CONTROL

The first step toward establishing Control is to determine:

• Where to start.

The answer is surprisingly simple and easy to implement: Start with what is ASKED FOR.

The second crucial step is to determine:

• What to do next.

The equations tell us what to do next when we learn to read what they are saying. This is not hard to do.

THREE PRINCIPLES

Three principles provide the instructions for reliable, common sense problem solving:

1. Start with what is ASKED FOR.
2. Ask, How would I FIND OUT?
3. Work left-right, top-down.

These fundamental problem solving concepts are independent of specific subject matter, applying equally well in all areas of study and application. They provide a strategy that makes reliable problem solving possible.

Any problem of substance will contain an assortment of given material and conditions, some of which may be irrelevant. As a result, confusion frequently exists about how to start.

I. Start with what is ASKED FOR.

This idea seems obvious, but failing to use the idea carefully and precisely is the most common cause of failing to solve problems reliably.

PUZZLES AS EXAMPLES

Puzzles, which are essentially algebra word problems, are a pure form of problem solving.

Puzzles require little knowledge of any particular subject and usually involve only the most elementary mathematical skills. They provide a splendid opportunity to understand problem solving and to think about problem solving (meta-problem-solving) without the need for specialized knowledge of subject matter.

For illustration we present here two puzzles which may seem to be too frivolous to be taken seriously. But, make no mistake about it, exactly the same process applies to the most serious, complex problems. A few examples in physics appear later. (Additional examples)

FIRST PUZZLE: HOW MANY APPLES?

If Tom has three times as many apples as Susan and Susan has one-fourth as many as Joe, who has four, how many does Mary have if she has two more than Tom?

Using the first principle, seek out what is ASKED FOR, ignoring everything else. Clearly, the puzzle ASKS FOR the number of apples Mary has. Where to start is now known:

Mary's = ?

Having identified what is ASKED FOR, we need to apply the second principle:

2. Ask, How would I FIND OUT?

In general, the answer will come from a law, definition or principle related to the question ASKED, using either

• a basic concept of the specific subject matter involved, or
• a basic premise that has become common everyday knowledge, such as, "The whole is the sum of its parts."

In order to focus clearly on the ideas leading to a solution, it is helpful (in fact, it is essential) to write a short verbal statement explaining the reasoning in advance of writing each equation. This produces solutions driven by ideas and thinking processes rather than by manipulation of equations.

In the case of puzzles, the answer to "How would I FIND OUT?" is usually contained within the puzzle itself. In this one the question is answered with the words, "Mary has two more than Tom."

Now, having identified what is ASKED FOR and having answered the question, "How would I FIND OUT?" we need to determine what to do next. The third principle makes it clear:

3. Work left-right, top-down.

Treat each unknown as you come to it, beginning at the left hand side of the equation and continuing to the right, working from the top to the bottom of any fraction in the expression.

For each unknown, repeat the process used to start the puzzle solution. Apply the first principle and identify what is ASKED FOR, thus producing a subproblem. In response to that subproblem, again ask yourself, "How would I FIND OUT?"

In order to provide structure to the presentation of the solution, all subproblems (along with the associated verbal reasoning) are indented from the left margin. Levels of indentation are similar to those used in such familiar nested structures as essay outlines and computer programming. Indeed this hierarchical approach is basic to critical thinking.

Returning to the apples problem, we have come to the first subproblem (note that verbal reasoning statements are indented to same level as the associated equations):

The first equation clearly ASKS FOR the number of apples Tom has.

Again we use the second principle of reliable problem solving, that is, "How would I FIND OUT?" Since puzzles are essentially content-free, we expect to find the answer in the statement of the puzzle itself. Indeed it TELLS us that, "Tom has three times as many apples as Susan," so

The sequence of "What is ASKED FOR, How would I FIND OUT?" has provided an additional step in the solution. What to do next is clear. The last equation written above ASKS FOR the number of apples Susan has. We FIND OUT from the puzzle.

The last equation written ASKS FOR the number of apples Joe has. We FIND OUT from the puzzle.

This equation makes no requests, which alerts us to the fact that we have solved the puzzle! No more information is needed. It remains only to substitute the indented results in the starting equation. To make the process more visual, the steps above are repeated below without the intervening words.

The first equation and the three indented equations constitute the "knowledge" part of solving the problem, technically referred to as "encoding" the problem. Obtaining the result is a mechanical process of successive substitution of the indented information into the starting equation. Such substitution can be carried out either by hand or by a computer.

Notes

1. For additional examples of puzzles and simple algebra problems, see Making Sense of Story Problems, published by SureMath Publishing, Inc., 1900 Virginia Ave. #C101. Ft. Myers, FL 33901-3332.

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2. The work presented here was done with the symbolic algebra program, SureMath, available from SureMath Publishing, Inc., 1900 Virginia Ave. #C101. Ft. Myers, FL 33901-3332.
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