Conclusions and Learning Accomplished

Traditional Thought vs. Creative Thought

By Gavin Wu

This paper was written as part of an individual directed study course directed by A. Klinger, Winter Quarter 1997

Individual Directed Study Course Description

About CGI Scripts

  Through the course of life and learning, people naturally develop set ways of thinking about certain things. In the field of mathematics for example, we learn that logic and common sense are the key ingredients to problem solving. That is, every mathematical problem has a formula or set algorithm which can aid in finding a solution. Unfortunately, we will find that not every problem in mathematics possesses a sequence of steps or instructions somewhere in print. Sometimes, a bit of creativity and free thought is the only way to a solution.

   Creativity is rarely a factor in traditional mathematical problem solving. Usually, one simply "applies a formula here," "takes the derivative there," and uses the algorithm to solve the rest. Performing this procedure over a span of several years can easily form the bad habit of "searching for the right steps" instead of actually "thinking it through".

   In the problems that follow, traditional problem solvers will likely give traditional answers. That is, some of these problems have more than one correct answer, but the traditional problem solver will only be able to get one of them. The creative thinker, on the other hand, will be able to get the traditional answers as well as some of the others.

   Question 1: What is half of eleven?

   Question 2: What is 1 / 2 + 1 / 3 ?

   Question 3: Arrange 9 apples in 4 baskets so that there are odd numbers of apples in each basket.

   (These questions are examples brought forth by Dr. Allen Klinger)

   These may seem easy, and to the traditional thinker, they are. Why? Because for them, there is only one answer. In actuality, each question has more than one answer.

Importance of Visuals

When faced with problems such as these, it is often difficult to grasp the concepts without visual help. This brings up another creative aspect in mathematical problem solving. How a person creates a visual directly determines the amount of restriction involved in the solving process. It can determine whether that person thinks traditionally or creatively. For example, suppose you encountered a problem involving the manipulations of many objects. Among the objects given to you were a teacup and saucer. Unfortunately however, you represented the teacup and saucer as a subset of your objects in this manner:

It would restrict you from seeing the other possibilities and combinations. So if the solution actually involved placing the saucer with other objects or in different orientations relative to the teacup, you would have a difficult time picturing them while staring at your "concrete" representation. If however, we represented the objects in this manner:

then we would clearly be able to think of more creative ways to manipulate them. Thus, the importance of our visual-spatial abilities is unquestionable. Although greatly influenced by our mental associations and capacities, it can be a powerful tool in the process of understanding. Simply drawing figures to a problem can help us to recognize our fixed mental associations, and steer us away from the traditional mode of thinking.

The effectiveness of a visual can determine the level of a student's understanding. Weak and sparse figures within text can cause confusion amongst readers. If a figure exists, it must be powerful and thorough. Anything less would be ineffective and even detrimental. If a reader had erroneous ideas about the contents of a particular text, then the figure would have to be able to clear any misconceptions. In other words, the visual would have to be self contained to a point that the reader would be able to pair the text to the visual and gain a full understanding of the concepts. The text can then serve to describe the visual in further detail, and form arguments from its contents. Dr. Klinger often stresses that the key to writing a paper lies within the descriptions and elaborations of figures and visuals. >