with a modest amount of headway. Problems in ethics stand in contrast to those more direct and simpler problems treated earlier. Are the problem solving heuristics brought to bear on these more difficult problems the same as for the more straightforward problems, or are we now on a radically different playing field? Finally, four advanced problem solving topics are presented in Chapter X. Each is an extension of an earlier activity. Robert R. Kadesch

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