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The rules presented here form a grammar of problem solving with a definitive syntax. There is no doubt about its validity as a reliable problem solving tool. It is also a subject area for additional research in the matters of cognition and logical thinking.


In the study of the intellectual processes involved in solving problems some basic mechanisms have been identified. (Sternberg 1984)

1. Selective encoding. Selective encoding requires sifting out relevant from irrelevant information. A useful tool for selective encoding, as used in problem solving, is answering the question, What is asked for? In a logical problem solving framework this is a recursive process.

The initial step of a problem solution consists of responding to the specific question asked, while ignoring all of the given material. This first step requires stating a concept that is the answer to the question asked quite independent of the details of the problem. A true, basic, simple response is the goal. In this way the process of selective encoding is automated.

Once an initial expression has been written, it will ask for more specific information. Further encoding of the problem statement thus occurs, frequently involving the use of additional concepts. In this way the process of selective encoding is further automated.

2. Selective combination. Selective combination involves combining selectively encoded information in such a way as to form a logically integrated solution. Selective combination is achieved by recognizing that the solution of a problem consists of solving a set of nested problems, that is, a number of subproblems, subproblems of the subproblems, and so forth.

The indented structure used in the examples here makes the logic of the problem solution visible. The process automates selective combination.

3. Selective Comparison. Selective comparison involves relating newly acquired information to information acquired in the past. Selective encoding and selective combination frequently generate the need to retrieve old information.

In a learning situation a problem solution may begin with either new knowledge or old knowledge. This emphasizes the importance of learning and understanding successive topics as they are encountered in the educational process. The purpose of homework and tests in a teaching situation is to reinforce and provide experience with new concepts, establishing those concepts as usable working tools and adding to one's knowledge.

In problem solving it is the question, "How would I FIND OUT?" that may be answered by either old, familiar knowledge or by new ideas just being introduced. This point often causes difficulties for students, who may try to generate solutions using only the new concepts, or only old tools they are comfortable with but which are inadequate for the new subject matter.


The answer to "What is ASKED FOR?" establishes the problem space identified in cognitive research. (Newell 1990) Going on with "How would I FIND OUT?" and continuing to work in left-right, top-down fashion implements the problem search and knowledge search in order to traverse the problem space and reach a solution reliably and efficiently. These ideas are well recognized by research in the learning processes.


The many criticisms of our education system today make it obvious that people who are really well grounded in their chosen fields will be in great demand. They can expect to receive commensurate satisfaction and reward for their work. On the larger scale, their contributions to the state of the nation and the world are of vital importance to all of us.

Problem solving is an essential part of learning to do scientific work and to apply scientific knowledge as compared with learning about scientific work. The problem solving principles presented here make it possible to generate problem solutions reliably.

Being assured of that success, it is then necessary to practice! Skills that can be gained only by practice are necessary for continued development of our advancing knowledge.

The goal is definitely worth the effort!


The Hydrogen Atom of Problem Solving

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A request is made by the problem; that is, it ASKS FOR something.

A response to that request is necessary.

A result is obtained, which may in turn ASK FOR additional information.

This process is applied recursively, using the three principles of reliable problem solving, until the solution is reached.

The three principles:

  1. Start with what is ASKED FOR. (Ignore the given material.)
  2. Ask, How would I FIND OUT? (What is it?)
  3. Work left-right, top-down. (Ignore the given material.)
SureMath Logo It is particularly important to understand that solving a problem is a smooth continuous process. It does not consist of listing givens, sketching pictures and identifying unknowns in some kind of fill-in-the-form manner. Problem solving consists of identifying what is asked for, then using knowledge to respond. This response in turn will request additional information in a smooth, continuous process that flows forward. Equations, words and pictures flow into the solution as needed. This is the same smooth, flowing process we use in all our activities whether it is riding a bicycle, writing an essay, reading a novel or playing a piano. Each successive action is a natural consequence of the previous one and a precursor to the next step. Solving a problem using mathematics is not a different intellectual process.

The forward flowing solution process described here is commonly referred to in the problem solving literature as backwards problem solving. It is sometimes called forward inference.

A problem is truly understood only after a solution is obtained and a conclusion developed.

A problem is solved only when one can answer the question,
"What does the result TELL me?"




Allen Newell, Unified Theories of Cognition, Harvard University Press, 1990, pp. 97 ff. Return to text.

Robert Sternberg, Beyond IQ, Cambridge University Press, 1984. Return to text.

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