Chapter III Mathematics and Nature Study
252. Aptitude for mathematics is not rarer than aptitude for other studies. That the contrary seems true, is owing to a belated and slighted beginning. But that mathematicians are seldom inclined to give as much time to children as they ought is only natural. The elementary lessons in combination and geometry are neglected in favor of arithmetic, and demonstration is attempted where no mathematical imagination has been awakened.
The first essential is attention to magnitudes, and their changes, where they occur. Hence, counting, measuring, weighing, where possible; where impossible, at least the estimating of magnitudes to determine, however vaguely at first, the more and the less, the larger and the smaller, the nearer and the farther.
Special consideration should be given, on the one hand, to the number of permutations, variations, and combinations; and, on the other hand, to the quadratic and cubic relations, where similar planes and bodies are determined by analogous lines.
Note.—This is not the place for saying much that might be said concerning that which renders early instruction in mathematics unnecessarily difficult. But it may be remarked in brief that some of these difficulties arise from the terminology, some from the teacher’s accustomed point of view, and some from the multiplication of varying requirements.
(1) The phraseology used forms an obstacle, even to the easiest steps in fractions. The fraction ⅔, for example, is read two-thirds, and, accordingly, ⅔ × ⅘, two-thirds times four-fifths, instead of, multiplication by two and by four, and division by three and by five. The fact is overlooked that the third part of a whole includes the concept of this whole, which cannot be a multiplier, but only a multiplicand. This difficulty the pupils stumble over. The same applies to the mysterious word square root, employed instead of the expression: one of the two equal factors of a product. Matters grow even worse later on when they hear of roots of equations.
(2) Still more might be said in criticism of the erroneous view according to which numbers are recorded as sums of units. This is true as little as that sums are products; two does not mean two things, but doubling, no matter whether that which is doubled is one or many. The concept of a dozen chairs is not made up of 12 percepts of single chairs; it comprises only two mental products,—the general concept chair and the undivided multiplication by 12. The concept one hundred men likewise contains only two concepts,—the general concept man and the undivided number 100. So, also, in such expression as six foot, seven pound, in which language assists correct apprehension by the use of the singular. Number concepts remain imperfect so long as they are identified with series of numbers and recourse is had to successive counting.
(3) In arithmetical problems the difficulty attaching to the apprehension of the things dealt with is confounded with that of the solution itself. Principal and interest and time, velocity and distance and time, etc., are matters which must be familiar to the pupils, and hence must have been previously explained, long before use can be made of them for practice. The pupil to whom arithmetical concepts still give trouble should be given concrete examples so familiar to him that out of them he can create over again the mathematical notion and not be compelled to apply it to them.
253. The measuring of lines, angles, and arcs (for which many children’s games, constructive in tendency, may present the first occasion) leads over to observation exercises dealing with both planes and spheres. Skill in this direction having been attained, frequent application must be made of it, or else, like every other acquirement, it will be lost again. Every plan of a building, every map every astronomical chart, may afford opportunities for practice.
These observation exercises are to be organized in such a manner that upon the completion of mensuration the way is fully prepared for trigonometry, provided that besides the work in plain geometry, algebra has been carried as far as equations of the second degree.
Extended discussions as to the place and value of the ratio idea in elementary arithmetic are found in “The Psychology of Number,” by McLellan & Dewey,27 and in “The New Arithmetic,” by W. W. Speer.28 The former work advocates early practice in measuring with changeable units, claiming that the child should early acquire the idea of number as the expression of the relation that a measured somewhat bears to a chosen measurer, and making counting a special case of measuring. Mr. Speer makes the ratio idea still more prominent by furnishing the school with numerous sets of blocks of various sizes and shapes with which to drill the pupils into instantaneous recognition of number as the ratio between two quantities. For an extended examination of these principles the reader may well consult Dr. David Eugene Smith’s able treatise on the teaching of elementary mathematics.29
Note.—It is now nearly forty years since the author wrote a little book on the plan of Pestalozzi’s A, B, C, of observation, and he has often had it used by teachers since. Numerous suggestions have been given by others under the title, “Study of Forms.” The main thing is training the eye in gauging distances and angles, and combining such exercises with very simple computations. The aim is not merely to secure keenness of observation for objects of sense, but, preëminently, to awaken geometrical imagination and to connect arithmetical thinking with it. Indeed, exercises of this sort constitute the necessary, although commonly neglected, preparation for mathematics. The helps made use of must be concrete objects. Various things have been tried and cast aside again; most convenient for the first steps are triangles made from thin hard-wood boards. Of these only seventeen pairs are needed, all of them right-angled triangles with one side equal. To find these triangles, draw a circle with a radius of four inches, and trace the tangents and secants at 5°, 10°, 15°, 20°, etc., to 85°. The numerous combinations that can be made will easily suggest themselves. The tangents and secants must be actually measured by the pupils; from 45° on, the corresponding figures, at first not carried out beyond tenths, should be noted, and, after some repetition, learned by heart. On this basis very easy arithmetical examples may be devised for the immediate purpose of gaining the lasting attention of the pupils to matters so simple. Observations relating to the sphere require a more complicated apparatus, namely, three movable great circles of a globe. It would be well to have such means at hand in teaching spherical trigonometry. Needless to say, of course, observation exercises do not take the place of geometry, still less of trigonometry, but prepare the ground for these sciences. When the pupil reaches plain geometry, the wooden triangles are put aside, and observation is subordinated to geometrical construction. Meanwhile arithmetic is passing beyond exercises that deal merely with proportions, to powers, roots, and logarithms. In fact, without the concept of the square root, not even the Pythagorean Theorem can be fully grasped.
“Herbart’s A, B, C, of Sense Perception,” together with a number of minor educational works, has been translated into English.30 It abounds in shrewd observations and ingenious devices, yet as a whole it represents one of those side excursions, which, though delightful to genius, is not especially useful to the world. To drill children into the habit of resolving a landscape into a series of triangles, may indeed be possible, but like any other schematization of the universe, is too artificial to be desirable. Nevertheless, a limited use of the devices mentioned in this section might tend to quicken an otherwise torpid mind.
254. But now a subject comes up that, on account of the difficulties it causes, calls for special consideration, namely, that of logarithms. It is easy enough to explain their use, and to render the underlying concept intelligible as far as necessary in practice—arithmetical corresponding to geometrical series, the natural numbers being conceived of as a geometrical series. But scientifically considered, logarithms involve fractional and negative exponents, as also the application of the Binomial Theorem. The latter, to be sure, is merely an easy combinatory formula so far as integral positive exponents are concerned, but, limited to these, is here of comparatively little use.
Now, since trigonometry in its main theorems is independent of logarithms, but is little applied without their aid, the question arises whether beginners should necessarily be given a complete and vigorously scientific course in logarithms, the highly beneficial instruction in trigonometry being postponed until after the successful completion of such a course, or whether the practical use of logarithms is to be permitted before accurate insight into underlying principles has been gained.
Note.—The difficulty encountered in this subject—undoubtedly one of those difficulties most keenly felt in teaching mathematics—is after all only an illustration of the injurious consequences of former sins of omission. If the geometrical imagination were not neglected, there would be ample opportunity, not only for impressing far more deeply the concept of proportion, demanded even by elementary arithmetic, but also for developing early the idea of function. The object lessons mentioned above have already illustrated the dependence of tangents and secants on angles. When these relations of dependence have become as familiar as may be expected after a half year’s instruction, sines and cosines also are taken up. But it is not sufficient to leave the matter here. Somewhat later, about the time when mensuration is introduced, the squares and cubes of natural numbers must be emphasized, and very soon committed to memory. Next it should be pointed out how by finding the differences of squares and cubes respectively, and then adding these differences, the original numbers may be obtained again. A similar treatment should be accorded to figurate numbers.
Small wooden disks, like checker-pawns, commend themselves for the purpose. By means of them various figures are found. The pupils are asked to indicate how many disks they need to construct one or the other kind of figures. A further step will be to show the increase of squares and cubes corresponding to the increase of the root, and to make this information serve as the preparation for the elementary parts of differential calculus. Now the time has come for passing on to the consideration of consecutive values of the roots, which are found to differ by quantities of continuously decreasing smallness as one progresses continuously through the number system. And so, after the logarithms of 1, 10, 100, 1000, etc., also of 1/10, 1/100, etc., have been gone over many times, forward and backward, the conception is finally reached of the interpolation of logarithms.
255. In schools where practical aims predominate, logarithms should be explained by a comparison of the arithmetical with the geometrical series, and the practical application will immediately follow. But even where recourse is had to Taylor’s Theorem and the Binomial Theorem, the gain to the beginner will not usually be very much greater. Not as though these theorems, together with the elements of differential calculus, could not be made clear; the real trouble lies in the fact that much of what is comprehended is not likely to be retained in the memory. The beginner, when he comes to the application, still has the recollection of the proof and of his having understood it. Indeed, with some assistance he would be able, perhaps, to again retrace step by step the course of the demonstration. But he lacks perspective; and in his application of logarithms it is of no consequence to him by what method they have been calculated.
What has been said here of logarithms may be applied more generally. The value of rigid demonstrations is fully seen only when one has made himself at home in the field of concepts to which they belong.
It is customary in American schools to take up elementary algebra and elementary geometry upon the completion of arithmetic, both algebra and geometry being anticipated to some extent in the later stages of arithmetic. The following paragraphs from the pen of David Eugene Smith31 indicate some of the advance in algebra since Herbart’s time:—
“The great revival of learning known as the Renaissance, in the sixteenth century, saw algebra take a fresh start after several centuries of complete stagnation. Tartaglia solved the cubic equation, and a little later Ferrari solved the biquadratic. By the close of the sixteenth century Vieta had put the keystone in the arch of elementary algebra, the only material improvements for some time to come being in the way of symbolism. For the next two hundred years the struggle of algebraists was for a solution of the quintic equation, or, more generally, for a general solution of an equation of any degree.
“The opening of the nineteenth century saw a few great additions to the theory of algebra. The first was the positive proof that the general equation of the fifth degree is insoluble by elementary algebra, a proof due to Abel. The second was the mastery of the number systems of algebra,—the complete understanding of the negative, the imaginary, the incommensurable, the transcendent. Other additions were in the line of the convergency of series, the approximation of the real roots of numerical equations, the study of determinants—all finding their way into the elements, together with the theories of forms and groups, which must soon begin to influence the earlier chapters of the subject.
“This hasty glance at the development of the subject is sufficient to show how it has been revolutionized in modern times. To-day it is progressing as never before. The higher culture is beginning to affect the lower; determinants have found place in the beginner’s course; graphic methods, objected to as innovations by some who are ignorant of their prominence in the childhood of science, are reasserting their rights; the ‘imaginary’ has become very real; the inheritances of the algebra-teachers’ guild are being examined with critical eyes, and many an old problem and rule must soon go by the board. It is valuable to a teacher to see what changes have been wrought so that he may join in the movement to weed out the bad, to cling to the good, and to reach up into the realm of modern mathematics to see if, perchance, he cannot find that which is good and usable and light-shedding for the elementary work.”
The true order of elementary mathematics, according to Dr. Smith, is substantially as follows:—
Elementary operations of arithmetic.
Simple mensuration, correlation with drawing, the models in hand:— Inductive geometry—the primitive form of the science.
Arithmetic of business and of science, using the simple equation with one unknown quantity wherever it throws light upon the subject.
Simple theory of numbers, the roots, series, logarithms.
Elementary algebra, including quadratic and radical equations.
Demonstrative plane geometry begun before the algebra is completed and correlated with it.
Plane trigonometry and its elementary applications.
Solid geometry. Trigonometry. Advanced algebra, with the elements of differentiation and integration.
“The student should then take a rapid review of his elementary mathematics, including a course in elementary analytic geometry and the calculus. He would then be prepared to enter upon the study of higher mathematics.”
256. Demonstrations taking a roundabout way through remote auxiliary concepts are a grave evil in instruction, be they ever so elegant.
Such modes of presentation are rather to be selected as start from simple elementary notions. For with these conviction does not depend on the unfortunate condition requiring a comprehensive view of a long series of preliminary propositions. Thus Taylor’s Theorem can be deduced from an interpolation formula, and this, in turn, from the consideration of differences, for which nothing is needed beyond addition, subtraction, and knowledge of the permutation of numbers.
The following account of imaginary and complex numbers by Dr. David Eugene Smith is so lucid that it is given at length:—
“The illustrations of the negative number are so numerous, so simple, and so generally known from the common text-books that it is unnecessary to dwell upon them.32 Debt and credit, the scale on the thermometer, longitude, latitude, the upward pull of a balloon compared with the force of gravity, and the graphic illustration of these upon horizontal and vertical lines—all these are familiar.
“But the imaginary and complex numbers have been left enshrouded in mystery in most text-books. The books say, inter lineas, ‘Here is √−1; it means nothing; you can’t imagine it; the writer knows nothing about it; let us have done with it, and go on.’ Such is the way in which the negative was treated in the early days of printed algebras, but now such treatment would be condemned as inexcusable. But there is really no more reason to-day for treating the imaginary so unintelligently than for presenting the negative as was the custom four hundred years ago. The graphic treatment of the complex number is not to-day so difficult for the student about to take up quadratics as is the presentation of the negative to one just beginning algebra.
“Briefly, the following outline will suffice to illustrate the procedure for the complex number:—
“1. Negative numbers may be represented in a direction opposite to that of positive numbers, starting from an arbitrary point called zero. Hence, when we leave the domain of positive numbers, direction enters. But there are infinitely many directions in a plane besides those of the positive and negative numbers, and hence there may be other numbers than these.
“2. When we add positive and negative numbers we find some results which seem strange to a beginner. For example, if we add +4 and −3 we say the sum is 1, although the length 1 is less than the length 4 or the length −3; yet this does not trouble us because we have considered something besides length, namely, direction; it is true, however, that the sum of 4 and −3 is less than the absolute value of either. This is seen to be so reasonable, however, from numerous illustrations (as the combined weight of a balloon pulling up 3 lbs., tied to a 4-lb. weight), that we come not to notice the strangeness of it; graphically, we think of the sum as obtained by starting from 0, going 4 in a positive direction, then 3 in a negative direction, the sum being the distance from 0 to the stopping-place.
“3. If we multiply 1 by −1, or by √−1 · √−1, or by √−1 twice, we swing it counter-clockwise through 180°, and obtain −1; hence, if we multiply it by √−1 once, we should swing it through 90°. Hence we may graphically represent √−1 as the unit on the perpendicular axis YY′, and this gives illustration to √−1, 2√−1, 3√−1, ··· −√−1, −2√−1, −3√−1, or, more briefly, ±i, ±2i, ±3i, ··· where i stands for √−1. We therefore see that i is a symbol of quality (graphically of direction), just as is + or −, and that −3 · 5i, i√5, etc., are just as real as −3 · 5, √5, etc. It is impossible to look out of a window −3 · 5 times as it is to look out −3 · 5i times; strictly, one number is as ‘imaginary’ as the other, although the term has come by custom to apply to one and not to the other.
“4. The complex number 3 + 2i is now readily understood. Just as 3 + (−2) is graphically represented by starting from an arbitrary zero, passing 3 units in a positive direction (say to the right), then 2 units in the opposite direction, calling the sum the distance from 0 to the stopping-point, so 3 + 2i may be represented graphically. Starting from 0, pass in the positive direction (to the right in the figure) 3 units, then in the i direction 2 units, calling the sum the distance from 0 to the stopping-place.
“Of course the question will arise as to the hypotenuse being the sum of the two sides of the right-angled triangle. But the case is parallel to that mentioned in paragraph 2; it is not the sum of the absolute values, any more than is 1 the sum of the absolute values of 4 and −3; it is the sum when we define addition for numbers involving direction as well as length.
“A simple illustration from the parallelogram of forces is often used to advantage.
“Suppose a force pulling 3 lbs. to the right (+3 lbs.) and another pulling 2 lbs. upwards (+2i lbs.); required the resultant of the two. It is evident that this is OP, i.e., OP = 3 + 2i.
“This elementary introduction to the subject of complex numbers shows that the ‘imaginary’ element is easily removed, and that students about to begin quadratics are able to get at least an intimation of the subject. This is not the place for any adequate treatment of these numbers: such treatment is easily accessible. It is hoped that enough has been presented to render it impossible for any reader to be content with the absolutely meaningless and unjustifiable treatment found in many text-books.”33
257. The pedagogical value of mathematical instruction, as a whole, depends chiefly on the extent to which it enters into and acts on the pupil’s whole field of thought and knowledge. From this truth it follows, to begin with, that mere presentation does not suffice; the aim must be rather to enlist the self-activity of the pupil. Mathematical exercises are essential. Pupils must realize how much they can do by means of mathematics. From time to time written work in mathematics should be assigned; only the tasks set must be sufficiently easy. More should not be demanded and insisted on than pupils can comfortably accomplish. Some are attracted early by pure mathematics, especially where geometry and arithmetic are properly combined. But a surer road to good results is applied mathematics, provided only the application is made to an object in which interest has already been aroused in other ways.
But the pupils ought not to be detained too long over a narrow round of mathematical problems; there must also be progress in the presentation of the theory. Were the only requisite to stimulate self-activity, the elementary principles might very easily suffice for countless examples affording the pupil the pleasure of increasing facility, and even the delight arising from inventions of his own, without giving him any conception of the greatness of the science. Many problems may be compared to witty conceits, which may be welcome enough in the right place, but which should not encroach on the time for work. There ought to be no lingering over things that with advancing study solve themselves, merely for the sake of performing feats of ingenuity. Incomparably more important than mere practice examples is familiarity with the facts of nature, and such familiarity renders all the better service to mathematics if combined with technical knowledge.
258. Even young children may very well busy themselves with picture books illustrating zoölogy, and later with analyses of plants which they have gathered. If early accustomed to this, they will, with some guidance, readily go on by themselves. At a later time they are taught to observe the external characteristics of minerals. The continuation of the study of zoölogy is beset with some difficulties on account of the element of sex.
Though industriously debated, there is no field of education more undecided both as to matter and method than nature work in the grades. Some scientists would teach large amounts of well-classified knowledge; others are content when they have secured a hospitable frame of mind toward nature. If a love for flowers and birds can be cultivated in children, the latter class are satisfied that the best result has been attained. Thus a discussion arises as to which is the more valuable, attitude or knowledge.
It is feared by some that any attempt to teach real science, even of an elementary kind, will result in a paralysis of permanent scientific interest. To this it is replied that a sentimental regard for æsthetic aspects of nature insures little or no true scientific interest.
Both sides are, in large measure, wrong; for, though apparently antagonistic in their aims, they make merely a different application of a common principle, which, if not wholly erroneous, is at least inadequate. Both parties assume that the end to be attained in nature study is something only remotely related to the pupil’s practical life. One would present nature for its own sake as scientific knowledge; the other would offer it for its own sake as a source of æsthetic or other feeling. The scientist often assumes that to a pupil a scientific fact or law is its own excuse for being. He thinks there must be a natural, spontaneous response to such a fact or law in the breast of every properly constituted child, so that, to imbue the mind with the scientific spirit, it is only necessary to expose it to scientific fact.
Perhaps, unfortunately for the normal child, this view is somewhat encouraged by the biographies of scientific geniuses. On the other hand, those who hold the poetic view of nature assume that there must be a native response to natural beauties in every child; so that the true method is to expose him to nature’s beauty, when rapture is sure to follow. Unfortunately again for the pupil, this view is also encouraged by the influence of the nature poets. The result is that natural science is presented as an end in itself—in the one case as scientific knowledge, in the other as the lovable in nature.
While it may be admitted that a few children will respond now to the one stimulus, now to the other, the great mass are not thrilled with rapture at nature’s beauty, nor are they fettered by scientific interest in her laws. To become an object of growing interest to children, nature must have a better basis than natural childish delight in the novel, or reverence for scientific law. The first of these is evanescent, the second feeble.
We may agree with the scientist as with the poet, that both science and poetic appreciation are desirable ends, but they cannot be imparted to the childish mind by didactic fiat.
If there is one service greater than another that Herbart has rendered to education, it is in bringing clearly to our consciousness the supreme importance of the principle of apperception, or mental assimilation, as a working basis for educative processes. So long as a fact or a principle or system of knowledge stands as an end in itself, just so long is it a thing apart from the real mental life of the child. Even a formally correct method of presentation, should it even appeal at once to all ‘six’ classes of interest, will fail to create more than a factitious mental enthusiasm. It is like conversation that is ‘made’ interesting; it may suffice to lighten a tedious hour, but it awakens no vital response. When, however, the natural love of novelty or inborn response to the true is reinforced by a sense of warm personal relationship, when the facts of forest, or plain, or mine, or animal life flood the mind with unexpected and significant revelations concerning either the present or the past in close personal touch with the learner, then instruction rests upon an apperceptive basis. Abstractions that before were pale, beauties that were cold, now receive color and warmth because they get a new subjective valuation that before was impossible.
A sedate sheep nibbling grass or resting in the shade, a skipping lamb gambolling on the green, are suitable objects of nature study. Their pelts, their hoofs, their horns, their wool, are worthy of note as scientific facts. A diluted interest may even be added by recitation of the nursery rhymes about “Little Bo-Peep” and “Mary had a Little Lamb.” But these are devices for the feeble-minded.
If the teacher can reveal to the pupil the function of wool in making garments for the race, and can lead him to repeat the processes by which, from time immemorial, the wool has been spun into yarn and woven into cloth; if, at the same time, the influence of this industry upon the home life, both of men and women, can be shown, the study of the sheep becomes worthy the attention even of a boy who can play foot-ball or of a girl who can cook. The literature of the sheep is no longer infantile or fatuous. We have a gamut reaching from Penelope to Priscilla. In the words of Professor Dewey: “The child who is interested in the way in which men lived, the tools they had to do with, the transformation of life that arose from the power and leisure thus gained, is eager to repeat like processes in his own action; to make utensils, to reproduce processes, to rehandle materials. Since he understands their problems and their successes only by seeing what obstacles and what resources they had from nature, the child is interested in field and forest, ocean and mountain, plant and animal.... The interest in history gives a more human coloring, a wider significance, to his own study of nature.”34
The conclusion arising from this argument is that nature study as an end in itself, or a thing apart from the real or imagined experiences of the pupil, is but a faint reflection of what it may become under a more rational treatment. In order of time, nature study in the earliest grades may indeed rest upon the mere delight of the childish mind in the new, the strange, the beautiful, and especially in the motion of live creatures, and may be reinforced by childish literature. When boyhood and girlhood begin, however, then the industrial motive, first in the home environment, then of primitive times, becomes the chief reliance for an abiding interest. In the reproduction of primitive processes, there is of necessity a historical element. When nature has attained a firm apperceptive basis through imitation of primitive industrial processes, and has obtained a historical background, then it may properly be further reinforced by literary reference. The poetic value of nature will now appeal to the mind with a potency that springs from inner life and experience; scientific law will now have some chance of appealing to the mind with something of the same reverence that Kant besought for the moral law. The true order of appeal in nature study is therefore as follows: For infancy, natural curiosity and delight in the movements of living creatures; for the age of boyhood and girlhood, imitation, real or imaginary, of processes depending upon natural objects and forces, together with historical and literary reference; secondarily, nature work may also appeal to youthful interest in natural law or beauty.
259. With the foregoing should be conjoined much attention to external nature, to the changes corresponding to the seasons, and to means of intercommunication.
Under this head belongs, on the one hand, observation of the heavenly bodies,—where sun and moon rise, how the latter waxes and wanes, where the north star is found, and what arcs are described by the brighter stars and the most conspicuous constellations.
Here belongs, on the other hand, technological knowledge, acquired partly through direct observation, partly through lessons in descriptive physical science. Technology ought not to be considered merely from the side of the so-called material interests. It furnishes very important connecting links between the apprehension of the facts of nature and human purposes. Every growing boy and youth should learn to handle the ordinary tools of the carpenter as well as rule and compasses. Mechanical skill would often prove far more useful than gymnastic exercises. The former benefits the mind, the latter benefit the body. With burgher schools should go manual training-schools, which does not mean that the latter must necessarily be trade schools. Finally, every human being ought to learn how to use his hands. The hand has a place of honor beside language in elevating mankind above the brute.
The foregoing store of information also enters into the study of geography; how, will appear in the next chapter.
The soundness of the foregoing remarks is witnessed by the rapid development of manual training-schools in the last decade, and the almost universal desire, if not practice, of providing considerable amounts of manual training for the pupils of the grammar grades. The girls usually have some form of sewing and cooking, while the boys have sloyd or other similar tool work in wood. The rationale of requiring girls to do carpenter work instead of the forms of manual exercise that especially pertain to their sex is not yet satisfactorily established.
260. On the observation of the heavenly bodies is based popular astronomy, which provides a test as to whether the mathematical imagination has been properly cultivated.
261. Elementary statics and mechanics will serve as an early introduction to physics, which combines with the easiest portions of chemistry. Long before physics is formally presented, it must be foreshadowed by many things stimulating the attention. Notice is directed to clocks, mills, the most familiar phenomena of atmospheric pressure, to electrical and magnetic toys, etc. In burgher schools, at least, so much must be said about buildings and machines as is necessary to incite to further study in the future. The same holds for the fundamental facts of physiology.
262. As often as a new topic for study is introduced, it is important to give prominence to some of the salient facts, and these must be accurately memorized. Moreover, pupils need to have practice in exact description. Where practicable, these descriptions are corrected by actually looking at the objects themselves.
Hasty and superficial observation of objects presented for inspection always calls for severe criticism; else collections and experiments become valueless. Nor should objects be shown too lavishly; pupils must often be told beforehand what they will have to look for. Frequently it may serve the purpose to employ successively good descriptions, pictures, and direct observation.
Notes
27 McLellan & Dewey, “The Psychology of Number,” International Education Series, D. Appleton & Co., New York, 1895.
28 Speer, W. W., “The New Arithmetic,” Ginn & Co., Boston, 1896.
29 Smith, David Eugene, “The Teaching of Elementary Mathematics,” Ch. V, The Macmillan Co., New York, 1900.
30 Eckoff, William J., “Herbart’s A, B, C, of Sense Perception,” International Education Series, D. Appleton & Co., New York, 1896.
31 Compare Smith, David Eugene, “History of Modern Mathematics,” in Merriman & Woodworth’s “Higher Mathematics,” Wiley, New York, 1896.
32 See Beman & Smith’s “Elements of Algebra,” p. 17.
33 For an elementary presentation of the subject, see Beman and Smith’s “Elements of Algebra,” Boston, 1900. For a history of the subject, see Beman and Smith’s translation of Fink’s “History of Mathematics,” Chicago, 1900, or Professor Beman’s Vice-Presidential Address before the American Association for the Advancement of Science, 1898, or the author’s “History of Modern Mathematics,” already mentioned.
34 Dewey, John, “The Aim of History in Elementary Education,” Elementary School Record, November, 1900, University of Chicago Press.