Organized Knowledge


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This page, and the one to which it links, are the main ones you need to break the problem solving barrier.

Quite simply, students cannot solve word problems reliably because they are presented with inconsistent models of problem solving that contradict the logical processes they have learned in other courses and in everyday life.

Problem solving continues to be taught using archaic problem solving methods. The significant reforms in education of recent years have not been applied to problem solving. The lack of use of standards set forth by the National Council of Mathematics (NCTM) is rather striking and provides a unique opportunity for advances in this area.

Students are not taught the problem solving process. The thinking part of problem solving is typically suppressed. They are primarily exposed to the result of the process. In the typical textbook the thought processes, the planning, the problem solver used to solve the problem are omitted. Only the results of that planning are displayed. Thus the very thing most needed, training in thinking, is omitted from the examples of problem solutions presented to the student.

This can be verified by examining any textbook and associated teaching material. It is simple to see that the solutions presented do not start at the point where the writer started in solving the problem. The thinking that forms the basis for the solution is not shown.

Students then feel that there is something different about problem solving that requires special aptitudes. This induced misconception leads to failure to develop facility in mathematical methods and failure to use mathematical methods effectively in subject matter courses such as physics, chemistry, business and other subjects.

These web pages directly address solving word problems in a reliable, confident manner. This is effected by using well established, logical processes that are well documented in the literature and are in common use in areas other than mathematical problem solving. The concepts are directly applicable in all subjects that use mathematics for problem solving.The logical structure is also applicable in non-mathematical problems solving.

The display below shows examples of the logical organization and presentation of knowledge normally used in effective communication.

Logical organization

The Chiastic Structure is from Chiastic Examples. Click the link for more examples. This is a page at the site maintained by Bill Ramey.

The organized, hierarchical communication process illustrated by the above examples is widely used and has extensive historical precedents. In view of this, the chaotic problem solution presentations typical of textbooks and teaching materials is enigmatic. Of equal concern is the lack of coherent problem solving instruction at the pre-college level. Though the NCTM provides sound standards describing the objectives for the teaching of mathematics, the problem solving instruction that has been implemented has no discernible relationship to the objectives in mathematics stated in the NCTM standards.

Some problem solution presentations typical of physics textbooks are shown below.

Chaotic solutions
Reliable problem solving is possible by simply replacing the presentations shown immediately above by using the same logical processes shown in the first display on this page. Two examples of doing this are shown below.

Logical solutions
You can cycle the three displays by clicking on them.

The solutions shown in the figure above and in the solutions presented at this web site illustrate the working backwards approach to problem solving.
Working backwards is a common-sense procedure within the reach of everybody and we can hardly doubt it was practiced by mathematicians and nonmathematicians before Plato.
p202 How to Solve It. G. Polya. Princeton University Press. 1945

The two articles by Alan Van Heuvelen in the American Journal of Physics (Volume 59, No. 10, 1991, pp 891-906) provided valuable, research-based insights into the difficulties students have in solving problems in physics and provides well developed means of addressing these difficulties. The concept-oriented approach in the problem solutions shown on these web pages serves to extend the ideas by embedding conceptualization directly in the flow of the problem solving process. A concept-oriented approach also serves to unify problem solving with other logical processes as illustrated by the first and last displays on this page. The second display is the type of problem solution students commonly encounter. They use these as models with disastrous results.

A paper by Fredrick Reif (Millikan Lecture 1994: Understanding and teaching important scientific thought processes , Am. J. Phys. 63 (1), January 1995) is an excellent description of the hierarchical nature of acquiring and organizing knowledge. The material in these web pages extends the knowledge so organized to hierarchical problem solving. This serves to unify problem solving processes with the well recognized logical processes used in acquiring and organizing knowledge as well as the logical processes used in other areas of communication shown in the first display on this page.

Sheila Tobias has investigated the students' difficulty with problem solving in physics. (See: They're Not Dumb, They're Different by Sheila Tobias.) It is clear from the data given, and the title of the book, that students are adept at the logical processes shown in the first display on this page. When they are confronted with the chaotic presentations shown in the second display, they are understandably confused.

Cognitive Architectures provides the definitive web based scholarly material on problem solving. Those interested in developing an in depth knowledge of problem solving and its current state will benefit from studying the material provided there. The cognitive architectures described are Prodigy, Homer, Soar . Material on Cognition and Intelligence is also available.

A philosophical understanding of the present state of problem solving can be gained by studying the material in the web pages of Principia Cybernetica. A very useful Table of Contents of the material is provided. Specifically see Problem-Solving for epistemological considerations with respect to problem solving.

Problem solving provides a natural environment for learning. It is indeed the primary mechanism by which advances in knowledge are made. Problem based learning (PBL) is widely used. The scope of this use is described on the web by Donald R. Woods. Additional examples of problem based problem solving (PBL) are shown at a University of Delaware site.

Though problem based learning, PBL, provides an environment well suited to learning concepts, it is of little value for problem solving per se. It is necessary to know how to solve problems in order to make optimum use of PBL. Concept based problem solving, CBPS, provides the how to needed to solve problems reliably. The solution of a problem is first addressed by identifying, or learning, the concept that provides the instructions for solving the problem. Commonly an hierarchy of concepts will be needed. These concepts are identified, or learned, as the problem solution evolves. Concept based problem solving, CBPS, is a necessary prerequist to problem based learning.

The problem solution process shown on this web site draws from concepts in the referenced material. The dependance on the ideas of Distributed Cognition (see Distributed cognition, G. Solomon (Ed.), New York, Cambridge University Press) will be evident to those familiar with cognitive research.

The problem solutions shown in these web pages also parallel the principles used in modern computer programming illustrated by Object Oriented Programming (OOPS). Computer science has implemented logical methods in computer programming. Using logical methods in problem solving will substantially improve the understanding of subject matter and effective use of mathematics to communicate this understanding.

The experts in any field most certainly know how to solve problems. The success of the scientific enterprise underscores this conclusion. The expert proceeds from basic concept, using logically connected steps, to the answer in the manner shown in the third display on this page. The process is a linear, hierarchical process. The conversion to the chaotic presentation shown in the second display on this page occurs in the processing of the expert's knowledge that occurs as it moves from author to the printed textbook. This is particularly true in pre-college and introductory college level textbooks.

It is rather surprising that this situation exists in light of the fact that how to solve problems is well understood. Substantial literature, as indicated above, exists. Fundamental to problem solving is the work done by G. Polya (How to solve it, Princeton University Press, 1945). Polya identifies four steps in the problem solving process. For more about Gyorgy Polya see Polya

  1. Understanding the problem.
  2. Devising a plan.
  3. Carrying out the plan.
  4. Looking back.
As Polya's examples show, understanding the problem consists of identifying what is asked for quite independent of the various conditions that might be imposed by the givens and constraints stated in the problem.

Following Polya for the second step, devising a plan consists of responding as directly as possible to what is asked for. This generally requires use of a law, definition or principle which is the answer to the question asked. The plan is typically omitted in textbooks and other teaching materials. Only the result of the plan is displayed.

Carrying out the plan consists of responding to requests made by the result of step 2. This may well be an embedded problem which requires the use of step 1, then step 2, then step 3, for the embedded problem.

Looking back consists of various checks, including common sense, of the result of using the first three steps. A powerful tool for looking back consists of answering the question "What does the result TELL me?"

In complex problems the four steps are used recursively. It is apparent that step 1 is sometimes not used carefully. It does not mean understanding all the details of the problem, such as listing the givens and conditions. It only means understanding what is asked for.

Care in using step 2 is also needed. Too frequently one will try to actually obtain the result while devising a plan. This can be done in simple problems and one can visualize the entire solution. As soon as problems become a bit more complex or have subtle variations, this fails. Step 2 means only stating the law, definition or principle which provides the instructions for answering the question asked and expressing it in mathematical form. This then tells one what to do next when using Polya's four steps.

Polya's process is expressed in different words (Selective encoding , Selective combination and Selective comparison ) by Sternberg (Beyond IQ , Robert Sternberg, Cambridge University Press, 1984). The problem solving process is also described by Newell (Unified Theories of Cognition , Allen Newell, Harvard University Press, 1990, p. 97 ff). Additional understanding of the process is available in Patterns, Thinking, and Cognition by Howard Margolis (The University of Chicago Press, Chicago, 1987).

The problems in these web pages repeatedly use the 4-step problem solving process described by Polya and the refinements available from the references above.

The link below provides the rest of the secret to problem solving. Don't miss it!

To complete your understanding of the use of Polya's four steps to effect reliable problem solving click here

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Copyright 1994. Howard C. McAllister