Publications of the University of Texas
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FREDERIC DUNCALF J. L. HENDERSON KILLIS CAMPBELL E. J. MATHEWS
F. W. GRAFF H.J. MULLER
C. G. HAINES HAL C. WEAVER
The University publishes bulletins four times a month, so numbered that the first two digits of the number show the year of issue, the last two the position in the yearly series. (For example, No. 2201 is the first bulletin of the year 1922.) These comprise the official publications of the University, publications on humanistic and scientific subjects, bulletins prepared by the Bureau of Extension, by the Bureau of Economic Geology, and other bulletins of general educational interest. With the exception of special numbers, any bulletin will be sent to a citizen of Texas free on request. All communications about University publications should be addressed to University Publications, University of Texas, Austin.
UHIVEHITY'O,TEXAS PJIESS , AUSTIH
University of Texas Bulletin
No. 2506: February 8, 1925
The Texas Mathematics Teachers' Bulletin
Volume X, No. 1
PUBLISHBD OY THE UNIVERSITY FOUR TIMES A MONTH, AND ENTERED A.I
JECOND·CLASS MATTER AT THE POSTOFFICE AT AUSTIN. TEXAS,
UNDER THE ACT OF AUGUST 24, 1912
The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preservation of a free govern· ment.
Sam Houston
Cultivated mind is the guardian genius of democracy. . It is the only dictator that freemen acknowl• edge and the only security that free· men desire.
Mirabeau B. Lamar
University of Texas Bulletin
No. 2506: February 8, 1925
The Texas Mathematics Teachers' Bulletin
Volume X, No. 1
Edited by
ALBERT A. BE:\:\ETT
Associate Profe;<: --i.e. n>1345" /rn 38
and for positive integral values of y 39104 n< --i.e. n< 134812; 2,. 29
The integral values of n between 134513/rn and 1348'0/w i.e., n= 1346, 1347, 1348 give the positive integral solutions of the equation. Assigning these valueR, we have
X=12, 50, 88 ?!=70, 41, 12
Mathematics Teachers' Bulletin
Ex. 3. Solve for positive integral values of J; and y 26;i_:+lly=164
(1) Find the com·ergents to 11) 26 (2 11'26. 22
4) 11 (2The partial quotients are 8
2, 2, 1, 3
3)4(1
The convergents are, 3
0 1 2 3 11 1)3(3 3
1 2 5 7 26
0
26X 3-11X7= 1
Multiply by 164, 26X 492-ll X1148= 164
26.xx Hu=l64 26 (:r-492) + 11 (y+l148) =0 ;;·-492 y-1-1148
---= --
-11 26
Let each fraction equal n.
:r-492 y_l._ 1148
---=n
-11 26 :r=492-lln y=26n-1148
For positive ,values of :r
492
n< -i.e. n < 44;/11
11
For positive value of !J 1148 n> --i.e. n> 44"/ 1e 26
As there is no integer intermediate in value between 44"/ ," and 44 -1 " there is no positive integral solu~ion of the equation.
INTUITION, MOTIVATION, REASONABLENESS
BY C. A. RUPP, UNIVERSITY OF TEXAS
About how much does a pound of tobacco cost? I hope you stop to think of a rational answer before you read my guess of a couple of dollars. At the same time I made that guess, I somehow became aware a number of qualifications, such as considering ordinary material, and estimating that the cost might be as low as a dollar, and possibly as high as four. The primary point of the initial query is to get you to consider the idea that the common run of events develop in us a feeling, an intuition, call it horse sense if you will, that enables us to give fairly reasonable answers to extraordinary question.
Is it possible for the average student to so develop his mathematical intuition that the statement of the problem arouses some scheme of attack whose result seems to him to be fairly reasonable? Naturally this question leads us so close to the frontier of our knowledge in psychoJogy that we are in some clanger of finding ideas so novel and so vague that we cannot understand them; but there is little sport in life for the timid. It would be well first to find out how much mathematical intuition the student has before we discuss its development.
One of my teachers was fond of saying that the only thing in mathematics more instructive than a contradiction was a paradox. As a bone of contention, therefore, I propose the statement that many dislike formal geometry because their intuition is already too well developed! I mean that most of the straight book theorems, and not a few of the originals have such a suspicious air of rectitude about them that it would be harder to persuade the class of their falsity than of their truth. Just try the experiment of asking your pupils to concoct a false theorem that shall be reasonably simple and yet not patently absurd.
(Incidentally the author would be very glad to add to hi~ store of good pseudo-theorems.) Granted, then, that many
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of the class have already developed such a feeling for geometry that proofs of what seem quite obvious theorems (obvious in this queer, dim, intuitional way) are regarded as attempts at gilding lilies, it would appear that a strange duty of the conscientious teacher is to inspire a healthy distrust of intuition.
A good way to do this is to show the class the l\fobius unilateral surface. Take a rectange ABCD about lO"xl", and make a half-twisted cylindrical ring by placing C on A and D on B, pasting or fastening the short ends of tho rectangle tQgether. Then try to cover one side of this ring with red crayon and the other with green, and see what happens. In Ball's iVlathematical Recreations you can find other unexpected properties of this surface. I might suggest cutting the ring down its central line.
If the class is one convinced that their intuition may badly fool them, there is some slight danger that they may try to dispense with it completely, and try to go it blind. To keep them traveling safely down the middle path is no slight task, as we all know. It seems to me that a class ought early to get ithe idea that the intuitions very often give us the clue to the solution and some information about the rationality of the result, and for these reasons are well worthy of cultivation; but also it should be impressed hard and often that the very vagueness of the intuitional process gives it an element of untrustworthiness that must always be borne in mind.
In the domain of algebra a feeling for the situation is less prevalent. The students find it harder to follow the teacher's explanations than they do in geometry because they have less adequate notions of the things involved. Hammering home the notion that algebra is just an expanded arithmetic and that all its formalism is a brief way of stating something which in ordinary words would be cumbersome (the occasional practice of translating the relations of an algebraic formula into more ordinary English helps to bring what is going on down from the clouds into
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the grasp of the student) enables the pupil to transfer his arithmetical intuitions, if any, to algebra.
An honest use of mathematical intuition is brought out in the search for a solution of a problem. By dint of experience gained by imitating one's way through the mazes of type problems already worked out in full by some one else, the learner comes to have a storehouse in his memory of possible ways and means ; in some mysterious manner these become welded into a method of procedure in such fashion that frequently one has more or less certainty that a particular problem will yield to a definite. method of attack. ·There is associated with this particular form of consciousness the power of generalization which enables one to see similarities in things apparently diverse, and now and again we have general methods that we can prove will operate successfully upon all problems of a determined class. More often we are content to realize that our general methods are very powerful, and we do not bother with the exact extent of their use.
It seems to me that one ought not to be content with a mere solution of the problem, but ought to be fully as interested in the way in which the solution was built up, seeing what were the clues that led to the adoption of the successful approach, and finding out what were the significant features of the problem that were characteristic of its class rather than of it itself as an individual. The general mass of ideas cognate to these I have partially expressed I call the motivation of a problem. I consider that a man has much to learn about a topic if he cannot trace its motif. The close scrutiny of the means employed in one problem will rather often come to your aid when you are stuck on the next. Students who have come to have intuitions about the possible ways of doing a problem, intuitions that are clear enough to denote the probable optimal methods, may be said to have acquired a very useful faculty, that of mathematical motivation. As far as that goes, much of this discussion would seem applicable to other fields.
Mathematics Teachers' Bulletin
There is a certain habit which, were it but the property of every pupil, would go far towards brightening the life of the teacher, and that is the habit of subjecting all results of inquiry to a rational examination as to their likelihood of being true, or of being reasonable. After having found what he considers the solution of a problem, the ordinary mortal is frequently apt, if he possesses little native interest in the subject, merely to hope that his work is correct. He
· rarely thinks of trying to get any kind of check on his work, least of all does he consciously endeavor to cultivate an attitude of mind which is inherently critical. It is by no means difficult, in the majority of problems which have a numerical answer, to go through a rough and ready two figure computation mentally, which will check the order of magnitude of the answer. Recently one of my freshman classes was computing at the blackboard the time it takes light to travel to us from the nearest fixed star whose parallactic angle was given. Most of them got the answer correct in miles, but in turning over to light years the would-be results ranged form eighteen weeks to twenty years. Not one of them had the ghost of an idea which limit was the more reasonable. Their background was still too sketchy. They had not formed the habit of improving the quality of their snap judgments. Returning to the original tobacco problem which stands at the head of this paper, we have an example of what I mean by a rational guess. It is by no means exceedingly hard to practice checking up rough guesses, and the constant revision of first impressions finally results in an ability to feel whether or not an answer is a reasonable one. A case in point is that of a young lad I know who added to his natural aptitude for mathematics this practice in trying to make rational guesses. It so happened that he had a chance to enter a contractor's office, and found there that despite his ignorance of the special problems of the trade he could very frequently estimate as closely as the old-timers by subjecting his rough first impressions to this check of rationality. Naturally as the work grew more
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familiar, his background grew better, and his judgments more accurate.
Many students show a distinct betterment in daily work after it has been dinned into them that t!iey can frequently weed out mechanical and even logical errors by wondering whether or not their ans•ver seems probable.
Summary.-It may well be that students can improve their mathematical powers by supplementing their formal work with common sense intuitions, paying particular at-· tention to the dominant guiding principles which enter in the solution of their problems, and constantly checking up on their ideas and results in such fashion that they will not care to submit a result which is more or less obviously ridiculous or improbable. ·
SYMBOLISM IN MATHEMATICS
BY ALBERT A. BENNETT, UNIVERSITY OF TEXAS
Summary:
I. General: (1) Mathematics counted among the natural sciences; (2) symbols not necessary for thought but needed for growth of science; ( 3) science to be distinguished as to use of symbols from ritualism on the one
·hand and from descriptive art on the other; ( 4) science classifies its symbols.
II. Mathematics in particular: (1) Mathematics almost a pure science of symbols; (2) mathematical existence;
(3) mathematics distinguished by type statements; (4) suitable notation makes formulas possible and computation convenient; (5) universal symbolic logic; (5) symbols are economical and aid in generalization; (7) symbols reduce computations to a mechanical art but cannot replace thought and ingenuity in mathematical progress.
The place of symbolism in mathematics can be rightly appreciated only when the question is treated in its larger aspect in relation to science in general. Mathematics is the first of the sciences in the antiquity of its inception, the purity of its logic, and the intricacy of its ramification. Philosophical speculations on the classification of knowledge has led some thinkers not intimately versed in any of the sciences to speak of mathEmatics as of a thing apart, not itself a science although related to all sciences. Whate\·er terms one may use, and it is fruitless to dispute over distinctions that ethers may draw, there seems to be no doubt that there iG· a close kinship in spirit and point of view between investigators in the more advanced fields of natural science usually so-called and those in mathematics. There is the same development of trained intuition, the same critical testing of surmises, the same ingenious experimental application of general principles, the same clarifying of fundamental concepts.
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Thought is possible without verpal or visual symbols. At least it is safe to say that for many dumb animals, a situation meets with a response that had we observed it presented to a human being, we would have called the resulting act the evident effect of thought. ·There is no real contribution made to the study by defining thought as a purely human phenomenon. Something•certainly has occurred and a new name for the mind process merely provides a distinction without a valid difference. Not only is this wordless reasoning possible, but ideas or at least information can be transmitted without language in the proper sen~e. Evidence is furnished by widely differing anim<'.l .::ornmunities even down to the insects, abundantly confirming this statement. Nevertheless such a thought cycle as receiving impressions, forming ideas, reasoning upon data secured, transmitting conclusions to others, is immeasurably facilitated by the use of language. And language is but symbols and their orderly use. Were there but a single extant language, one might imagine perhaps that the word used as a symbol had also some intrinsic relation to the object denoted, but the multiplicity of words in any one language and even more the diversity of tongues reminds us constantly of the purely symbolic character of speech and this applies even when a word shows traces of onomatopoeia.
Mathematics, like every other science, employs language, and so indulges continuously in symbolism. But science may be distinguished in its use of symbolism from two other types of human expression, namely, ritualism and descriptive art. By ritualism, I mean here the use of traditional symbols whose meaning has been largely lost to the lay mind. The essence of a magical formula is that the tokens themselves may be trivial or meaningless but the whole incantation has an unreasoned potency. The mysteries of priestly practice are intended to be felt and not analysed by the layman. They must be such as to suggest mysterious origins transmitted by immutable traditions from antiquity. The symbols must be seen or heard and
Mathematics Teachers' Bulletin 21
recognized but their interpretation must be among the arcana of hierarchy hidden perhaps even from the high priest himself. In descriptive art, on the other hand, as I shall use the term, the individual symbols are noted and analytically appreciated chiefly by the artist. The aim is to convey ideas not explicitly enunciated, to suggest more than is expressed, to hint at depths, at brilliance, at grandeur, or grace or simpilicty that is not present save by implication from the total composition. The boundlessness of the ideas to be conveyed, and the inherent limitations of the medium employed determine the scope for the true artist whether in drama, architecture, painting, sculpture, or music. Ritualism and descriptive art are like science governed by certain general esthetic principles in so far as they correspond to expressive forms of mental activity; they all show symmetry, proportion, and so forth. But science takes a middle course between the sub-significance of ritualism and the super-significance of descriptive art. In science the symbol must accurately correspond to the thing symbolized. It must have no traditional and occult connotation on the one hand nor must it hint at imaginary wealth of meaning on the other. Religion may teach the universal potency of some sacred symbol, and even a prosaic ejaculation like "amen" may take on a special odor of sanctity. Science must discard or modify the sense of any ancient concept that no longer fits into the texture of our present-day theories of the universe. The astronomer no longer speaks of the seven crystal spheres of the heavens, and though he talks of planets, he does not intend to suggest aimless wandering. The physicist no longer refers to the heat substance, phlogiston, and when he speaks of atoms he knows that they are not inherently indecomposable. The zoologist no longer discusses the distribution of dragons, basilisks, fauns .and centaurs, but when he talks of the innominate bone he knows that he has named ~ L But it is not sufficient that the scientist rejects obsolete terms or modifies them to correspond to modern theory.
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He must equally guard his language from unjustified inferences, from deductions that are not intended. One of the simplest conditions needed for securing this is to observe the precept, one name for one thing. Imaginative literature revels in a rich vocabulary. Fine shades of meaning may be expressed and an attitude of mind engendered that transcends direct expression. Precisely the same literal thought may be investured in the sonorous cadence of classical phraseology, or hewed out in rough bare outlines in Anglo-Saxon words, and the reader or hearer feels at once a difference that he may be at a loss to account for. Fashions and modes are apparent in the historical trend of scientific research but the body of science cannot vary with whime. It is international in its claims and nonte~poral in its principles. Though a single bug, or bird or blossom may have in a hundred digerent localities more than as many names, the technical label of the species is except in isolated gaps of ignorant confusion, unique. One cannot be surprised that childhood names for familiar objects should under other circumstances come to be applied by simple analogy to somewhat similar things, particularly by those whose language is influenced by immediate practical contingencies only. It is perhaps inevitable that "buttercup" should mean any yellow flower of a certain general size and shape, that the calla should be called a lily, and flycatchers be known as hawks, or that asparagus should be widely corrupted to "sparrow-grass," while such facts as that the pear is a rose, the onion is a lily, the spider, is an emancipated shrimp, the squirrel little more than a treeclimbing rat, and the earth only a planet such as Mars, should be passed over without realization or interest by most people.
Science classes its symbols. Not only must each name correspond to a fairly well-defined concept-significant distinctions leading to the introduction of new terms,-but in the classification of knowledge that raises learning to a science these terms become necessarily classified. The irttroduction of a system of nomenclature at once elastic and
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comprehensive is one of the first steps in the establishment of any science. But the system must be more than thes1:. It must reflect in its own structure the relations within the universe of data with which the science deals. The scientist whether he be mathematician, biologist, chemist, physicist, or what not, faces continuously the slowly developing problem of a suitable system of notation.
But let us examine more intimately those questions which relate to mathematics and which identify its distinguishing attributes, and drop all further discussion of the commcn features that scientific inquiry universally displays.
Mathematics comes near to being a pure science of symbols. While symbols are essential in the practical discu