We mentioned in class an article I had seen in Scientific American magazine many years ago.  A psychologist wanted to test how mathematicians think, so he devised the following experiment which was conducted by a graduate assistant.  People, one at a time, were brought in a gym to do an experiment.  These people were volunteers and included both professional mathematicians and non-mathematicians.  The gym was, let us say, 25 feet long.  About 5 feet from one wall was a red line going across the width of the gym floor, and on the opposite wall was a large nail.  The assistant conducting the experiment held five rings, such as those used in a carnival “ring-toss” game.  He told the volunteer (one at a time) that he/she was to play a game whereby the goal was to get the rings on the nail as quickly as possible.  The assistant then said “I will show you how it might be done,” and, standing behind the red-line, proceeded to toss the rings, one at a time across the gym floor, trying to get them on the nail.  The first time maybe only one or two landed on the nail.  The assistant then ran across the gym floor, picked up the remaining rings, and ran back to the other end, stood behind the line, and again tried to toss the rings on the nail.  Each time some rings missed, he would run to the other end, pick them up, return to the initial place behind the red line, and continue to toss the rings until all were on the nail.  The assistant further explained that the score of the volunteer was based upon the time.  He said explicitly that the volunteer could use any method he or she wished to use, as long as it was understood the object of the game was to get all five rings on the nail as quickly as possible.  He then asked if there were any questions.

When the volunteers had each performed the task and the results of the experiment were analyzed, the experimenter discovered a very clear division between the mathematicians and non-mathematicians.  All the mathematicians had times less than 10 seconds, and all the non-mathematicians had times over two minutes!.  Why?  Because the mathematicians took an entirely different approach to the problem.  While all the non-mathematicians had mimicked the assistant and tried to toss the rings onto the nail, the mathematicians heeded the instructions very carefully—in which it was said any method could be used as long as he got the rings on the nail as quickly as possible—and they simply ran to the nail and deposited all the rings onto it directly without throwing them.

In short, the non-mathematicians were deceived by the red line and the method used by the assistant.  It had no meaning with regards to the game.  They falsely assumed it did and thereby made it much harder.  I have termed the coin “red-lining” to refer to this false type of mental processing whereby a person makes a problem harder than it really is by attaching incorrect ideas to the problem based upon previous associations.

Often when we have to make a decision, we will use past experiences or associations to color the situation with which we are current dealing.  In fact, we may add to, subtract from, or alter the facts of, the case in front of us based upon these associations.  The type of associative thinking that enables us to change the facts is called “red-lining.”  Red-lining may be useful in many “human” or interpersonal situations.  For example, one may have a feeling in walking down a street at night that a certain character down the block is dangerous or to be avoided, and accordingly, one may cross to the other side of the street.  Objectively, there may be no evidence that there is something wrong, but intuitively, one may feel something is wrong and chose to be careful based upon this intuitive feeling rather than on purely objective evidence.  This decision is certainly a good precaution in this case even if there is a lack of objective evidence to support this decision.  In the world of mathematics, however, the thinking we use is to be based upon purely objective standards only.

The experiment mentioned in class about getting the rings on the nail is a classical example of red-lining.  All the mathematicians, and only the mathematicians, crossed the red-line on the floor (by running) to hang the rings on the nail.  The non-mathematicians, literally saw the red line on the floor, and saw how the experimenter stood behind the red-line and tried to toss the rings onto the nail.  Although the experimenter said nothing about the red-line, and therefore, the red-line played no real role in the problem, by association from past experiences and the experimenter’s method, most people “added” the red-line to the problem thinking (in error) that it was forbidden to cross the red line, and therefore made it more difficult to solve.

Another example is the nine-dot problem.  Students were challenged to connect nine dots placed at the centers of the cells in a tic-tac-toe board by drawing four straight lines, without taking the pencil off the paper and without retracing any lines.  Most people mentally draw a “red line” (square) around the nine dots and cannot solve the problem because they do not draw lines that go past the border of this square.  In truth, though, the problem set no conditions about such limitations, and can easily be done if the lines go beyond this square.  Symbolically, this problem may be said to typify the need to “think outside the box.”  Avoiding redlining often means thinking outside the box.  Using abstract thinking is NOT the same as avoiding redlining, but it can be useful in helping one remove the non-essential elements from the problem (such as a red line) and think outside the box.

Similarly, another challenge given to the class was to make four equilateral triangles out of six match sticks, with each side of each triangle being exactly one match stick (not less or more).  This problem cannot be done in two dimensions, but can be done easily by using three dimensions (by building a pyramid with a triangular base, called a tetrahedron).  The red-lining here is the unnecessary restriction to two dimensions.

Of interest here is that I have often found many students may present alleged solutions by using half-match sticks as the length of a side.  It has often happened that when I point out that they must use a full match stick, they deny that I had so said when I first gave the problem.  This denial has taken place so often, that I actually tape-recorded once the lesson wherein I assigned this problem.  I clearly stated that condition no less than five times.  Yet, at the next class, many students—in fact, almost, the entire class—were very adamant that I had not stated that condition.  I then played back the tape.  Most students were nonchalant about it.  They said sort of neutrally that they had made a mistake.  I found it amazing.  First, they did not hear something said five times.  Secondly, they were very adamant and even upset that I had claimed I had so stated, and some even swore I had not stated this condition.  Thirdly, it did not seem to bother them very much that they had made such a mistake.  I know I would feel upset if I had made such a serious error that I was adamant about something that had been said five times, and I had totally missed it.  Part of being a good student in math—or just a good student—is to pay attention.  Mathematics is more detailed than most other subjects, and missing just one idea can be critical.  It is important to get all the ideas exactly straight.  One should neither omit one idea (by accident or not paying attention), and one should not add an idea, based upon past experiences or other reasons.  Extreme accuracy in communication—both in speaking and in listening–is an important skill that must be mastered before one can be successful in a math course.

Summarizing, most people use mental processing they call thinking, but mathematicians do not consider to be true thinking because true mathematical thinking is totally objective, whereas the mental processing used by most people involves subjective elements, the addition of which can actually make a problem appear to be harder than it really is.