Mathematics and Visual Thinking
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This page contains a 1985 article on Mathematics and Visual Thinking written by Gene Maier. It was initially published in Washington Mathematics, Spring 1985 and then reprinted in Oregon Mathematics Teacher, September 1985.
The Mathematics and Visual Thinking article is Copyright © 2003 by Eugene A. Maier, Ph.D. Used by permission. The article is protected against editing changes by readers. However, readers can comment about the article and contribute their own ideas through use of the discussion choice in the top menu bar.
This article is one of a collection of three closely related articles:
Mathematics and Visual Thinking.
Other writings by Eugene Maier that are available online can be found at:
http://www.mathlearningcenter.org/resources/gene/archive
http://www.mathlearningcenter.org/resources/gene/play-on-numbers/
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Cite this article as:
Maier, Eugene A. (1985). Mathematics and VisualThinking. Gene's Corner and Other Nooks & Crannies: Perspectives on Math Education. Salem, OR: The Math Learning Center. Retrieved from http://iae-pedia.org/Mathematics_and_Visual_Thinking.
Contents |
Visual Thinkers
School mathematics has never been kind to visual thinkers. Carl Jung, whose propensity for the world of dreams and other realms of visual thought led to major contributions in analytical psychology, had this to say about his mathematical training:
- I felt a downright fear of mathematics class. The teacher pretended that algebra was a perfectly natural affair, to be taken for granted, whereas I didn’t even know what numbers really were. They were not flowers, not animals, not fossils; they were nothing that could be imagined, mere quantities that result from counting. To my confusion, these quantities were now represented by letters, which signified sounds…Why should numbers be expressed by sounds?…a, b, c, x, y, z, were not concrete and did not explain to me anything about the essence of numbers…
- As we went on in mathematics, I was able to get along more or less by copying out algebraic formulas whose meaning I did not understand, and by memorizing where a particular combination of letters had stood on the blackboard.… Thanks to my good visual memory, I contrived for a long while to swindle my way through mathematics.
Thus, Jung’s visual gifts, rather than being used to provide insight about mathematics, were used to reproduce configurations he had seen on the blackboard, giving the pretense that he had some understanding of the matter. In reality, however, he had no idea where his algebra teacher got the letter he scribbled on the blackboard, nor why he did it. The young Jung was so intimidated by his incomprehension that he dared not ask any questions and ultimately, he reports, “Mathematics classes became sheer terror and torture to me.” (1)
Other Visual Thinkers
Other visual thinkers have fared better in mathematical matters, but as a result of their own determination and an abandonment of school methods. Freeman Dyson has this to say about his colleague Dick Feynman, who ultimately won a Nobel prize for his work in theoretical physics:
- (Dick) said that he couldn’t understand the official version of quantum mechanics which was taught in textbooks, and so he had to begin afresh from the beginning. This was a heroic enterprise.… At the end, he had a version of quantum mechanics he could understand. He then went on to calculate with his version of quantum mechanics how an electron could behave.… Dick could calculate these things a lot more accurately, and a lot more easily, than anybody else could. The calculation that I did…took me several months of work and several hundred sheets of paper. Dick could get the same answer calculating on a blackboard for half an hour.…
- We talked for many hours about his private version of physics, and I finally began to get the hang of it. The reason Dick’s physics was so hard to grasp was that he did not use equations. Since the time of Newton, the usual way of doing theoretical physics had been to begin by writing down some equations and then to work hard calculating solutions of the equations.… Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the picture gave him the solutions directly, with a minimum of calculation. It was no wonder that people who had spent their lives solving equations were baffled by him. Their minds were analytical; his mind was pictorial. My own training…had been analytical. But as I listened to Dick and stared at the strange diagrams that he drew on the blackboard I gradually absorbed some of his pictorial imagination and began to feel at home in his version of the universe. (2)
Feynman diagrams have become a standard mechanism for thinking about electron behavior. Sketches of them now appear in textbooks, helping physics students draw on their visual, as well as their analytic, faculties to assist them in learning.
Still others managed to translate the school version of things into their own imagery and used that to make sense out of school mathematics. Seymour Papert, Professor of Mathematics at MIT, describes his experiences as a child.
- Before I was two years old I had developed an intense involvement with automobiles. The names of car parts made up a substantial portion of my vocabulary: I was particularly proud of knowing about the parts of the transmission system, the gearbox, and most especially the differential. It was, of course, many years later before I understood how gears work; but once I did, playing with gears became a favorite pastime.… I became adept at turning wheels in my head and at making chains of cause and effect.… I found particular pleasure in such systems as the differential gear.…
- I believe that working with differentials did more for my mathematical development than anything I was taught in elementary schools. Gears, serving as models, carried many otherwise abstract ideas into my head. I clearly remember two examples from school math. I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x + 4y = 10) immediately invoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend. (3)
Papert, who also holds an appointment as Professor of Education, goes on to relate how his love affair with gears led to his formulation of what he considers “the fundamental fact about learning: Anything is easy if you can assimilate it to your collection of models. If you can’t, anything can be painfully difficult.… What an individual can learn, and how he learns it, depends on what models he has available.” (4) He suggests several reasons for the effectiveness of gears in helping him grasp mathematical ideas:
- First, they were part of my natural “landscape,” embedded in the culture around me. This made it possible for me to find them myself and relate to them in my own fashion.
- Second, gears were part of the world of adults around me and through them I could relate to those people. Third, I could use my body to think about the gears. I could feel how gears turn by imaging my body turning. This made it possible for me to draw on my “body knowledge” to think about gear systems. And finally, because, in a very real sense, the relationship between gears contains a great deal of mathematical information, I could use the gears to think about formal systems…the gears serves as an “object-to-think-with.” I made them that for myself in my own development as a mathematician. (5)
The need for models and images on which to hang one’s mathematical thinking has been stressed by others. Robert Sommer, Professor of Psychology and Environmental Studies at the University of California, Davis, maintains that “new math failed because of its bias towards abstraction and its devaluation of imagery.” (6) He claims that it tried to develop understanding at the expense of the senses. Sommer states:
- A mathematical statement leaves the hearer cold when it evokes no images or associations. It is as if the words were uttered in a foreign language. Indeed, mathematics is often taught as if it were a foreign language, with only the most arbitrary connection between symbols and objects.… The problem is not the symbols themselves, but that our teaching of arithmetic detaches numbers from the stuff of life…I have seen otherwise intelligent students turn in bizarre arithmetic solutions which they never would have considered acceptable if they had been using words instead of symbols. It was as if they were stringing together foreign terms according to some set of rules, without any ideas what the words meant…
- Emptying ideas of their sensuality does not produce meaningful learning or discovery, as some of its (the new math’s) proponents maintained, but mechanical and arbitrary learning. What must be criticized is not abstraction itself, which is too much a part of the human mind to be discarded, but abstraction at the expense of the senses. (7)
Bringing School Mathematics Back to its Senses
And that brings us to our goal: to bring school mathematics back to the senses. We are calling the process for doing this “visual thinking.” This may be a misnomer because we have more than the sense of sight in mind. A more appropriate name might be “sensual thinking,” but this has connotations we want to avoid.
For our purposes, “visual thinking” shall mean at least three things: perceiving, imaging and portraying. Perceiving is becoming informed through the senses: through sight, hearing, touch, taste, smell, and also through kinesthesia, the sensation of boy movement and position. Imaging is experiencing a sense perception in our mind or body that, at the moment, is not a physical reality. Portraying is depicting a perception by a sketch, diagram, model, or other representation.
Whereas the previous quotations might suggest that some individuals are visual thinkers and others are not, everyone to a greater or lesser extent, engages in visual thinking. Our dreams attest to that. And visual thinking, when nurtured and developed, can play a significant role in the development of mathematical understanding and in the creative and insightful use of mathematics in other areas.
There are those who claim visual thinking is primary and vital for all but the most routine and stereotyped thought processes. Robert McKim, in Experiences in Visual Thinking, (8) quoting psychologists Jerome Bruner, Abraham Maslow, Ulric Neisser, and others, suggests that visual thinking is the primary thinking process. It provides the content for the secondary process of rational, analytic, symbolic thought. This secondary process is vital also, for without it our thoughts would remain imprecise and incommunicable. However, to rely solely on this secondary mode of thought in teaching mathematics can lead to the situations described above in the quotations from Jung and Sommer. Large doses of visual thinking experiences are recommended for all learners.
In his book McKim stresses the importance of visual thinking in every field. He specifically mentions mathematics, an area in which one might suppose that symbolic thought is the dominant thinking mode. He mentions the study of Jacques Hadamard (9) in which Hadamard concluded that the most creative mathematicians were visual thinkers. Perhaps the most celebrated instance cited by Hadamard is that of Albert Einstein; a letter from him is reproduced in the appendices of Hadamard’s book. Einstein writes, “The words or the language, as they are written or spoken, do not seem to play any role in the mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs which can be “voluntarily” reproduced and combined.… [These] elements are, in my case, of visual and some of muscular type. Conventional words or signs have to be sought for laboriously in a secondary stage.”
We may encounter few future Einsteins in our classrooms, but we can provide all our students with experiences in visual thinking that lead to increased understanding, enjoyment, and meaningful use of mathematics.
Notes
1. The quotes are from Carl Jung, Memories, Dreams, Reflections (Vintage Books, Random House, New York, 1965) pages 27–29.
2. Dyson, Freeman, Reflections (The World of the Scientist Part II). The New Yorker, August 13, 1979, pp. 68–70.
3. Papert, Seymour, Mindstorms (Basic Books, New York, 1980) pp. vi–vii.
4. Ibid. p. vii.
5. Ibid. p. 11.
6. Sommer, Robert, The Mind’s Eye (Dell Publishing Co., 1978) p. 73.
7. Ibid. pp. 76, 77 & 81.
8. McKim, Robert, Experiences in Visual Thinking (Brooks/Cole, 1972).
9. Hadamard, Jacques, The Psychology of Invention in the Mathematical Field (Dover, 1954).
About the Author
Quoting from The Math Learning Center Website:
- Dr. Eugene Maier is past president and cofounder of The Math Learning Center, and professor emeritus of mathematical sciences at Portland State University. Earlier in his career, he was chair of the Department of Mathematics at Pacific Lutheran University and, later, professor of mathematics at the University of Oregon.
- He has a particular interest in visual thinking as it relates to the teaching and learning of mathematics. He's coauthor of the Math and the Mind's Eye series and has developed many of the mathematical models and manipulatives that appear in Math Learning Center curriculum materials.
- He has directed fourteen projects in mathematics education supported by the National Science Foundation and other agencies, has made numerous conference and inservice presentations, and has conducted inservice workshops and courses for mathematics teachers throughout the United States and in Tanzania.
- Gene Maier was born in Tillamook, Oregon and is a lifelong resident of the Pacific Northwest.
