Charles G. Moore

There was no dearth of competent scholars who visited and observed the American Indian tribes in their homelands in the 1800s. Among them are found the names of Stewart Culin, Washington Matthews, Alfred C. Hadden, Henry R. Schoolcraft and John W. Powell. Their observations were astute and their writing was carefully done. This paper, however, investigates the possibility that they have failed to recognize in a widespread activity--the making of string figures--evidence of mental characteristics that can impact positively upon modern American Indian mathematics students.

The scholars of that era were primarily anthropologists, ethnologists or linguists, but they were not mathematicians. They saw and recorded those events, attitudes and objects they considered to be of importance in their fields. Illustrative of the concept is the following statement by Washington Matthews (1889). In the course of describing the moccasin game played by the Navajos, he wrote:

Matthews described in detail the implements of the game. There were eight moccasins; a roundish stone or pebble about an inch and a half in diameter, a chip blackened on one side to toss and 102 counters each about 9 inches long made of stiff slender yucca leaves, two of which were notched on the margins. These details were important to him and his colleagues but a mathematician would have dismissed them as inconsequential and considered the rules of the game to be of paramount importance.

That the making of string figures, also known as Cat's Cradle, was widespread among the Indian tribes of the period was amply documented by statements such as the following by Stewart Culin (1902).

In Cambridge, England, in 1905, Alfred C. Haddon wrote for the introduction to String Figures and How To Make Them by Jayne (1906):

Evidence exists that writers dealing with string figures in the late 1800s suspected that the activity may have possessed some significance that was escaping them. Franz Boas (1888) wrote:

Rivers and Haddon (1902) observed of string figures, "The existing data are too slight to indicate how far these string figures are of value in ethnological study."

The attitude of most of the earliest writers toward string figures was that they simply constituted an "amusement." But a hint of contradiction can be detected. Writers simultaneously referred to them as merely amusing and also admitted that they were very complicated. As an example, we read from Haddon and Rivers (1902):

The prevailing attitude toward string figures as being merely an amusement was not conducive to serious thought or study, and the situation did not improve after a statement appeared in the American Anthropologist in 1959 denying the figures of even the dignity of being classified as a game. Roberts, Arth and Bush (1959) stated in an article entitled "Games and Culture":

It is not surprising that no further mention of string figures were made in the American Anthropologist since 1959.

The literature does yield indications inferring that researchers did consider the making of string figures at least a mental activity. In 1910 Father Berard Haile wrote in An Ethnological Dictionary of the Navaho Language published by Franciscan Friars (1910):

But Haile did not speculate upon what cognitive processes were occurring in the minds of those children whose attention was being "riveted" by the activity. Caroline Furness Jayne (1906) observed:

The hypothesis to be explained and supported in this paper is as follows:

If the hypothesis can be shown to be tenable, the potential impact upon the mathematical education of modern American Indian youth will be positive. The knowledge that their ancestors, even from the preliterate era, engaged in mathematics-like thought will tend to relieve any existing sense of alienation from the subject. Those students will be less inclined to conclude that they are operating more from a position of disadvantage than their Anglo classmates who can claim such stellar mathematicians as Newton, Leibnitz and Pascal as part of their heritage.

The issue of a possible link between string figures and mathematical thought came to the attention of this writer while teaching an elective evening course in recreational mathematics. The students were nonmathematics majors who enrolled for the purpose of earning two credit hours to complete a liberal studies mathematics requirement (Moore, 1983). Some rope tricks were presented in connection with concepts from knot theory and topology and intersecting with the literature dealing with knots, books were found describing string figures. Thinking that string figures might be a viable adjunct to knot theory and topology, additional books dealing with the topic were studied. Those included Fun With String by Joseph Leeming (1940), String Figures and How To Make Them by Caroline Furness Jayne (1906), and Fun With String Figures by W.W. Rouse Ball (1920).

Having never attempted to make a string figure before, this writer studied the exceedingly technical instructions in order to learn a few of the simple forms and to teach them to the evening class. Through both the demonstrations and the dittoed illustrations and instructions the students learned to make the basic figures more quickly, but only after careful reading, observation and intense concentration. As they began to experience the satisfaction of success they voiced some observations, and the following are quotes: "Learning to make string figures is a lot like learning a mathematical process," and "Learning string figures feels like learning mathematics."

The instructor had not thought of the relationship but was immediately interested and asked the students to attempt to identify the ways in which the learning of string figures were similar to the learning of mathematics. They had some difficulty identifying specific properties that described what they claimed they had felt. Examples of the thoughts they listed are:

The instructor determined to investigate the intriguing concept further. The first breakthrough occurred with the realization that he had heard the name W.W. Rouse Ball before. W.W. Rouse Ball was the English mathematician who wrote the important book, Mathematical Recreations and Essays, fast published in London in 1892. Here then was a connection between string figures and mathematics.

Ball (1911) first included a chapter on string figures in the fifth edition of the book, and in the preface he stated somewhat apologetically:

When Ball's statement is read carefully it is seen that he is saying that string figures are connected with mathematics. Of course, he used the qualifiers "only indirectly" and nowhere explained what he saw the connection to be. It may be conjectured that Ball's apologetic tone and lack of explanation were due to a currently perceived snobbism of the mathematical community and he was reluctant to subject himself to possible criticism.

H.S.M. Coxeter, the world's best known geometer, was delighted with Ball's book and after Ball's death he undertook a revision. In the preface to the eleventh edition Coxeter stated:

Evidently Coxeter thought the relation between string figures and mathematics was more tenuous than did Ball. It is not recorded whether Coxeter ever tried to master any of the figures.

Since the purpose of this article is to show that the thinking involved in the invention and construction of string figures is mathematical in nature it is appropriate to identify those properties which are inherent in mathematical activity. To do so, we turn to the book What Is Mathematics by Courant and Robbins (1961). They say:

The involvement in the making of string figures can easily be shown to involve the three elements of the preceding statement. The participant certainly has to actively assert his/her will to learn to make the figures. The concentration upon the completion of a sequence of well-defined operations actively involves the contemplative reason and the motivation for completing the figures is the desire for the fulfillment derived from the viewing and exhibiting of a completed form and admiring the aesthetical appeal of its symmetries or the degree to which it represents a familiar object.

Each pair of antithetical forces will now be investigated.

We shall consider first analysis and construction. In geometry the construction (the actual drawing) of a figure is carried out after the analysis is performed. The more general term, synthesis, shall be used in place of construction. Both analysis and synthesis can be identified upon consideration of the invention of the figures and the assumption is made that each figure was first invented by some individual.

Jayne (1906) concluded Chapter IX of her book, String Figures and How To Make Them, with a section entitled, "Figures Known Only from Their Finished Form." Therein she shows 106 beautiful and intricate figures. About them she says:

An example of one of Jayne's figures "known only from the finished form" is shown in Figure 2.

Clearly, what Jayne is in need of is an analysis of the figures. The process of executing the correct set of movements in the correct order to produce the completed figure is the synthesis. To be able to reproduce the figures at will Jayne would be required to find both the analysis and synthesis. She deemed the analysis to be "practically impossible." That some preliterate South Sea Islander or Eskimo or Indian did perform the synthesis is evident from the obvious fact that the figures exist. Contemplate now on how that might have occurred.

Let us grant the majority of the early writers their assumption that the figures were originally created by an individual idly amusing himself/herself by making random movements on his/her loop of string or sinew. As a result of the random movements, he/she noted on the hands a figure that was noteworthy with respect to its beauty or its resemblance to a familiar object such as a bear, rabbit, fish or tepee. The maker of the figure next faced the problem of deciding how to reproduce it at will. The problem had been solved by the preliterate inventor of each of Jayne's 106 completed figures. It was the inventor's problem to perform an analysis of the figure, i.e., look at the figure and recall the last movement, then the one preceding that until arriving at the opening movement.

To carry out such an analysis, although difficult and requiring great concentration, was not as difficult for the original inventor as it was for Jayne and her contemporaries who had to look at the figure "cold" and attempt to discover how it was made. The original discoverer had the advantage of having performed the needed movements in the immediate past and the first moves to be found in the process of making an analysis could be done from memory. Further, it has been observed that the memories of preliterate peoples were, of necessity, highly developed. The discoverer of the figure could have mastered the last two or three moves and then moved to the previous two or three, working then toward the final figure, each time mastering a part of the analysis and synthesis at each stage until finally the analysis and synthesis of the entire figure had been achieved. Perhaps the analysis was soon forgotten and only the synthesis was taught to others, but the hundreds of intricate figures observed in primitive societies imply that a great many preliterate people had utilized the concepts of analysis and synthesis on the road to original mastery of the figures.

Jayne's book is a masterful example of technical writing. Her synthesis of the figures, the step-by-step verbal instructions, are clear and unambiguous. Yet, to follow them to the desired conclusion requires careful study. Still, the inventors of those eye-catching figures performed the analysis mentally, without the aid of written instructions.

It might be argued that most of the American Indians learned the figures by observation instead of analysis, learning only the sequences of physical movements. Even if this be true, of interest is the following statement from the book Thinking Goes to School: Piaget's Theory in Practice, by Furth (1974):

We now turn to the antipodal properties of generality and individuality. That generality was present in the development of string figures is evident from the many statements indicating certain figures were made throughout the world. Following is an example by Kathleen Haddon (1902):

The inference that individuality has been at work in the history of string figures can be drawn from the wide variety of figures that have been created. Not only were certain figures observed generally throughout the world, but diverse societies have developed others, reflecting their view of their particular environment. Haddon (1902) continues:

The essence of a system of deductive logic resides in two elements, the accepted assumptions (axioms) and the conclusions (theorems and corollaries) based upon and derived from the assumptions. Upon examination it is seen that the sets of various figures derived from the same opening position possess the elements of a deductive system. Over half of the figures in Jayne's book are developed from the same opening (Opening A). Various operations are then performed in a specific sequence starting with Opening A (the assumption) and culminating in a desired figure (a conclusion or theorem).

A new theorem that is easily derived once a given theorem has been established is called a corollary of the given theorem. The same situation prevails in string figures. Many figures are achieved in a very few moves starting with a given figure that has been constructed. As one example, Ball (1920) describes the method of constructing the pig figure, "The Porker." The first instruction for The Porker is "First, make Little Fishes." The Porker is developed in a very few additional moves. The elements of the opening position were the axioms of the system, Little Fishes was a theorem based upon the opening position and The Porker was a corollary based upon the theorem Little Fishes. Little Fishes and The Porker are shown in Figure 3.

Intuition was the force that motivated the inventors of the original figures to search for interesting patterns. Their intuition told them that since others had achieved success in discovering and mastering new figures they might be successful also. Further, they had ample time and opportunity to develop that intuition. Jayne (1906) speculates:

After intuition had motivated the discovery of a figure that was deemed to be a "keeper," the elements of analysis and synthesis (as discussed earlier) were brought into play to ensure the reproduction of the figure.

It has been the purpose of this article to show that the process of the creation and mastery of string figures possess the elements associated with mathematical thought as defined by Courant and Robbins, i.e., 1) logic-intuition, 2) generality-individuality, and 3) analysis-synthesis. There is, nevertheless, no universally agreed upon definition of mathematics. It would possibly be advisable to have accepted the "gut-level" statement of the (non-mathematics major) students in the recreational mathematics class when they said, "Learning these string figures feels like learning a mathematics process," for Courant and Robbins (1961) concluded their discussion on the definition of mathematics by saying:

The members of American Indian communities still participate in the making of string figures. They enjoy sharing them and learning new ones and they react with great interest to the hypothesis that this entertaining activity which has so long been a part of their culture may be related to the cognitive processes associated with mathematical thought.

Ball, W.W. Rouse. (1971). Fun With String Figures, Dover Publications, Inc., New York. This is an unabridged reproduction of the third edition of the work published by W. Heffer and Sons, Ltd., Cambridge, England in 1920 under the title An Introduction to String Figures.

Ball, W.W. Rouse. (1911). Mathematical Recreation and Essays. Macmillan and Co., Limited, St. Martin’s Street, London. Fifth Edition.

Ball, W.W. Rouse & Coxeter, H. S.M. (1938). Mathematical Recreations and Essays. University of Toronto Press. Eleventh Edition.

Boas, Franz. (1888). "The Central Eskimo." Sixth Annual Report of the Bureau of Ethnology, 1884-85, Washington.

Courant, Richard & Robbins, Herbert. (1961). What Is Mathematics. Oxford University Press, New York.

Culin, Stewart. (1975). Games of the North American Indians. Dover Publications, Inc., New York, p. 761. This is an unabridged republication of the accompanying paper, "Games of the North American Indians," of the Twenty-Fourth Annual Report of the Bureau American Ethnology to the Smithsonian Institution, 1902-1903, by W. H. Holmes, Chi originally published by the Government Printing Office in 1907.

The Franciscan Fathers (1910). An Ethnologic Dictionary of the Navaho Language. The Franciscan Fathers, Saint Michaels, Arizona, p. 488.

Furth, Hans G. & Wachs, Harry (1974). Thinking Goes to School; Piaget’s Theory in Practice. Oxford University Press, New York, Part III, Thinking Games, p. 71.

Haddon, Alfred C. (1890). "The Ethnography of the Western Tribes of the Torres Straits." Journal of the Anthropological Institute, XIX, p. 361.

Haddon, A.D. & Rivers, W.H.R. (October, 1902). "A Method of Recording String Figures and Tricks." Man, 109, pp. 146-153.

Haddon, Kathleen. String Figures for Beginners. W. Heffner and Sons, Cambridge.

Jayne, Caroline Furness (1962). String Figures and How To Make Them. Dover Publications, Inc., New York. This is an unabridged reproduction of the work first published by Charles Scribner’s Sons in 1906 under the former title, String Figures.

Leeming, Joseph (1974). Fun With String. Dover Publications, Inc., New York. This Dover edition, first published in 1974, is an unabridged and unaltered reproduction of the work originally published in 1940 by J.B. Lippincott Company.

Matthews, Washington (1889). "Navaho Gambling Songs." The American Anthropologist, vol. 2, p. 2.

Moore, Charles G. (1983). "Just for Fun, Teach a Course in Recreational Mathematics." The AMATYC Review, Vol. 5, no. 1, pp. 44-49.

Roberts, John M., Arth, Malcolm J. & Bush, Robert R. (August 1959). "Games in Culture." American Anthropologist, vol. 61, no. 4, p. 597.