The Teaching of Geometry Part 5

It will be found that the course just laid down, excepting the sixth book of it only, is not of much greater extent, nor very different in point of matter from that of Euclid, whose "Elements" have at all times been justly esteemed a model not only of easy and progressive instruction in geometry, but of accuracy and perspicuity in reasoning.

De Morgan's effort, essentially that of a syllabus-maker rather than a textbook writer, although it was published under the patronage of a prominent society with which were a.s.sociated the names of men like Henry Hallam, Rowland Hill, Lord John Russell, and George Peac.o.c.k, had no apparent influence on geometry either in England or abroad. Nevertheless the syllabus was in many respects excellent; it rearranged the matter, it cla.s.sified the propositions, it improved some of the terminology, and it reduced the number of essential propositions; it had the a.s.sistance of De Morgan's enthusiasm and of the society with which he was so prominently connected, and it was circulated with considerable generosity throughout the English-speaking world; but in spite of all this it is to-day practically unknown.

A second noteworthy attempt in England was made about a quarter of a century ago by a society that was organized practically for this very purpose, the a.s.sociation for the Improvement of Geometrical Teaching.

This society was composed of many of the most progressive teachers in England, and it included in its membership men of high standing in mathematics in the universities. As a result of their labors a syllabus was prepared, which was elaborated into a textbook, and in 1889 a revised syllabus was issued.

As to the arrangement of matter, the syllabus departs from Euclid chiefly by separating the problems from the theorems, as is the case in our American textbooks, and in improving the phraseology. The course is preceded by some simple exercises in the use of the compa.s.ses and ruler, a valuable plan that is followed by many of the best teachers everywhere. Considerable attention is paid to logical processes before beginning the work, such terms as "contrapositive" and "obverse," and such rules as the "rule of conversion" and the "rule of ident.i.ty" being introduced before any propositions are considered.

The arrangement of the work and the number of propositions in plane geometry are as follows:

Book I. The straight line 51 Book II. Equality of areas 19 Book III. The circle 42 Book IV. Ratio and proportion 32 Book V. Proportion 24 ---- Total for plane geometry 168

Here, then, is the result of several years of labor by a somewhat radical organization, fostered by excellent mathematicians, and carried on in a country where elementary geometry is held in highest esteem, and where Euclid was thought unsuited to the needs of the beginner. The number of propositions remains substantially the same as in Euclid, and the introduction of some unusable logic tends to counterbalance the improvement in sequence of the propositions. The report provoked thought; it shook the Euclid stronghold; it was probably instrumental in bringing about the present upheaval in geometry in England, but as a working syllabus it has not appealed to the world as the great improvement upon Euclid's "Elements" that was hoped by many of its early advocates.

The same a.s.sociation published later, and republished in 1905, a "Report on the Teaching of Geometry," in which it returned to Euclid, modifying the "Elements" by omitting certain propositions, changing the order and proof of others, and introducing a few new theorems. It seems to reduce the propositions to be proved in plane geometry to about one hundred fifteen, and it recommends the omission of the incommensurable case.

This number is, however, somewhat misleading, for Euclid frequently puts in one proposition what we in America, for educational reasons, find it better to treat in two, or even three, propositions. This report, therefore, reaches about the same conclusion as to the geometric facts to be mastered as is reached by our later textbook writers in America.

It is not extreme, and it stands for good mathematics.

In the United States the influence of our early wars with England, and the sympathy of France at that time, turned the attention of our scholars of a century ago from Cambridge to Paris as a mathematical center. The influx of French mathematics brought with it such works as Legendre's geometry (1794) and Bourdon's algebra, and made known the texts of Lacroix, Bertrand, and Bezout. Legendre's geometry was the result of the efforts of a great mathematician at syllabus-making, a natural thing in a country that had early broken away from Euclid.

Legendre changed the Greek sequence, sought to select only propositions that are necessary to a good understanding of the subject, and added a good course in solid geometry. His arrangement, with the number of propositions as given in the Davies translation, is as follows:

Book I. Rectilinear figures 31 Book II. Ratio and proportion 14 Book III. The circle 48 Book IV. Proportions of figures and areas 51 Book V. Polygons and circles 17 ---- Total for plane geometry 161

Legendre made, therefore, practically no reduction in the number of Euclid's propositions, and his improvement on Euclid consisted chiefly in his separation of problems and theorems, and in a less rigorous treatment of proportion which boys and girls could comprehend.

D'Alembert had demanded that the sequence of propositions should be determined by the order in which they had been discovered, but Legendre wisely ignored such an extreme and gave the world a very usable book.

The princ.i.p.al effect of Legendre's geometry in America was to make every textbook writer his own syllabus-maker, and to put solid geometry on a more satisfactory footing. The minute we depart from a standard text like Euclid's, and have no recognized examining body, every one is free to set up his own standard, always within the somewhat uncertain boundary prescribed by public opinion and by the colleges. The efforts of the past few years at syllabus-making have been merely attempts to define this boundary more clearly.

Of these attempts two are especially worthy of consideration as having been very carefully planned and having brought forth such definite results as to appeal to a large number of teachers. Other syllabi have been made and are familiar to many teachers, but in point of clearness of purpose, conciseness of expression, and form of publication they have not been such as to compare with the two in question.

The first of these is the Harvard syllabus, which is placed in the hands of students for reference when trying the entrance examinations of that university, a plan not followed elsewhere. It sets forth the basal propositions that should form the essential part of the student's preparation, and that are necessary and sufficient for proving any "original proposition" (to take the common expression) that may be set on the examination. The propositions are arranged by books as follows:

Book I. Angles, triangles, parallels 25 Book II. The circle, angle measure 18 Book III. Similar polygons 10 Book IV. Area of polygons 8 Book V. Polygons and circle measure 11 Constructions 21 Ratio and proportion 6 ---- Total for plane geometry 99

The total for solid geometry is 79 propositions, or 178 for both plane and solid geometry. This is perhaps the most successful attempt that has been made at reaching a minimum number of propositions. It might well be further reduced, since it includes the proposition about two adjacent angles formed by one line meeting another, and the one about the circle as the limit of the inscribed and circ.u.mscribed regular polygons. The first of these leads a beginner to doubt the value of geometry, and the second is beyond the powers of the majority of students. As compared with the syllabus reported by a Wisconsin committee in 1904, for example, here are 99 propositions against 132. On the other hand, a committee appointed by the Central a.s.sociation of Science and Mathematics Teachers reported in 1909 a syllabus with what seems at first sight to be a list of only 59 propositions in plane geometry. This number is fict.i.tious, however, for the reason that numerous converses are indicated with the propositions, and are not included in the count, and directions are given to include "related theorems" and "problems dealing with the length and area of a circle,"

so that in some cases one proposition is evidently intended to cover several others. This syllabus is therefore lacking in definiteness, so that the Harvard list stands out as perhaps the best of its type.

The second noteworthy recent attempt in America is that made by a committee of the a.s.sociation of Mathematical Teachers in New England.

This committee was organized in 1904. It held sixteen meetings and carried on a great deal of correspondence. As a result, it prepared a syllabus arranged by topics, the propositions of solid geometry being grouped immediately after the corresponding ones of plane geometry. For example, the nine propositions on congruence in a plane are followed by nine on congruence in s.p.a.ce. As a result, the following summarizes the work in plane geometry:

Congruence in a plane 9 Equivalence 3 Parallels and perpendiculars 9 Symmetry 20 Angles 15 Tangents 4 Similar figures 18 Inequalities 8 Lengths and areas 17 Loci 2 Concurrent lines 5 ---- Total for plane geometry 110

Not so conventional in arrangement as the Harvard syllabus, and with a few propositions that are evidently not basal to the same extent as the rest, the list is nevertheless a very satisfactory one, and the parallelism shown between plane and solid geometry is suggestive to both student and teacher.

On the whole, however, the Harvard selection of basal propositions is perhaps as satisfactory as any that has been made, even though it appears to lack a "factor of safety," and it is probable that any further reduction would be unwise.

What, now, has been the effect of all these efforts? What teacher or school would be content to follow any one of these syllabi exactly? What textbook writer would feel it safe to limit his regular propositions to those in any one syllabus? These questions suggest their own answers, and the effect of all this effort seems at first thought to have been so slight as to be entirely out of proportion to the end in view. This depends, however, on what this end is conceived to be. If the purpose has been to cut out a very large number of the propositions that are found in Euclid's plane geometry, the effort has not been successful. We may reduce this number to about one hundred thirty, but in general, whatever a syllabus may give as a minimum, teachers will favor a larger number than is suggested by the Harvard list, for the purpose of exercise in the reading of mathematics if for no other reason. The French geometer, Lacroix, who wrote more than a century ago, proposed to limit the propositions to those needed to prove other important ones, and those needed in practical mathematics. If to this we should add those that are used in treating a considerable range of exercises, we should have a list of about one hundred thirty.

But this is not the real purpose of these syllabi, or at most it seems like a relatively unimportant one. The purpose that has been attained is to stop the indefinite increase in the number of propositions that would follow from the recent developments in the geometry of the triangle and circle, and of similar modern topics, if some such counter-movement as this did not take place. If the result is, as it probably will be, to let the basal propositions of Euclid remain about as they always have been, as the standards for beginners, the syllabi will have accomplished a worthy achievement. If, in addition, they furnish an irreducible minimum of propositions to which a student may have access if he desires it, on an examination, as was intended in the case of the Harvard and the New England a.s.sociation syllabi, the achievement may possibly be still more worthy.

In preparing a syllabus, therefore, no one should hope to bring the teaching world at once to agree to any great reduction in the number of basal propositions, nor to agree to any radical change of terminology, symbolism, or sequence. Rather should it be the purpose to show that we have enough topics in geometry at present, and that the number of propositions is really greater than is absolutely necessary, so that teachers shall not be led to introduce any considerable number of propositions out of the large amount of new material that has recently been acc.u.mulating. Such a syllabus will always accomplish a good purpose, for at least it will provoke thought and arouse interest, but any other kind is bound to be ephemeral.[32]

Besides the evolutionary attempts at rearranging and reducing in number the propositions of Euclid, there have been very many revolutionary efforts to change his treatment of geometry entirely. The great French mathematician, D'Alembert, for example, in the eighteenth century, wished to divide geometry into three branches: (1) that dealing with straight lines and circles, apparently not limited to a plane; (2) that dealing with surfaces; and (3) that dealing with solids. So Meray in France and De Paolis[33] in Italy have attempted to fuse plane and solid geometry, but have not produced a system that has been particularly successful. More recently Bourlet, Grevy, Borel, and others in France have produced several works on the elements of mathematics that may lead to something of value. They place intuition to the front, favor as much applied mathematics as is reasonable, to all of which American teachers would generally agree, but they claim that the basis of elementary geometry in the future must be the "investigation of the group of motions." It is, of course, possible that certain of the notions of the higher mathematical thought of the nineteenth century may be so simplified as to be within the comprehension of the tyro in geometry, and we should be ready to receive all efforts of this kind with open mind. These writers have not however produced the ideal work, and it may seriously be questioned whether a work based upon their ideas will prove to be educationally any more sound and usable than the labors of such excellent writers as Henrici and Treutlein, and H. Muller, and Schlegel a few years ago in Germany, and of Veronese in Italy. All such efforts, however, should be welcomed and tried out, although so far as at present appears there is nothing in sight to replace a well-arranged, vitalized, simplified textbook based upon the labors of Euclid and Legendre.

The most broad-minded of the great mathematicians who have recently given attention to secondary problems is Professor Klein of Gottingen.

He has had the good sense to look at something besides the mere question of good mathematics.[34] Thus he insists upon the psychologic point of view, to the end that the geometry shall be adapted to the mental development of the pupil,--a thing that is apparently ignored by Meray (at least for the average pupil), and, it is to be feared, by the other recent French writers. He then demands a careful selection of the subject matter, which in our American schools would mean the elimination of propositions that are not basal, that is, that are not used for most of the exercises that one naturally meets in elementary geometry and in applied work. He further insists upon a reasonable correlation with practical work to which every teacher will agree so long as the work is really or even potentially practical. And finally he asks that we look with favor upon the union of plane and solid geometry, and of algebra and geometry. He does not make any plea for extreme fusion, but presumably he asks that to which every one of open mind would agree, namely, that whenever the opportunity offers in teaching plane geometry, to open the vision to a generalization in s.p.a.ce, or to the measurement of well-known solids, or to the use of the algebra that the pupil has learned, the opportunity should be seized.

FOOTNOTES:

[32] The author is a member of a committee that has for more than a year been considering a syllabus in geometry. This committee will probably report sometime during the year 1911. At the present writing it seems disposed to recommend about the usual list of basal propositions.

[33] "Elementi di Geometria," Milan, 1884.

[34] See his "Elementarmathematik vom hoheren Standpunkt aus," Part II, Leipzig, 1909.

CHAPTER VII

THE TEXTBOOK IN GEOMETRY

In considering the nature of the textbook in geometry we need to bear in mind the fact that the subject is being taught to-day in America to a cla.s.s of pupils that is not composed like the cla.s.ses found in other countries or in earlier generations. In general, in other countries, geometry is not taught to mixed cla.s.ses of boys and girls. Furthermore, it is generally taught to a more select group of pupils than in a country where the high school and college are so popular with people in all the walks of life. In America it is not alone the boy who is interested in education in general, or in mathematics in particular, who studies geometry, and who joins with others of like tastes in this pursuit, but it is often the boy and the girl who are not compelled to go out and work, and who fill the years of youth with a not over-strenuous school life. It is therefore clear that we cannot hold the interest of such pupils by the study of Euclid alone. Geometry must, for them, be less formal than it was half a century ago. We cannot expect to make our cla.s.ses enthusiastic merely over a logical sequence of proved propositions. It becomes necessary to make the work more concrete, and to give a much larger number of simple exercises in order to create the interest that comes from independent work, from a feeling of conquest, and from a desire to do something original. If we would "cast a glamor over the multiplication table," as an admirer of Macaulay has said that the latter could do, we must have the facilities for so doing.

It therefore becomes necessary in weighing the merits of a textbook to consider: (1) if the number of proved propositions is reduced to a safe minimum; (2) if there is reasonable opportunity to apply the theory, the actual applications coming best, however, from the teacher as an outside interest; (3) if there is an abundance of material in the way of simple exercises, since such material is not so readily given by the teacher as the seemingly local applications of the propositions to outdoor measurements; (4) if the book gives a reasonable amount of introductory work in the use of simple and inexpensive instruments, not at that time emphasizing the formal side of the subject; (5) if there is afforded some opportunity to see the recreative side of the subject, and to know a little of the story of geometry as it has developed from ancient to modern times.

But this does not mean that there is to be a geometric cataclysm. It means that we must have the same safe, conservative evolution in geometry that we have in other subjects. Geometry is not going to degenerate into mere measuring, nor is the ancient sequence going to become a mere hodge-podge without system and with no incentive to strenuous effort. It is now about fifteen hundred years since Proclus laid down what he considered the essential features of a good textbook, and in all of our efforts at reform we cannot improve very much upon his statement. "It is essential," he says, "that such a treatise should be rid of everything superfluous, for the superfluous is an obstacle to the acquisition of knowledge; it should select everything that embraces the subject and brings it to a focus, for this is of the highest service to science; it must have great regard both to clearness and to conciseness, for their opposites trouble our understanding; it must aim to generalize its theorems, for the division of knowledge into small elements renders it difficult of comprehension."

It being prefaced that we must make the book more concrete in its applications, either directly or by suggesting seemingly practical outdoor work; that we must increase the number of simple exercises calling for original work; that we must reasonably reduce the number of proved propositions; and that we must not allow the good of the ancient geometry to depart, let us consider in detail some of the features of a good, practical, common-sense textbook.

The early textbooks in geometry contained only the propositions, with the proofs in full, preceded by lists of definitions and a.s.sumptions (axioms and postulates). There were no exercises, and the proofs were given in essay form. Then came treatises with exercises, these exercises being grouped at the end of the work or at the close of the respective books. The next step was to the unit page, arranged in steps to aid the eye, one proposition to a page whenever this was possible. Some effort was made in this direction in France about two hundred years ago, but with no success. The arrangement has so much to commend it, however, the proof being so much more easily followed by the eye than was the case in the old-style works, that it has of late been revived. In this respect the Wentworth geometry was a pioneer in America, and so successful was the effort that this type of page has been adopted, as far as the various writers were able to adopt it, in all successful geometries that have appeared of late years in this country. As a result, the American textbooks on this subject are more helpful and pleasing to the eye than those found elsewhere.

The latest improvements in textbook-making have removed most of the blemishes of arrangement that remained, scattering the exercises through the book, grading them with greater care, and making them more modern in character. But the best of the latest works do more than this. They reduce the number of proved theorems and increase the number of exercises, and they simplify the proofs whenever possible and eliminate the most difficult of the exercises of twenty-five years ago. It would be possible to carry this change too far by putting in only half as many, or a quarter as many, regular propositions, but it should not be the object to see how the work can be cut down, but to see how it can be improved.

What should be the basis of selection of propositions and exercises?

Evidently the selection must include the great basal propositions that are needed in mensuration and in later mathematics, together with others that are necessary to prove them. Euclid's one hundred seventy-three propositions of plane geometry were really upwards of one hundred eighty, because he several times combined two or more in one. These we may reduce to about one hundred thirty with perfect safety, or less than one a day for a school year, but to reduce still further is undesirable as well as unnecessary. It would not be difficult to dispense with a few more; indeed, we might dispense with thirty more if we should set about it, although we must never forget that a goodly number in addition to those needed for the logical sequence are necessary for the wide range of exercises that are offered. But let it be clear that if we teach 100 instead of 130, our results are liable to be about 100/130 as satisfactory. We may theorize on pedagogy as we please, but geometry will pay us about in proportion to what we give.

And as to the exercises, what is the basis of selection? In general, let it be said that any exercise that pretends to be real should be so, and that words taken from science or measurements do not necessarily make the problem genuine. To take a proposition and apply it in a manner that the world never sanctions is to indulge in deceit. On the other hand, wholly to neglect the common applications of geometry to handwork of various kinds is to miss one of our great opportunities to make the subject vital to the pupil, to arouse new interest, and to give a meaning to it that is otherwise wanting. It should always be remembered that mental discipline, whatever the phrase may mean, can as readily be obtained from a genuine application of a theorem as from a mere geometric puzzle. On the other hand, it is evident that not more than 25 per cent of propositions have any genuine applications outside of geometry, and that if we are to attempt any applications at all, these must be sought mainly in the field of pure geometry. In the exercises, therefore, we seek to-day a sane and a balanced book, giving equal weight to theory and to practice, to the demands of the artisan and to those of the mathematician, to the applications of concrete science and to those of pure geometry, thus making a fusion of pure and applied mathematics, with the latter as prominent as the supply of genuine problems permits. The old is not all bad and the new is not all good, and a textbook is a success in so far as it selects boldly the good that is in the old and rejects with equal boldness the bad that is in the new.

Lest the nature of the exercises of geometry may be misunderstood, it is well that we consider for a moment what const.i.tutes a genuine application of the subject. It is the ephemeral fashion just at present in America to call these genuine applications by the name of "real problems." The name is an unfortunate importation, but that is not a matter of serious moment. The important thing is that we should know what makes a problem "real" to the pupil of geometry, especially as the whole thing is coming rapidly into disrepute through the mistaken zeal of some of its supporters.

A real problem is a problem that the average citizen may sometime be called upon to solve; that, if so called upon, he will solve in the manner indicated; and that is expressed in terms that are familiar to the pupil.

This definition, which seems fairly to state the conditions under which a problem can be called "real" in the schoolroom, involves three points: (1) people must be liable to meet such a problem; (2) in that case they will solve it in the way suggested by the book; (3) it must be clothed in language familiar to the pupil. For example, let the problem be to find the dimensions of a rectangular field, the data being the area of the field and the area of a road four rods wide that is cut from three sides of the field. As a real problem this is ridiculous, since no one would ever meet such a case outside the puzzle department of a schoolroom. Again, if by any stretch of a vigorous imagination any human being should care to find the area of a piece of gla.s.s, bounded by the arcs of circles, in a Gothic window in York Minster, it is fairly certain that he would not go about it in the way suggested in some of the earnest attempts that have been made by several successful teachers to add interest to geometry. And for the third point, a problem is not real to a pupil simply because it relates to moments of inertia or the tensile strength of a steel bar. Indeed, it is unreal precisely because it does talk of these things at a time when they are unfamiliar, and properly so, to the pupil.

It must not be thought that puzzle problems, and unreal problems generally, have no value. All that is insisted upon is that such problems as the above are not "real," and that about 90 per cent of problems that go by this name are equally lacking in the elements that make for reality in this sense of the word. For the other 10 per cent of such problems we should be thankful, and we should endeavor to add to the number. As for the great ma.s.s, however, they are no better than those that have stood the test of generations, and by their pretense they are distinctly worse.

It is proper, however, to consider whether a teacher is not justified in relating his work to those geometric forms that are found in art, let us say in floor patterns, in domes of buildings, in oilcloth designs, and the like, for the purpose of arousing interest, if for no other reason.