QUESTION
(by Reneè M.):Hello, my name is Reneè M. I am writing to you because I am in need of
information on the Gestalt theory for a research paper. The paper is for a math
class, and even though I have read a lot of material, I cannot understand how the
theory relates to mathematics. I saw your web page and found it helpful, but I would
very much appreciate any information you could send me to help me understand the
concept further. Thank you so much for taking the time to read this. I will be anxiously
waiting for a response!
ANSWER
(by Prof. Edith H. Luchins):We think that the best reference is Max WERTHEIMERs Productive Thinking,
Harper, 1945, with the enlarged edition in 1959, edited by his son, Michael WERTHEIMER.
The book has many references to teaching math. e.g., to teaching the formula for
the area of a parallelogram in a manner that is blind to the structure of the figure
(contrastructural) or that takes the structure into account (prostructural); different
methods of finding the sum of the interior or exterior angles of a polygon; and
different methods of finding the sum of arithmetic series. A thesis is that methods
are needed that help the learner to grasp the structure whether it be of a geometric
figure or an arithmetic series, etc.
More examples are given in our volumes that reconstruct Max WERTHEIMERs seminars
at the New School for Social Research: Abraham S. LUCHINS & Edith H. LUCHINS,Wertheimer's
Seminars Revisited: Problem Solving and Thinking, State University of New
York, 1970.
Fifty years ago we co-authored a paper: "A Structural Approach to the
Teaching of the Concept of Area in Intuitive Geometry", J. Educational
Research, Vol. 7, 528-533, 1947. Others of our papers that may interest you
include "Geometric Problem Solving Related to Differences in Sex and Mathematical
interests", J. Genetic Psychology, 134, 255-269, 1979, which describes
some geometric figures that WERTHEIMER used in his demonstrations and compares female
and male students' reactions to the problems and to hints given to help see the
structure of the problems and get insight into their solutions. Another is "Students'
Misconceptions in Geometric Problem Solving", Gestalt Theory, 7,
No. 2/1985, 66-77. Our recent surveys of experiments on the water jar problems
(arithmetical volume-measuring tasks) included a section on educational implications
suggested by WERTHEIMER. Discussed are such concepts as rote drill, often used in
teaching arithmetic, and how it may lead to Einstellung Effects or mental sets which
may foster mechanization, habituation, and stereotyped approaches to problems rather
than productive thinking. Of possible interest may be a paper on computer solutions
of the water jar problems summarized in Gestalt Theory, 18, No. 2/1996,
143-147, and a submitted manuscript on visualization of geometric objects and the
role of computers.
There are also educational implications for teaching mathematics in our articles,
"The Influences of Thinking, Surveyability, and Illustrations on Einstellung
Effects," Gestalt Theory, 9, No. 1/1987, 17-27, and "Computer
Simulation, Algorithms, and Heuristics in Einstellung Effects," Genetic,
Social, and General Psychology Monographs,115(1), 49-80, 1989. Also relevant
are "The Einstein-Wertheimer Correspondence on Geometric Proofs,"
The Mathematical Intelligencer, 12(2), 35-43, 1990, and "Task Complexity
and Order Effects in Computer Presentation of Water-Jar Problems," Journal
of General Psychology, 118(1), 45-72, 1991.
I apologize for so many references to our work. Abraham S. LUCHINS was WERTHEIMERs
research assistant at the New School, and has been active in promulgating the educational
implications of Gestalt Psychology. Let me mention his invited address at the New
York Academy of Sciences, "On Some Aspects of the Creativity Problem in
Thinking", Annals of the New York Academy of Sciences, Vol. 91,
Art. 1, pp 128-141, 1960. Also of possible interest is his paper "Implications
of Gestalt Psychology for AV (Audio Visual) Learning", which appeared in
AV Communication Review, Vol. 9, No. 5, 7-31, 1961. In it he notes:
"Gestalt psychology suggests not so much what pupils should learn
but how they should learn. It is concerned not only with responses, but also
with the processes that lead to responses....the same overt response may result
from a process of understanding involving organization and insight into structure
as from a process of memorization in which stimulus and response are arbitrarily
connected. However, consequences of learning through understanding and organization
seem to differ from the consequences of learning through memorization."
This issue of the AV Communication Review describes approaches to AV learning
in different schools of psychology, with interesting comparisons among them drawn
by the editor, Wesley C. MEIERHENRY.
Last but not least let me mention George KATONAs book, Organizing and
Memorizing, New York: Columbia University Press, 1940, and Catherine
STERNs Children Discover Arithmetic: An Introduction to Structural Arithmetic,
New York: Harper, 1949, based on Gestalt principles; an earlier text is Principles
of Mental Development: A Textbook in Educational Psychology, New York: Crowell,
1932 (reprinted 1962), especially Chapter XXV which deals with Learning of Arithmetic.
Also of interest is "A Note on the Circle" of Erwin LEVY,
Gestalt Theory, 12, pp 116-122, 1990.
I hope this long answer is of some help and not completely off the mark.
Edith H. Luchins
Professor Emerita of Mathematics
(Prof. E.H. Luchins died November 18, 2002 - click
here for the E.H. Luchins page.)
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