College Teaching Part 14

If we seek the most logical limit that may be a.s.signed the realm of earth-science, as distinguished from that of celestial science, of which it is the home unit, it may be found at that borderline _within which_ any pa.s.sive body obeys the call of the earth, as against the call of the outer worlds, and _without which_ such a pa.s.sive body obeys the call of the outer worlds, the call of the sun in particular.

This limit is the _dynamic dividing line_ between the kingdom of the earth and the kingdom of the outer heavens. This boundary, according to Moulton, incloses a spheroid whose minimum radius is about 620,000 miles, and whose maximum radius is about 930,000 miles. We may, then, conveniently say that the earth's sphere of control stretches out a million kilometers from its center and that this defines its true realm. At the same time, this defines the logical limit of the earth's ultra-atmosphere and appears to mark a zone of exchange between the ultra-atmosphere of the earth and the ultra-atmosphere of the sun. It thus appears to imply the place and the mode of an exchange of vital elements upon which probably hangs the wonderful maintenance of the earth's atmosphere for many millions of years and the equally wonderful regulation of the essential qualities of the atmosphere so that these have always remained within the narrow range subservient to terrestrial life. It is needless to add that this regulation also conditions the present intellectual status of the thinking factor among the inhabitants of the earth out of which--may I be pardoned for saying?--has grown the present educational discussion.

If this last shall seem to squint toward special pleading, let it be considered that, as we see things, it is precisely those views that take hold of the issues upon which our very being and all its activities depend, that serve best to train youth to broad views and penetrating thought. Such thinking seems to me to form the very essence of a really liberal education.

Not only is this definition of the sphere of geology comprehensive, but it has the special merit of being _dynamic_, rather than material.

Such a dynamic definition comports with the view that earth-study should center on the forces and energies that actuated its evolution, since these are the most vital feature of the evolution itself. It is important to form adequate concepts of the energies that have maintained the past ongoings of the earth not only, but that still maintain its present activities and predetermine its future. It is the study of the forces and the processes of past and of present evolutions that const.i.tute the soul of the science, rather than the apparently fixed and pa.s.sive aspects of the earth's formations and configurations which are but the products of the processes that have gone before. Even the apparent pa.s.siveness of the geologic products is illusive, for they are in reality expressions of continued internal activities of an intense, though occult, order. These escape notice largely because they are balanced against one another in a system of equilibrium which pervades them and gives them the appearance of fixity. To serve their proper functions as sources of higher education, the concepts of the const.i.tution of the earth should penetrate even to these refined aspects of physical organization and should bring the whole into harmony with the most advanced views of the real nature of physical organisms. This removes from the whole terrestrial organism every similitude of inertness and gives it a fundamental refinement, activity, and potency of the highest order. To form a true and consistent concept, the enveloping earth-science must be a.s.sumed to embrace, potentially at least, the essentials of all that was evolved within it and from it, with, of course, due recognition of what was added from without.

_The history of the earth should therefore be taught in college courses as a succession of complex dynamic events, great in the past and great in future potentialities._

The formations and configurations left by the successive phases of action are to be studied primarily as the vestiges of the processes that gave them birth, and hence as their historic credentials. They are to be looked upon less as the vital things in themselves, than as the _record_ of the events of the time and as the forerunners of the subsequent events that may be potential in them. And so, primarily, the geologic records are to be scrutinized to find _the deeper meanings which they embody_, whether such meanings lie in the physical, the biological, or the psychological world.

=Geology the means of developing scientific imagination of time and s.p.a.ce=

Turning to specific phases of the subject, it may first be noted that geology is singularly suited to develop clear visions of vast stretches of time; it opens broad visions of the panorama of world events, a panorama still pa.s.sing before us. While the celestial order of things no doubt involves greater lapses of time, these are not so easily realized, for they are not so well filled in with a succession of records of the pa.s.sing stages that make up the whole. But even the lapses of geologic time are greater than immature minds can readily grasp; however, their _powers of realization_ are greatly strengthened by studying so protracted a record, built up stage upon stage. The very slowness with which the geologic record was made, as well as the evidences of slowness in each part of the record, help to draw out an appreciation of the immensity of the whole. The round period covered by the more legible range of the geologic record rises to the order of a hundred million years, perhaps to several hundred million years. The large view of history which this implies has already come to form the ample background on which are projected the concepts of the broader cla.s.s of thinkers; such largeness of view will quite surely be held to be an indispensable prerequisite to the still broader thinking of the future for which the better order of students are now preparing.

While this is preeminently true of the concept of time, the concept of s.p.a.ce is fairly well cultivated by geologic study, though far less effectively than is done by astronomical study. Astronomy and geology work happily together in contributing to largeness of thought.

The study of the origin and early history of the earth brings the student into touch with the most far-reaching problems that have thus far called forth the intellectual efforts of man. If rightly handled, these great themes may be made to teach the true method of inquiry into past natural events whose vastness puts them quite beyond the resources of the laboratory. This method finds its key in a search for the history of such vast and remote events by a scrutiny of the vestiges these events have left as their own automatic record. This method stands in sharp contradistinction to simple speculation without such search for talismanic vestiges, a discredited method which is too often supposed to be the only way of dealing with such themes. To be really competent in the field of larger and deeper thinking, every courageous mind should be able to cross the threshold of any of the profound problems of the universe with safe and circ.u.mspect steps, however certain it may be that only a slight measure of penetration of the problem may be attainable. A well-ordered mind will remain at once complacent and wholesome when brought to the limit of its effort by the limit of evidence. The problem of the origin of celestial worlds, of which the genesis of the earth is the theme of largest human interest, is admirably suited to give college students at once a modest sense of their limitations and a wholesome att.i.tude toward problems of the vaster type. Without having acquired the power to make prudent and duly controlled excursions into the vaster fields of thought, the mind can scarcely be said to have been liberalized.

=Geology a means of training in thinking in scientific experiences=

From the very outset, the tracing of the earth history forces a comprehensive study of the co-workings of the three dominant states of matter ma.s.sively embodied in the atmosphere, the hydrosphere, and the lithosphere, the great terrestrial triumvirate. The strata of the earth are the joint products of these three elements and const.i.tute their lithographic record. These three cooperating and contending elements not only bring into view the three typical phases of physical action, but they present this action in such t.i.tanic aspects as to force the young mind to think along large lines, with the great advantage that these actions are controlled by determinate laws, while the causes and the results are both tangible and impressive.

While there is a large cla.s.s of tangible and determinate problems of this kind, embracing s.h.i.+ftings of matter on the earth's surface, distortions of strata, and changes of bodily form, there are also problems of a more hidden nature such as internal mutations. These give rise to mathematical, physical, and chemical inquiries while at the same time they call into play the use of the scientific imagination and are thus rich in the possibilities of training. Thus in varied ways geological work joins hands with chemical, physical, mechanical, and mathematical work.

When life first appears in the record, there is occasion to raise the profound question of its origin, and with this arises a closely related question as to the nature of the conditions that invited life, which leads on to the further question, what fostered the development of life throughout its long history? While the obscurity of the earliest record leaves the question of origin indeterminate for the present, duly guarded thought upon the subject should foster a wholesome spirit toward inquiry in this vital line as well as a hospitable att.i.tude toward whatever solution may finally await us. In all such studies the student should be invited to look to _the vestiges left automatically by the process itself_ for the answer, and he should learn to accept the teachings of evidence precisely as it presents itself. So also when a problem is, for the present, indeterminate, it is peculiarly wholesome for the inquirer to learn to rest the case where the light of evidence fails, and to be complacent in such suspension of judgment and to wait further light patiently in serene confidence that the vestiges left by the actuating agencies in their constructive processes are the surest index of the ultimate truth and are likely to be sooner or later detected and read truly.

=Relation of geology to botany, zoology, psychology, and sociology=

In the successive records of past life impressed on strata piled one upon another until they form the great paleontologic register, there is an ample and a solid basis for the study of the historic evolution of life. With this also go evidences of the conditions that attended this life progress and that gave trend to it. This record of the relations of life to the environing physical conditions forms one of the most stimulating fields of study that can engage the student who seeks light on the great problems of biological progress. Here geology joins hands with botany and zoology in a mutual helpfulness that is scarcely less than indispensable to each.

Following, or perhaps immediately attending, the introduction of physiological life, there appeared signs of sentient life. The preservation of certain of the sense organs, taken together with the collateral evidences of sense action, as early as Cambrian times, furnish the groundwork for a historical study of the progress of sentient life, eventuating in the higher forms of mental life. Here the problems of geology run hand in hand with the problems of psychology. The limitations of the evidence bearing on psychological phenomena, while regrettable, are not without some compensation in that they center the attention on the simpler aspects of the protracted deployment of the psychological functions.

In addition to the clear evidences of psychic action, in at least its elementary forms, there appeared early in the stratigraphic records intimations of some of the relations.h.i.+ps that sentient beings then bore to one another; and this relations.h.i.+p gives occasion to study the primitive aspects of sociological phenomena. If nothing more is learned than the important lesson that sociology is not a thing of today, not an untried realm inviting all kinds of ill-digested projects, but on the contrary is a field of vast and instructive history, the gain will not be inconsiderable. There are intimations of the early existence and effective activity of those affections that precede and that cl.u.s.ter about the parental relations.h.i.+p, the nucleus of the most vital of all the sociological relations.h.i.+ps. In contrast to the affections, there are distinct evidences of antagonistic relations, of pursuit and capture, of attack and defense; there were tools of warfare and devices for protection. In time, a wide-ranging series of experiments, so to speak, were tried to secure advantage, to avoid suffering, to escape death, and to preserve the species. There were even suggestions of the cruder forms of government. The many stages in the evolution of the various devices, as well as the stages of their abandonment, that followed one another in the course of the ages recorded the results of a mult.i.tude of efforts at sociological adjustment. They raise the question whether a common set of guiding principles does not underlie all such relations.h.i.+ps, earlier and later, whatever their rank in our scale of valuation. And so this great field of inquiry--too narrowly regarded as merely humanistic--comes into view early in the history of the earth. The geological and the sociological sciences find in it common working ground. If the geologic and the humanistic sciences are given each their widest interpretation and their freest application, the advantage cannot be other than mutual.

It is perhaps not too much to say that studies in the physiological, the psychological, the sociological, and the allied fields necessarily lack completeness if they do not bring into their purview the data of their common historical record traced as far back as it is found to contain intimations of their actual extension.

It is customary to speak of the geologic ages as though they were wholly past; they are, indeed, chiefly past as the record now stands, but time runs on and earth history continues; the processes of the past are still active, and they are likely to work on far into the future. And so geologic study links itself fundamentally into all such present terrestrial interests as take hold of the distant future. The forecast of the earth's endurance, attended by conditions congenial to life and to the mental and moral activities, hinges on a sound insight into the great actuating forces inherent in the earth, together with those likely to come into play from the celestial environment. All human interests, in so far as they are dependent on a protracted future, center in the prognosis of the earth based on its present and its past. The latest phases of geologic doctrine prophesy a long future habitability of the earth. They thus give meaning and emphasis to the deeper purposes sought in all the higher endeavors, not the least of which is education, particularly those phases of education that lead to effects which may be handed down from age to age.

=Standard for selecting subject matter for the general college course: select fundamentals or that which bears on fundamentals=

Out of all this vast physical, biological, and psychological history, the things to be selected for substance of thought and for service in mental training in a college course are, first of all, those that are either fundamental in themselves, or that have vital bearings on what is fundamental. These are chiefly the great dynamic factors, the agencies that gave trend to the master events, the forces that actuated the basal processes by which the vast results were attained.

The material formations and the surficial configurations that resulted are to be duly considered, to be sure, for they form the basis of interpretation and they are, besides, the repositories of economic values of indispensable worth; but, as already urged, in a course of intellectual training, these are to be regarded rather as the relics of the great agencies and the proofs of their actions, than as the most vital subjects of study, which are the agencies themselves. As already remarked, the geologic formations are to be treated rather as the credentials of the potencies that reside in the earth organism, than as the vital things themselves. The vestiges of creation and the footprints of historical progress embody the soul of the subject; they const.i.tute the chief source of inspiration to those who aspire to think in large, deep ways of really great things. It is of little value, from the viewpoint of liberal culture, to know that there is a certain succession of sandstones, shales, and limestones; that professional convention has given them certain names, more or less infelicitous in derivation and in phonic quality; but it is of vital consequence to learn how and why these relics of former processes came to be left as they were left, and thus came to be witnesses to the history of the far past. It was a wise thing, no doubt, that the fathers of geology strongly insisted that there should be a rigorous and rather literal adhesion to the terrestrial record in all earth studies, because in those times of transition from the loose, more or less fantastic thought that marked the adolescent stage of the human race, it was imperative that students should stick close to the immediate evidence of what had transpired, and should withhold themselves from much enlargement of view based on the less tangible evidences; but at the present stage, when the general nature of the earth's history has been firmly established, it would be an error on the part of those who seek for the most liberalizing and broadening values of the science, to treat the record merely as a material register of immediate import only, to the neglect of the less tangible but more vital teachings immanent in its great forces and processes.

The seeker of liberal culture should direct his attention to the great events, and, above all, to the larger and deeper meanings implied by these events.

And so--may I be pardoned for reemphasizing?--the teacher of geology whose essential purpose is liberal training, leading to broad and firm knowledge and to sound processes of thought, will critically observe the distinction between geology taught appropriately from the collegiate point of view, and geology taught specifically from the professional and technical points of view. In these latter, specific details in specific lines are important, and may even be essential, but it is the function of the college teacher of geology _to select_ from the great ma.s.s of material of the science such factors as are basal, vital, and talismanic. He will give these emphasis, while he neglects the mult.i.tude of details that lack significance as working elements or as landmarks of progress, whatever their value in other relations. This selection is equally important, whether applied to the great physical processes that have shaped the earth into its present configuration, or to the great chemical and mineralogical processes that have determined its texture and its structure, or to the great biological and psychological processes that have given trend to the development of its inhabitants.

Even if the undergraduate course in geology is pursued less for the purpose of liberal culture than as a means of preparing for a professional career as an economic geologist, no essential departure from an effort to master first the basal features and the broader aspects of the science, especially the dynamic aspects, is to be advised. The shortest road to _declared success_ in professional and economic geology lies through the early mastery of its fundamentals.

No doubt immediate and apparent success may often be sooner reached by a narrower and shallower study of such special phases of the subject as happen just now to be most obviously related to the existing state of the industries; but industrial demands are constantly changing--indeed, at present, rather rapidly--and new aspects follow one another in close succession. These new aspects almost inevitably spring from the more basal factors as these rise into function with the progress of experience or the stress of new demands. Those who have sought only the immediate and the superficial, at the expense of the basal, and especially those who have neglected to acquire _the power and the disposition to search out the fundamentals_, are quite sure to be left among the unfortunates who trail behind; they are little likely to be found among those who lead at the times when leaders.h.i.+p counts. In the judgment of those master minds that lead in affairs and that take large and penetrating views, the lines along which the most vital contributions to economic interests are being made connect closely with basal studies of the actuating agencies that condition great enterprises. In the judgment of the writer, it is a false view to suppose that any short, superficial study of so vast a subject as the const.i.tution and history of the earth can result in economic competency. In so far as time for study is limited, it should be concentrated on the great underlying factors that const.i.tute the essentials of the science. It is here a.s.sumed that men who care to take a college course at all are seeking for a large success and are ambitious for a high personal career. If they look ultimately to professional work in economic lines, they may safely be advised that the straight road to declared success lies in a search for the vital forces, the critical agencies, and the profound principles that make for great results, not along the by-paths whose winding, superficial courses are turned hither and thither by advent.i.tious conditions whose very nature invites distrust rather than confidence.

=Evaluations of methods of teaching=

Turning to some of the more formal phases of treatment, three types of work are presented: (1) the use of nature's laboratory, the world itself, (2) the use of the college collections and laboratories, and (3) the use of the literature of the subject.

(1) Fortunately, there is no place on the face of the earth where there is not some natural material for geologic study, for even in the most artificialized locations geological processes are active. In crowded cities these processes may be easily overlooked, but yet they are susceptible of effective use. Within easy access from almost every college site there are serviceable fields of study, and these, in any live course, will be a.s.siduously cultivated. They may be relatively modest in their phenomena; they may seem to lack that impressiveness which has played so large a part in the popular notion of the content of geology, but they may nevertheless serve as most excellent training grounds for young geologists. If students are so situated as to be brought at the beginning of study under the influence of very impressive displays of geologic phenomena--precipitous mountains, rugged cliffs, deep canons, and the like--there is danger that their mental habits may become diffusive rather than close and keen; the emotions may be called forth in wonder rather than turned into zest in the search for evidence. If students are to be trained to diligence in inquiry and to the highest virility in inference and interpretation, it is perhaps fortunate for them if they are located where only modest records of geological processes are presented for study. In such regions they are more likely to be led to scrutinize the field keenly, sharply, and diligently for data on which to build their interpretations. The scientific use of their imaginations is all the better trained if, in their endeavor to build up a consistent concept of the whole structure that underlies their field, they are forced to project their inferences from a few out-crops far beneath the cover of the adjacent mantle that shuts off direct vision. Few teachers have, therefore, any real occasion to long for richer fields than those accessible to them, if they have the tact to render these fertile in stimulus and suggestion.

(2) Laboratory work upon the material collected in the field work, as well as laboratory work upon the college collections, are essential adjuncts. Ample provisions for this supplementary work, however modest the appointments, are important and can usually be secured by ingenuity and diligence in spite of financial limitations.

Both field and laboratory work should be well correlated with one another and with the systematic work on the text that guides the study, so that each shall whet the edge of the other and all together accomplish what neither could alone.

(3) The text selected should be such as lends itself, in some notable degree at least, to the general purposes set forth above. It should be supplemented, so far as may be, by judicious a.s.signments for reading and for special study. Lectures may be made a valuable aid to the discussions of the cla.s.sroom, but with college cla.s.ses they can rarely be made an advantageous subst.i.tute for the discussions. Lecturing, so far as used, is best woven informally into the cla.s.sroom discussions.

Supplementary lecturettes may be advised if they are of such an informal sort that they may almost unconsciously take their start from any vital point encountered in the course of discussion, may run on as far as the occasion invites, and may then give way again to the discussion with the utmost informality. Such little partic.i.p.ations in the work of the cla.s.sroom, on the part of the teacher, are likely to be cordially welcomed. At the same time, if well done, they will set an excellent example in the presentative art as also in an apt organization of thought.

=Organization of courses=

If the stated course in earth-science is limited to the junior and senior year by the existing requirements of the curriculum of the inst.i.tution or by the rulings of its officers--as is not uncommonly the case at present--it is relatively immaterial whether the sections of the course are marshaled under the single name "geology" or whether they are given separate t.i.tles as sub-sciences, provided the special subjects are arranged in logical sequence and in consecutive order.

If, on the other hand, the teacher's choice of time and relations is freer, the more accessible phases of earth study, now well organized under the name of "physiography," form an excellent course for either freshmen or soph.o.m.ores. It opens their minds to a world of interesting activities about them which have probably been largely overlooked in previous years. It gives them substance of thought that will be of much service in the pursuit of other sciences. It has been found that it is not without rather notable service to young students as the basis of efforts in the art of literary presentation, a felicity to which teachers of this important art frequently give emphatic testimony. The secret seems to lie in the fact that physiography gives varied and vivid material susceptible of literary presentation, while the fixed qualities of the subject matter control the choice of terms and the mode of expression.

If geography and physiography are given in the earlier years, the course in historical geology, as well as the study of the more difficult phases of geological processes, of the principles of dynamic geology, together with mineralogy, petrology, and paleontology, may best fall into the later years, even if some interval separates them from the geography and physiography.

One hundred and twenty cla.s.sroom hours, or their equivalent in laboratory and field work, are perhaps to be regarded as the irreducible minimum in a well-balanced undergraduate course, while twice that time or more is required to give a notably strong college course in earth-science.

A consideration of the sequences among the geological sub-subjects, as also among the subjects that are held to be preliminary to the earth-sciences, is important, but it would lead us too far into details which depend more or less on local conditions. In the experience of American teachers it appears to have been found advisable to put geological processes and typical phenomena to the front and to take up geological history afterwards. The earlier method of taking up the history first, beginning with recent stages and working backward down the ages,--once in vogue abroad,--has been abandoned in this country. It was the order in which the science was developed and it had the advantage of starting with the living present and with the most accessible formations, but this latter advantage is secured by studying the living processes, as such, first, and turning to the history later. This permits the study of the history in its natural order, which seems better to call forth the relations of cause and effect and to give emphasis to the influence of inherited conditions.

Respecting antecedents to the study, the more knowledge of physics, chemistry, zoology, and botany, the better, but it is easy to over-stress the necessity for such preparation, however logical it may seem, for in reality all the natural sciences are so interwoven that, in strict logic, a complete knowledge of all the others should be had before any one is begun, a _reductio ad absurdum_. The sciences have been developed more or less contemporaneously and progressively, each helping on the others. They may be pursued much in the same way, or by alternations in which each prior study favors the sequent one. They may even be taken in a seemingly illogical order without serious disadvantage, for the alternative advantages and other considerations may outweigh the force of the logical order, which is at best only partially logical. It is of prime importance to stimulate in students a habit of observing natural phenomena at an early age. It may be wise for a student to take up physiography, or its equivalent, early in the college course, irrespective of an ideal preparation in the related sciences. It is unfortunate to defer such study to a stage when the student's natural apt.i.tude for observation and inference has become dulled by neglect or by confinement to subjects devoid of naturalistic stimulus. To permit students to take up earth-science in the freshman and soph.o.m.ore years, even without the ideal preparation, is therefore probably wiser than to defer the study beyond the age of responsiveness to the touch of the natural environment. The geographic and geologic environment conditioned the mental evolution of the race.

It left an inherited impress on the perceptive and emotional nature, only to be awakened most felicitously, it would seem, at about the age at which the naturalistic phases of the youth's mentality were originally called into their most intense exercise.

T. C. CHAMBERLIN _The University of Chicago_

VIII

THE TEACHING OF MATHEMATICS

=Recent changes and some of their sources=

In recent years the teaching of mathematics has undergone remarkable changes in many countries, both as regards method and as regards content. With respect to college mathematics these changes have been evidenced by a growing emphasis on applications and on the historic setting of the various questions. To understand one direct source of these changes it is only necessary to recall the fact that in about 1880 there began a steady stream of American mathematical students to Europe, especially to Germany. Most of these students entered the faculties of our colleges and universities on their return to America It is therefore of great importance to inquire what mathematical situation served to inspire these students.

The German mathematical developments of the greater part of the nineteenth century exhibited a growing tendency to disregard applications. It was not until about 1890 that a strong movement was inaugurated to lay more stress on applied mathematics in Germany.[3]

Our early American students therefore brought with them from Germany a decided tendency toward investigations in mathematical fields remote from direct contact with applications to other scientific subjects, such as physics and astronomy, which had so largely dominated mathematical investigations in earlier years.

This picture would, however, be very incomplete without exhibiting another factor of a similar type working in our own midst. J. J.

Sylvester was selected as the first professor of mathematics at Johns Hopkins University, which opened its doors in 1876 and began at once to wield a powerful influence in starting young men in higher research. Sylvester's own investigations related mainly to the formal and abstract side of mathematics. Moreover, "he was a poor teacher with an imperfect knowledge of mathematical literature. He possessed, however, an extraordinary personality; and had in remarkable degree the gift of imparting enthusiasm, a quality of no small value in pioneer days such as these were with us."[4]

=Influence of researches in mathematics on methods of teaching=