REPHRASING

Often a problem or situation seems overly complex or hard to understand, simply because the words someone else uses to des­cribe it to you are complicated, vague, or confusing.

Here is an example of a simple problem.

EXAMPLE PROBLEM 6-1 (REPHRASING)

How much is two-thirds of one-half?

If you recall your junior-high school math, you remember that this problem calls for a routine procedure dealing with the manipulation of fractions.

You say something like: “Two-thirds of one-half... that means I have to multiply two fractions together. Let me see— what’s that procedure? Multiply the top number of one fraction by the top number of the other and use the product as the top number of the new fraction. Then, multiply the bottom number of one by the bottom number of the other and use that product as the bottom number of the new fraction. So, two times one equals two—the top number of the answer; and three times two equals six—the bottom number of the answer. I come up with two over six, or two sixths. Now I see that two sixths can be reduced to one third. So the answer is: two-thirds of one-half is one-third.”

Fairly simple yes? Now consider this problem: How much is half of two-thirds? The answer fairly jumps out at you: Half of two-thirds is one-third, of course. This is exactly the same problem you just solved, simply worded in reverse order. Just by changing the way we stated the problem, by rephrasing it, we have made it simpler, easier to understand, and easier to solve. Rephrasing is one of the most powerful thinking strate­gies available to you, and it is fairly easy to start using it more frequently.

Resolve right now that you will never again accept another person’s phrasing of a problem situation unless you have thought about various other ways to phrase it, and determine that the other person’s way can help you to think about it clearly.

Let’s try another example of rephrasing, this time with a simple problem that invites creative thinking as well as sequen­tial thinking.

EXAMPLE PROBLEM 6-2 (REPHRASING)

A man playing golf drove his first ball so well that it rolled right onto the green. From a distance, something about the ball looked peculiar. When he arrived at the pin, he noticed that the ball had rolled right into an empty paper bag that had appar­ently blown onto the green. With the ball inside the bag, he couldn’t figure out how to sink the putt for his birdie. Then he suddenly realized how to do it. What was his solution? Think about it before you read further.

How did you make out? In any case, let’s tune in on the thoughts of our friend the logical thinker, to see how he or she would attack it.

“Hmm... I don’t want to waste a shot getting the ball out of the bag so I can knock it into the hole. What shall I do? Let me see-I want to get the ball into the hole. First, though, I want to get it out of the bag. (!)0r maybe I just want to get rid of the bag (rephrasing)... I’ve got it! I’ll Ught a match, set fire to the paper bag, and let it bum to ashes. Then I’ll just blow away the ashes and sink the putt without any hindrance.”

In this imaginary dilemma, we can see that rephrasing the problem helped to find a workable solution. It enabled the thinker to make the creative leap from trying to hit the ball to getting rid of the bag. This is a good illustration of the intimate connections that can and should exist between “logical” think­ing and “creative” thinking.

Make it a habit to talk your thoughts out loud in a problem-solving situation whenever possible. If you’re alone, or with compatible people, just start verbalizing different thoughts or fragments of thoughts. This kind of loud thinking gets your mind working in a sequential mode, and helps you start moving toward a solution, however erratic or uncertain that motion might be.

Let’s try another example of rephrasing, this time with a problem stated in a very intricate, slippery way.

EXAMPLE PROBLEM 6-3 (REPHRASING)

A man is standing before a portrait hanging on the wall. He points to the portrait and makes the following statement about the person represented in the portrait: “Brothers and sisters have I none, but this man’s father is my father’s son.” Your problem is to find out who the person in the portrait is, i.e. how is he related to the speaker? Take a pen and paper and analyze the situation, making a special effort to rephrase the problem or parts of the problem so you can deal with it more clearly. Don’t read any further until you feel you have an answer, or until you’re ready to see how a skilled logical thinker would attack it.

Okay? Now, let’s listen in on the thoughts of a person skilled in logical thinking. Again the editorial symbol (!)is a signal that points out something the thinker is about to do that will help him or her understand the problem more clearly.

“Hmm, let me see... the thing is worded in a kind of circular way—it doesn’t hold still. It seems to turn back on itself, without offering a clear-cut place to start. I’ll see if I can (!)change a few of the key terms to make it more under­standable (rephrasing). ‘Brothers and sisters have I none’ tells me that the speaker is an only child. Ifhe is an only child, then the phrase ‘my father’s son’ can only refer to the speaker him­self (stepping). I can (!)substitute these simpler terms for the confusing statements given in the problem. I can express it in the following way: ‘I am an only child. This man’s father is me.’ So, the person represented in the portrait is the speaker’s son.”

How did your answer compare to the one given here? How successful were you at rephrasing the problem? Please don’t be discouraged if you found this problem difficult, or if you didn’t come up with the correct solution.

In particular, notice that the logical thinker did not simply jump at the answer in a single leap. As we followed the internal reasoning process, we could see a definite sequence of steps. First, the thinker clarified the role of the person speak­ing—he was an only child. This fact, clearly understood, paved the way to the if-then reasoning process that concluded that the speaker was talking about his own son.

This step-wise nature of logical thinking is the single most important point you need to understand. People who are skilled at what is commonly called intuitive thinking, but have trouble with logical thinking, all suffer from the same handicap: lack of patience with small steps. The intuitive thinker likes to move in large, exciting jumps—flashes of insight, inspirations, global con­cepts, overall impressions, notions that feel right. He or she has very little patience with the disciplined, linear, step-by-step procedure of organizing the elements of a situation and moving from one known fact to another. While the free-wheeling intuitive process is ideal for certain kinds of problems, it fails dismally in the face of problems that demand a logical, sys­tematic approach. The intuitive thinker must learn to respect the disciplined, linear approach and to capitalize on its advantages in those situations where it works best.

Now, it’s time for you to try your hand at a few practice problems that invite you to use the technique of rephrasing. As you approach the following problems, keep in mind that a clear statement of what you are trying to do is half the battle. What is the outcome you want? What are you trying to find out? Try to simplify, reduce, restate, paraphrase, or explain the problem from various angles until you find one that will help you really understand it.

Practice Problem 6-1 (Rephrasing)

A mouse was nibbling on a long, straight piece of cheese. One- third of an hour after he started, his friend joined him, nibbling at the other end. It took another one-third of an hour for the two of them to finish eating the cheese, both nibbling at the same speed. If they had started nibbling together, and the second one quit at the point when only the middle one-third remained, how long would it take the first one to eat the rest of the cheese, and how long in all would it take them to finish it?

Practice Problem 6-2 (Rephrasing)

Three children were suspected of Steahng an apple pie from the kitchen. When asked which one did it, they answered as follows.

1. A said, “I didn’t take it.”

2. B said, “A is lying.”

3. C said, ”B is lying.”

If only one of their statements is true, which child took the pie? Hint: Try testing all three assumptions in turn, i.e. that A did it, then that B did it, and then that C did it.

Solutions to Problems

Practice Problem 6-1 (Rephrasing):

The second situation is, for all practical purposes, the same as the first. One mouse can eat one-third of the cheese in one-third of an hour. When they worked together, they ate two-thirds of it in one-third of an hour. In the second situation, it would take the two of them one-third of an hour to eat two-thirds of the cheese, and it would take the first one one-third of an hour to finish the remaining one-third of it. In either case, it takes them two-thirds of an hour to eat the cheese.

Practice Problem 6-2 (Rephrasing):

If only one of the statements is true, then any assumption about which boy took the pie that results in more than one true statement must be invalid. Let’s test the possibilities. If A took the pie, then statement number 1 is false; that makes statement number 2 true; and statement number 3 must be false. This arrangement is logically consistent. A might be guilty; let’s check on the other boys.

If B took the pie, then statement number 1 is true, state­ment number 2 is false, and statement number 3 is true. There can only be one true statement, so B did not take the pie.

If C took the pie, then statement number 1 is true, state­ment number 2 is false, and statement number 3 is true. This is also invalid.

Therefore, boy A took the pie.