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.

[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.

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
Smith[31] 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:--

1. Elementary operations of arithmetic.

2. Simple mensuration, correlation with drawing, the models in
hand:--

Inductive geometry--the primitive form of the science.

3. Arithmetic of business and of science, using the simple equation
with one unknown quantity wherever it throws light upon the
subject.

4. Simple theory of numbers, the roots, series, logarithms.

5. Elementary algebra, including quadratic and radical equations.

6. Demonstrative plane geometry begun before the algebra is
completed and correlated with it.

7. Plane trigonometry and its elementary applications.

8. 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.”

[31] Compare Smith, David Eugene, “History of Modern Mathematics,” in Merriman & Woodworth’s “Higher Mathematics,” Wiley, New York, 1896.

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:--

5 4 3 2 1 | 1 2 3 4 5
|___|___|___|___|___|___|___|___|___|___|
|
− 0 +

“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]

[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.

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.

[34] Dewey, John, “The Aim of History in Elementary Education,” _Elementary School Record_, November, 1900, University of Chicago Press.

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.