“Among the branches of this subject which it is proposed to omit are some which have survived from an epoch when more advanced mathematics was scarcely known in our schools, e.g., cube root, duodecimals; so far as any useful principles are embodied in them, they belong to algebra, and can be taught by algebraic methods with such facility, that there is no longer any sound reason for retaining them in the arithmetical course.”
I do not insist on algebra for all; it gives the same sort of mental discipline that arithmetic does, and so, educationally, is not of special value. Geometry, on the other hand, gives a different kind of training—opens a different set of ideas. Many girls, therefore, do not learn algebra, especially those who come late with no clear ideas about arithmetic. Those who have been taught arithmetic well from the beginning can be led on to use algebraic symbols and letters very early.
As soon as a pupil has gone through the course I have recommended, she is ready to take up algebra in a systematic way—I shall suppose she has already been familiarised with the use of letters as general symbols.
How and when can we best introduce mathematical teaching? We have to do at present in girls’ schools with many who have come to the age, say of fifteen or sixteen, with no mathematical teaching except a very slight knowledge of arithmetical processes. For these it seems to me more important to give the mental training afforded by some initiation into geometrical ideas and methods, than to teach algebra.
Beginnings in the kindergarten.For the children (and they are happily a rapidly increasing number) who have had good teaching in the kindergarten, one may frame a course more approaching the ideal. Children can be quite early familiarised with geometrical forms and figures, and learn some of their simpler properties in connection with the drawing and modelling lessons.
Practical geometry.The Conference on Mathematics, called by the Committee of Ten, U.S.A., recommends that children from the age of ten should have some systematic instruction in concrete or experimental geometry. “The mere facts of plane and solid geometry should be taught, not as an exercise in logical deduction and exact demonstration, but in as concrete and objective a form as possible; the simple properties of similar plane figures and solids should not be proved, but illustrated and confirmed by cutting up and rearranging drawings and models. The course should include the careful construction of plane figures by the eye and by the help of instruments, the indirect measurements of heights and distances by the aid of figures drawn to scale, and elementary mensuration plane and solid.”
A small book by Paul Bert, First Elements of Experimental Geometry (Cassell), is very suggestive, and would throw much interest into the subject. Spencer’s Constructive Geometry may be referred to, but it is not altogether satisfactory. A useful and practical book is Geometry for Kindergarten Students, by Pullar (Sonnenschein).
Geometry before algebra.I consider that geometry should be preferred to algebra in order of time, because, as I have said, arithmetic gives the same kind of mental training as algebra, whereas from geometry the learner gains a unique mental discipline.
Its educational value.Thus the learner is taught to frame a definition; he has to put before the imagination the abstract generalised idea, and then describe, in words clear and precise, what is in the mind. Each proposition begins with a general statement regarding what is to be proved, or to be done, and compels us to have a clear idea of what we are going to talk about before we begin. The sub-enunciation makes us bring the general into the region of the particular, and infer the general from it. We must for the demonstration select certain relations relevant to the subject and omit all others, and we must be ready to give a reason for every assertion. Thus geometrical teaching trains the judgment and forms a most useful and logical habit of mind. One finds the tendency is greatly checked to use words without any clear idea of their meaning, to plunge into a subject without having set in order in the mind, what is the matter to be discussed, or the problem to be solved, and order is introduced into the general work in all other subjects of study.
Leads up to the region of ideas.But geometry has still higher uses in the process of mental development. It is, so to speak, the link between the real and the ideal; as Professor Cayley has said, “imaginary objects are the only realities, the οντως οντα, in regard to which the corresponding physical objects are as the shadows in the cave”;[23] if, on the one hand, it opens the gates of science, on the other it leads us to philosophy, and so Plato is said to have placed over the door of the Academy, “Let none enter here ignorant of geometry”.
[23] Presidential Address, Brit. Assoc.
To study geometry is to enter a new path, and we do not see at first to what heights it leads, upwards to the universe of ideas; ideas are nothing for sense, and yet they are the most necessary things for the everyday life we lead. Thus, a point, though it exists not, yet as a thought-dynamic is—it moves and traces out lines which do not exist, and yet give us direction, and are of most practical use; by them we calculate the height of real things, we guide our ships, we find paths in the heavens. Again, moving lines give us planes, and these, which exist only in thought, as they move, form what we call solid figures, i.e., something which occupies space.
Forming definitions.Of course, no one who is grounded in the principles of real education, would think of letting children begin by learning definitions; they must be made to put their vague notions into words; and it will be well for them to see how difficult this is, e.g., in the case of a straight line, an angle, though the notion is quite clear to the mind’s eye. It is surprising to those who have not taught the subject how long it takes girls, who have not been trained to exactness, to bring out, e.g., the definition of a circle. They will say, all lines drawn from the centre are equal; or all lines drawn from the centre to the circumference are equal.
No child should be allowed for a long time to see a Euclid. Each proposition must be treated as a rider, and a copious supply of riders provided in addition; the child helped to discover the solution or the proof, then set to write it; if wrong it must be gone over again and again; it will take a long time to get through a very few propositions thus, but later all is easy.
Methods of teaching.It appears from the report of the Oxford Local Examinations, August, 1897, that the methods of the dark ages still prevail in too many schools; we read: “In many cases candidates who wrote out correctly all propositions for the first six books sent up attempts at problems that can only be described as grotesque, and showed their complete failure to understand the subject, giving the unpleasing impression that all they knew was learned by heart”.
Euclid.As a formal introduction to Euclid for young pupils, I know nothing better for the teacher to study and use than Bradshaw’s First Step. Many others might be named. The Harpur Euclid is good (Longmans), and Books I. and II., by Smith and Bryant, may be specially recommended. Still I regret that the text-book in England is Euclid; its inconsistencies are manifest; we stand alone in keeping it. Yet a good workman will make the best of his tools, and there are editions which remedy many of the defects. One would, however, hope that some day Societies for the Improvement of Geometrical Teaching and Reformed Spelling will rejoice together. It does seem an anachronism not to have an angle as large as 180°; to use the circle, and think of a circumference, yet refer to no other loci, and work out in a cumbrous manner the propositions of Books III. and IV.—to talk of lines touching and not make use of limits. The more a teacher knows of the higher mathematics, and looks forward for the pupil, the better will he teach the rudiments. The treatment of the subject by Professor Henrici (London Science Class-books, Longmans) seems excellent, but I do not know how far it would answer for young beginners. I should be glad to have the experience of some who have tried it. The professor derives the notion of a point from a solid, particular figures from infinite planes, and proceeds generally in an inverse direction from that of Euclid; the nomenclature is admirably compact, and must result in a large economy of thinking power—the notion of a locus is introduced early, and the methods employed lead up to the modern or projective geometry.
I once spent some time at Zurich, a town especially remarkable for its intellectual activity, and chiefly for its mathematical school. Through the kindness of Professor Kinkel and other friends, I easily obtained permission to be present at various lessons in the Polytechnic and Canton School. I found the method there similar to that which we follow. The pupils used as a text-book Wolff’s Taschen-buch, a duodecimo of less than 300 pages, which contains the principal results in pure mathematics and the applied sciences, but no demonstrations. I heard a lesson given in the Canton School. Professor Weileman first read the proposition; it was the same as Euclid, XI. 2: to draw a perpendicular to a plane from a given point without it. About a dozen held out their hands to show they were ready to demonstrate. The professor selected one, who took his place at the board, and, subject to correction, worked the problem. The professor gave as little direct instruction as possible, appealing rather to the class. I was much struck with the eager interest that the class (I think it was Class II. B) took in the work. The next proposition (in Wolff) afforded much amusement. The demonstrator jumped to the conclusion that the lines required to complete the construction would meet, and could not be made to see he had assumed what required proof. Other members of the class offered to take the matter up; he was accordingly superseded by No. 2, who having surmounted this difficulty, also broke down before he reached the end. No. 3 therefore took his place at the board. Thus were the reasoning and inventive powers of the boys developed, and a keen interest awakened; there was no weariness, no apathy.
I make a few remarks on what may seem to some trivial matters, yet which are of importance to beginners.
In giving the proof at the board, there is no need to use three letters, and drag children by their help round every angle; we can write a Greek letter or a number, as we constantly do in trigonometry, or we could colour the angles; say the red is equal to the blue, and let the children write out the propositions in an abbreviated form first; or we might adopt the convenient and concise plan of Professor Henrici: let capitals stand for points, small letters for lines, and let angles be represented by the small letters with ∠ prefixed. Thus we have line PQ or a; PR or b; and ∠ QPR or ∠ ab; anything to avoid tediousness is good; children are so bored by verbosity.
Riders need not be always mere lines without any human or scientific interest. Suppose instead of saying—From two points to draw lines to a given line, which shall make equal angles with the given line, we say—Let CD be a mirror or a wall, a ray or a ball strikes it at P, draw the direction it will take after—or, There is a big house A, and a little house B, near a river—the man in B has to fetch water for A daily, where should he draw the water so as to go the shortest possible distance?
The method of determining the distance of the moon can be made clear long before a child is able to conceive the trigonometrical ratios, and if we are able to arouse an interest in astronomy, we may excite ardour in some which will make hard thought and work delightful. The distant prospect of the mountain top has a wonderful power of leading us on. The writer can never forget the joyful enthusiasm with which she threw herself into the study of mathematics in consequence of hearing courses of lectures on astronomy from Mr. Pullen of Cambridge, Gresham Professor of Astronomy, and the late Vice-Chancellor of Cambridge has described to her the power which the first realisation of the wonders of the boundless universe had over him when a boy of fourteen.
Mr. Glazebrook has suggested that some insight may be given to those who have no high mathematical ability into what seems so marvellous to the uninitiated, the development of curves from equations.
Algebra.The close relation between algebra and geometry becomes apparent in Euclid, Book II., but this might be shown somewhat earlier by methods such as those recommended by Mr. Wormell in the first pages of Plotting or Graphic Mathematics. We can see by a figure that 1 + 2 + 3 + 2 + 1 = 32, and lead the pupil on to the general proposition which is in constant use, when treating of falling bodies.
Or we can show similarly that the sum of an arithmetical series equals a + l2.
As regards the formal introduction of generalised arithmetic or algebra, one cannot lay down any limit of age, owing to the very untrained state in which girls come to secondary schools, but with children who have been taught thoroughly the principles of arithmetic up to fractions, it is easy to introduce literal symbols and so prepare the way: this should be done much earlier than is usual.
Children well taught in arithmetic might perhaps begin the subject formally about thirteen, and I think it well for the first term to drop arithmetic altogether, so as to get as much time as possible for overcoming the initial difficulties, and making use of the zeal which a new study gives; but of course every good teacher of arithmetic will train his pupils to use letters for numbers very much earlier. There is a good deal put into arithmetic books, which would be much better dealt with by algebraical methods, and should be postponed, e.g., involution and evolution, and much time should be saved by omitting long sets of examples on weights and measures, etc., and giving sums to be worked out mechanically with large numbers. As in arithmetic, it is extremely important to give an insight into the composition of quantities, so that de-composition may be easy, subsequent mechanical work in multiplication, division, involution, etc., minimised, and the pupil reach sooner the more attractive branches of the subject, and feel the power it gives.
Mixed mathematics.If children have acquired early a fair knowledge of geometry and algebra, they may, say at sixteen, be ready to pass on to those branches in which the alliance of the two is most intimate, and which are so closely correlated with all the teaching in mechanics and physics. It takes most girls some time to assimilate the ideas of the trigonometrical ratios, and it is fatal to hurry them.[24] Those who are able to proceed further, and enter upon the study of co-ordinate geometry, usually take great delight in it; and it is well, too, to lead them gradually on by some such books as Proctor’s Easy Lessons in the Differential Calculus, to form some idea of what a powerful instrument the Calculus is, before they actually make use of it or formally study it; it takes time for a new method to infiltrate the mind of an ordinary student.
[24] I may add that there is an interesting chapter in Herbart’s A B C of Sense-Perception, in which he works out trigonometrical ratios on the basis of his philosophical system: this chapter would interest those teaching mathematics.
Historical method.Finally, I would once more recommend that, whenever it is possible, pupils should be led along the path of discovery pursued by original investigators, both in physics and applied mathematics; I have found the interest of logarithms greatly increased by this method.[25]
[25] Professor Salford (Monographs on Education and Health) insists on the importance of teaching logarithms as a part of scientific arithmetic. “Often logarithms are first taught in connection with trigonometry, and the average pupil does not learn the difference between a logarithmic and a natural sine; there is no cure for this confusion but to teach logarithms where they belong and to apply them to purely arithmetical problems.” He advises the introduction of logarithms “as soon as the pupil has reached in algebra the proposition am × an = am × n, and he should be shown that the practical method of dealing with powers and roots is the logarithmic. Teachers will then abstain from annoying young pupils with difficult and needless problems solved in the antiquated manner; they will learn how to calculate a compound interest table, an excellent exercise in itself, as well as a labour-saving contrivance in arithmetic. The reason why logarithms are so little appreciated, is that teachers of arithmetic have not as a rule really learned their use; they go on wasting time in arbitrary exercises in evolution, interest, etc., done by tedious methods, and do not appreciate how instinctively the best calculators employ logarithms.”
Professor Lodge’s popular book, Pioneers of Science, is very much appreciated by the young, and I may quote à propos evidence given by Dr. Bryce of Glasgow before the Royal Commission of 1864:—
“Pure mathematics cultivates the power of deductive reasoning, and as soon as boys are capable of forming abstract ideas, and grasping general principles, as soon as they have got correct notions of numbers, and an accurate knowledge of the essential parts of arithmetic, and have made some progress in geometry, then natural philosophy may be advantageously taught. I speak on this matter from experience. My relative and colleague, who had charge of the mathematical department in the Belfast Academy, introduced natural philosophy as part of the work of all the mathematical classes. After these classes had gone a certain length in geography and algebra, he took up the elements of natural philosophy two days in the week, as part of the work of every mathematical class. He began with simple experiments, and according as the progress of the boys in Euclid and algebra admitted of it, more mathematical views of natural philosophy were introduced. The great advantage of the study of physical science is that, when properly taught, it interests boys in intellectual pursuits generally. For instance, Newton’s great discovery, the identity of the power which retains the moon in her orbit with terrestrial gravity, was being explained to a class of from twelve to eighteen boys. The teacher did not tell them the result; he enumerated the phenomena by which Newton arrived at it, taking care to present them in the order most likely to suggest it. As fact after fact was marshalled before them, they became eager and excited more and more, for they saw that something new and great was coming; and when at last the array of phenomena was complete, and the magnificent conclusion burst upon their sight, the whole class started from their seats with a scream of delight. They were conscious that they had gone through the very same mental operation, as that great man had gone through. The consciousness of fellowship with so great a mind was an elevating thing, and gave them a delight in intellectual pursuits. An unusual proportion of those boys who passed through the Belfast Academy during the twenty years that I was able to have natural and physical science taught on those principles, have, as men, been distinguished and successful; and they owe it, I am convinced, in a large degree to the taste for intellectual pursuits thus formed.”
As Rosencranz expresses it, there may be distinguished three epochs:—
I. The intuitive—I use the word with the German meaning of sense-perception.
II. The imaginative, during which the developing mind is more accustomed to dwell on mental images, is less passive to impressions, more active in calling them up, in fashioning them anew.
III. The logical, during which the impulse is to harmonise the world without and the world within, to fit all things into a scheme of space and time, of order and law.
Regarding these, we may ask what is the thought-material in which the developing mind may best work successively—or if we take the same material, in what varying way shall we deal with it? The near objects which the children can touch and taste and see objectively, these are the first things which call forth the attention, that self-activity by which the mind fastens on its prey, and converts percepts into concepts; as the jelly fish catches the floating prey in its tentacles, and absorbs it into its substance, so the child stores up experiences and memories which enrich all future percepts.
Botany.What subject of systematic study can be better suited to the child then, than that which calls out its sense of wonder and beauty, and which in harmony with its own restless nature is ever changing; in which is found endless variety with underlying order? Surely the world of flowers is specially suited for teaching the little ones. How the colours and forms delight them—has not the first sight of a flower remained with many of us through life, “a joy for ever”? It is for us to teach how to observe, so that the memories shall be not mere vague impressions, but clear-cut, accurate, lasting: all the senses must combine to give unity and completeness to the sense-concept, so that the child may feel the beauty, enter into loving sympathy with Nature, and perfect that “inward eye, which is the bliss of solitude”. Children should be led to form collections, by which the first observations may be repeated and fulfilled; they should also learn to draw, so that not merely the individual, but the essential, the typical may be brought into clearness; we should, too, encourage in them the desire to co-operate with Nature in making the earth beautiful, and call out the affections towards the Unseen Giver of all good things.
These are a few of the reasons why botany in its simplest forms is fit nourishment for the child. The hard names, the intricate divisions into classes and orders, the physiology of growing plants can be touched on only lightly; but the power of observation can be greatly developed, and the main facts of classificatory botany can be taught, and teaching full of interest given as regards structure, growth, seed distribution and relations to the insect world. Mrs. Bell’s Science Ladders form a good introduction. When we have exhausted our material, so far as the little child is capable of understanding, it is better to turn to some fresh subject; we may later, when the mind is ripe for these things, take the subject up again. Children whose eyes have been opened, will be able to go into the country, and note down the things they have seen. Diaries I have seen quite beautifully kept by poor children taught at the House of Education at Ambleside. The children knew the different buds as they came out on the trees, and watched the delicate and deepening tints, saw the leaf-buds develop into leaves, and the opening of the flowers.
Zoology.Elementary botany should, I think, be followed by a year of zoology (say at ten years old), treated in a simple way; the teacher should dwell not upon the internal structure, but on what presents itself to the eye, beginning with living creatures that the children are familiar with, or can get to know—domestic animals, “beasties” from garden and pond, caterpillars and birds, tadpoles and dragon-flies—they should have their menageries, and watch the creatures’ habits. Especially suited to women is the work of observing insect life, and there are worlds for us to discover, if we, as we walk round our garden, have eyes to see.
The animal world too is specially calculated to develop the affections rightly. The character of the human being is too complex, too far above the understanding of the child, and as long as he is dependent, he should not be exercised in observing and chronicling the doings of those whom he cannot yet understand. It is something to give him objects, on which he can exercise his powers of criticism and observation. So too the sense of responsibility may be fostered towards those who depend upon him, and are in his power.
Astronomy.These two sciences bring the child into contact with things on the earth; he might next lift up his eyes to the heavens. It delights the child to learn the names of the constellations, and trace their forms, to notice the movements of the planets, the changing aspect of the sky as the years go round. The sense of the greatness of the universe gradually dawns on him, and the awe and reverence for that power and wisdom which is revealed in the heavens, prepares the way for those deeper teachings which belong to religion. Especially stimulating is astronomy to the developing reflective powers, from the number and variety of problems it suggests; and yet it is not altogether baffling, for the child can be led on to draw conclusions respecting the movements and distances of the heavenly bodies; very early he can be shown how to solve such questions by simple processes, and thus the mathematical passion awakened; surely most of us can remember the first time that our soul really ascended into the seventh heaven. I have heard a mathematician describe what it was to him—how at fourteen he fled from the school into the fields to be alone.
Physical geography.And what next? There is something near to the child, which he can touch, which lies at his feet, a magic book with mysterious characters, in which he reads of infinite time; let him open the pages of the great rock-book, and gather the relics of the past. Geology will help him to observe in a new way; astronomy and geology (I use it in the sense of earth-history) are more suited than the two first to the beginning of the reflective period, because there is nothing to be done to alter the objects of the two last sciences—whereas we can do much, and observe the effect of our doings on plants and animals.
Physiography, including geology and all that has to do with the phenomena of Nature included under the head of physical geography, would claim a two years’ course and unify the subjects already touched on: the pupil will learn many facts on physical science.
And now the girl, say about fifteen, with an increasing power of abstraction and reflection, and a greater knowledge of mathematics, will be ready to receive more formal and definite instruction regarding what we call matter and force—elementary physics; the subjects of light and heat, electricity or chemistry might be selected; the girl is becoming the woman—the reflective powers are gaining the ascendant—she is longing to interpret more than to gain ever more knowledge, she understands something of physics and chemistry; let her return now to her first study and carry it still further, see the mysteries of life revealed in the flower, take physiological botany, the chemical changes produced by the physical processes, watch the plants as they grow, and trace the relation of flower and insect, plant and animal—recognise that all-embracing intelligence working in all, which has harmonised not only the outward things, but the intelligence of every living creature, and made each able more or less to know the laws of their life and to obey them. The developing and deepening religious instinct will find utterances from heaven in these earthly things, hear the voice of God among the trees of the garden. Later still we can pass into the inner temple, treat of physiology, show how marvellous is the living tabernacle of the soul, how fitted for our temporary abode.
It is objected by some that physiology should not be studied because it involves the whole circle of sciences, whilst others regard it as the most necessary and fundamental branch of instruction. Experienced teachers know that much of great educative and practical value can be given on the lines of Mrs. Bell’s Laws of Health, and brought home to comparatively uneducated people by the tracts of the Ladies’ Health Society, and we all know how important it is for those who are growing into womanhood, that the subject should be treated with the wisdom and judgment and reverence which it demands.
On the later stages of the teaching of natural science I do not propose to dwell. Those who take up science as a speciality will have to limit the field, and others will be guided by circumstances, but whatever special line they may follow later, such a course of study must surely have nourished the powers of the mind, developed the sympathies, disciplined the character, enlarged the horizon beyond the petty concerns which occupy the whole attention of the uneducated of all sorts and conditions. The woman who has really thought about these things, when she travels will see things with different eyes, she will understand enough to profit by the companionship of able and thoughtful men, and later perhaps to share it may be a man’s work as Miss Herschel, and Mrs. Huggins, and Mrs. Proctor, and Mrs. Marshall, and Mrs. Sidgwick and many more—to be the friend of her brothers and the first teacher of her sons—and she will surely have learned the first lesson of wisdom, the humility which knows that all we know is to know that our knowledge is as nothing in the presence of the Infinite, that if any man think that he knows, he knows nothing as he ought to know it.
I have worked out the order in detail in respect to science; it will be enough to touch very briefly on the parallel teachings in other subjects, which must also be taught scientifically.
Take, e.g., language. The child is ever observing and imitating; restless activity characterises the child.
The teacher has to perfect the observing powers by insisting on right pronunciation, as I have shown in another chapter, first in English, then in another language; knowledge is first empirical.
Next will follow, not grammatical definitions and rules to be learned, but the discovery of classification, just as in the case of botany, through observation—the discovery of rules inductively; then, when the need is felt for a shortening of the process, the collections made by grammarians may be produced, as the book of dried specimens, say of ferns, which the child had not time and opportunity to collect for herself. Afterwards will come reading and reflection upon the relationship of words, like the systems of scientific classification of flowers, and later the age of poetry and philosophy. It is the giving the grammatical abstractions to children who are at the stage of observation merely, which creates the distaste for school learning; it is the giving dead languages at a time when children are at the active, intuitive age, and have not the powers of thought necessary to disentangle the classical authors, that makes so much of our teaching a failure.
So with history. First the simple tales, e.g., Jack and the Giant—no complications of character there—good and bad, black and white—stories of fairies and hobgoblins, beings so unlike ourselves, that we are not troubled too much with moral scruples; they are like dream people. Then old-world heroes, in whom the moral emerges—not the priggish boys and girls, to cramp the character, but boys and girls, writ large. Then passing from the individual to the general, the specimen to the species, we have family life enlarged to the state under a kingly constitution, as in ancient patriarchal times, the first teachings of which are best gathered from the Old Testament. As in the nature teachings we shall lead children to feel underlying all, the sense as of an unseen presence, a King of Kings ruling the course of this world, leading and guiding the mind of man to work with Him as in the nature realm. And lastly in the highest teachings, which have to do, not with the objective surroundings, but with the man himself, with his thoughts and aspirations, with the expression of these in literature, in art, in ethics, and politics, and philosophy, the student will find enough to develop the highest powers of thought, as he wrestles with the problems of life, when he has reached the later period of study.
And the same order is observed in religion. The objective first—the Divine acts seen in nature, in the acts of the good, in the punishment of evil; at first the thought of God is more objective, since it must be so in the early life of the child under parental government. Later more subjective, through conscience. Sin is at first regarded chiefly as an act against a loving person, later it is felt to be the degradation of our nature, or that of others, by taking in a poison as it were; or as ἁμαρτια, the frustration of the true ends of our being, the exclusion from the light and life and joy of the Divine presence, which is the soul’s sunlight, into outer darkness—the conceptions formed will be different, the underlying truths one, the thoughts will pass from the physical to the panpsychical, and later to the highest conceivable by us—the anthropomorphic, stripped of the transitory and the finite, but embracing all those eternal things by which we know that we are more than creatures of time, since we gladly throw from us all that would then be our highest good, for the things which eye sees not and ear hears not, but which can come to us by revelation only of the spiritual; things which all men, in all ages, have felt to be the best, whatever their actions may have been, truth, love, righteousness, justice, the eternal things.
Introduction.The biological sciences deal with the manifestations of life. This distinguishes them at once from the physical and chemical sciences; not, indeed, that it is possible to understand the life of any organism without some knowledge of physics and chemistry; thus to explain intelligibly the circulation of the blood some acquaintance with mechanics is necessary, but organisms have certain properties which belong to them from the very fact of their being endowed with life; the inherent properties of protoplasm, its contractility, irritability, etc., are all vital properties due to the presence of life.
The first point then that a teacher of biology has to decide in order to teach this subject rightly is: What is it possible to teach about life? Is this nineteenth century with its marvellous electrical discoveries any nearer the secret of life? Although it may fairly be claimed that the manifestations of life are better understood, yet scientists will be the first to confess that what life itself is still remains a mystery; therefore the teacher of biology must never be satisfied without arousing in the minds of his pupils a growing consciousness of the limitations of knowledge, the basis of true reverence. Any teaching of science, not only of biology, which fails to do this is defective.
Development of observation (a) in class and home work.The teacher of biology then will desire first of all to develop a reverent attitude of mind, so that the facts of life may be understood aright. Observation of vital phenomena is by no means an easy thing; it needs much accuracy, constant patience and minute attention to detail. In school teaching the foundations of accurate observation ought to be laid. Botany affords much scope for this. In planning lessons, in choosing specimens for home work, the teacher should aim at developing this faculty. A lesson on a buttercup may very well be followed by home work on a marsh marigold. The two plants belong to the same order and have great similarity in structure, but certain important differences; the tendency of unobservant pupils will be to conclude that the same description will apply to both, and possibly nectaries will be described as present on the sepals of the marsh marigold instead of on the carpels, etc. As a rule, home work should demand original observation on the part of the pupils; it should not be a mere repetition of what has been done in class; thus, supposing the sweet-pea has been worked through in class, clover may be set for home work, provided of course that the class is sufficiently advanced.
Then, as regards the observation of vital phenomena, it is possible to show that plants, like animals, take in oxygen. The details of “Garreau’s experiment” can be contrived even in schools where there is no physiological laboratory; with a water plant such as Anacharis, the evolution of oxygen in the making of starch can be demonstrated; and with such a simple thing as yeast growing in sugar and water, it is easy to show that carbonic acid gas is given off by fungi; more elaborate experiments are necessary to demonstrate the evolution of this gas by green plants. The teacher should always point out any similarity of process in plants and animals; transpiration of plants should be compared with the perspiration of animals, so that after a few lessons on the physiology of plants, it is possible to indicate the essential differences between plants and animals as far as they are known.
In zoology, as in botany, the teacher should aim at developing the power of observation, but zoology is a much more difficult subject to teach well; for it is not always possible to get animals for observation, consequently lessons in zoology are often dry; they are wanting in that living interest which comes not from book study, but from watching the animal itself. Where, however, this has been done, keen interest is aroused. A teacher who has spent hours off the coasts of Devonshire, pulling sea-anemones out of the crevices of the rocks, or watching them expand their tentacles and draw them in, will give a very different lesson from one who has merely read about a sea-anemone.
A class, having lessons in zoology, should have access to an aquarium, which can be kept in the class-room, and in planning a course on this subject, especially for young children, it is most important to choose those types which can be observed. In a first year’s course for children of ten or eleven, preference should be given to the habits of the animals, and structure introduced only so far as is necessary to explain habit. Living specimens for lessons may be obtained from aquaria in Jersey, Birmingham and elsewhere.
(b) By means of field work.It is not possible, however, to do all that ought to be done in developing observation within the limits of an hour a week in a schoolroom. The teacher of botany or zoology should be willing to organise expeditions into the country for botanising or pond grubbing. Here we have a Field Club, consisting of three or four sections: botanical, geological, zoological, archæological. The teacher of each subject is naturally the leader of the section, and is thus able to arouse a keener interest than is possible in the class-room alone. A yearly conversazione, when collections are exhibited, gives zest to the working of the sections, brings all the members of the club together, and affords an opportunity for obtaining a lecture from some original worker. It is found that if 200 belong to a school society of this kind, each member subscribing one shilling a year, a conversazione can be held, and prizes for collections given out of the funds of the society; each member bears in addition her share of the expense of an expedition; but the less expensive and the nearer home these are, the better.
(c) Through a museum.An excellent means of arousing a real interest in science lessons, and of developing the observation, is to have a school museum. That part of the museum devoted to natural history should combine two functions; it should have perfect specimens of the chief types of animal life arranged morphologically; for instance, the covering organs, such as scales of fishes, feathers of birds, hair of animals, should be grouped together, so that the homology of these organs can be seen at a glance; secondly, the museum should have surplus specimens specially intended for teaching purposes. One specimen will not serve these two purposes; for the only way of preserving any specimen in its perfection is to keep it under lock and key in a glass case, which must be air- and dust-tight. As soon as a specimen is taken out and passed about from teacher to teacher and from class to class, it will inevitably get damaged, as the curator of many a school museum can testify.
What share can the pupils take in the museum work? They may furnish specimens, but here the difficulty is to get them perfect enough; children require to be trained to aim at a standard of perfection, and in this particular the school museum may do valuable work; at the same time if the curator demands too much, the ardour of the children becomes damped; so it is sometimes well to accept an imperfect specimen, and put it in the museum until a more perfect one is forthcoming. Pupils can also do much useful work in making diagrams and drawings; every specimen in the science portion of the museum should be drawn, and parts explained by means of an accompanying diagram. Reference may here be made to the scheme at the end of this paper for a specimen museum case, illustrating the flowering plant. It has been drawn up on the lines of the Natural History Museum at South Kensington, where, as is well known, great attention is paid by Sir William Flower to the homology of organs. This scheme has been carried out in our museum; almost every specimen has been illustrated with a drawing done by pupils, the scientific explanation being written by the teacher. In the first instance, as the case was being arranged, specimens and diagrams were merely pinned, not gummed, so that as the work progressed it was possible to alter and improve upon the first arrangement.
(d) Use of microscopes.In connection with the development of observation, a word may be said about the use of the microscope in schools. Every school should have at least one microscope, if even it has only one or two powers; a great deal can be done with a 1-inch and 2-inch objectives. At present many girls take the course required by the University of Oxford for the Senior Local without having seen a single structure under the microscope. This ought not to be, especially now that microscopes are so inexpensive (a microscope with 1-inch and 1⁄4-inch objectives can be obtained for £3 6s.).
There is considerable difficulty in managing microscope work with large classes; not more than two pupils, or at the most three, can work at a microscope at the same time, and where there are only one or two microscopes in a school, the simplest plan is for the teacher of botany to have pupils out singly, whilst the rest of the class are doing paper work at their desks. Lantern slides are an immense help in class work, but they cannot altogether take the place of the microscope, and it is very important that elder pupils likely to do anything at science should learn to manipulate the microscope.
Order of lessons.In no subject is it more necessary to plan lessons carefully than in science, for not only does the development of the observing faculty depend on a right sequence, but the scope of science is ever widening.
Biology alone includes at the present time subdivisions which hardly existed thirty years ago. Teachers of botany now have to find time for vegetable morphology, histology and physiology, for the life-histories of plants as well as for the descriptions necessary to classification. At the same time there are other considerations, besides a right sequence, which must be borne in mind in planning a course. Theoretically, it would be best in botany to begin with a description of the plant as a whole; root, stem, leaf, flower, branch, and the relation of these parts to each other, should be the subject of the first lessons. But children of ten or eleven could hardly be expected to be interested in learning that a leaf is a lateral appendage of a stem, and a branch an axillary outgrowth, whereas they are fascinated by flowers, and enjoy lessons about the visits of insects to flowers, etc. Undoubtedly with young children it would be wiser to begin with the flower and gradually lead up to the plant as a whole. The teacher, too, must be guided to some extent at any rate by his own individuality. In a subject as wide as botany some minds are attracted by one part, some by another; one teacher can be so luminous in his account of structure and its adaptation to function that the children are in their turn interested, especially if minute structure is seen through the microscope, and the delight of drawing forms part of the lesson. Another teacher revels in classification, and loves to point out the resemblances between plants of one order and those of another.
There must be, and it is almost impossible to over-emphasise this, a certain sequence, a certain gradation, a definite plan, on which the lessons are arranged; but this plan, this sequence should be the teacher’s own, it should be the outcome of his own individuality; he will best teach what most interests him, hence he had better follow his own order than that of any text-book, however excellent. In higher classes, where the work is arranged on examination lines, the teacher has a definite syllabus for his guidance; but even in this case there is play for his individuality, and nothing can dispense with this. He must be always reading the new books on his subject; he must keep himself in touch with the new work that is being done through visiting museums, botanical gardens, working in laboratories, etc., so as to be keen about his subject, otherwise his lessons will be dull and lifeless, and the unforgivable sin in a teacher is dulness.
Science cultivates the faculties of imagination and reasoning.Although teachers of biology will naturally attach much importance to the development of observation, it is very necessary to remember that observation is only a means to an end, not an end in itself. If teachers aim only at cultivating the faculty of observation, they are likely to produce pupils who will make good collectors (a work not to be despised), but nothing more. The accurate observation of facts is absolutely necessary, but it is by no means the only thing to be done in science teaching. The power of generalisation, from the facts collected, should follow if science is to advance at all. It may be thought that this cannot be done in school work, but surely some attempt should be made in this direction, for it is most necessary that pupils should be taught to understand, to some extent at any rate, when a generalisation is sound and when unsound. This is specially the case in teaching physiology; for instance, pupils are most interested in hearing something of the cell theory of the body, and can quite appreciate the bearing of the discovery, that the walls of the capillary blood-vessels are composed of cells, on this theory.
Science is not a matter merely of memory and accurate observation, it needs considerable reasoning power and much imagination, for without the power of seeing resemblances in facts, i.e., true induction, progress is impossible. The theory of evolution, which has revolutionised not only science, but the whole thought of the present day, could never have been formulated had Darwin and Wallace been mere observers, however accurate, and in this connection a science teacher may be allowed to bear witness to the importance of the Humanities in the training of the mind. As a scholar of Shrewsbury Grammar School, Darwin had little training in science, but possibly without the mental discipline of the classics, he would have been unable to accomplish what he did for science in later life; for the higher walks of science require much imagination. In science lessons pupils may be called on to devise experiments for themselves, to invent diagrams, to find out resemblances, to note dissimilarities, in order to develop the faculty of imagination. Speaking very generally, in younger classes the aim of the teacher will be to cultivate the faculty of observation, in the upper to develop not only observation, but the imagination and power of reasoning.
Time—one hour.
In a previous lesson the structure of the seed of bean, maize and sunflower has been given.
Material required:—
A. Seedlings of bean, maize and sunflower, ten days old; one of each kind for each pupil.
B. Seedlings of the above, three weeks old.
C. Seedlings grown in different media; water, sawdust, soil.
1. The Seedlings of the Broad Bean should first be examined.
(a) The radicle, observed in the seed, has given rise to the primary root, on which possibly lateral roots have begun to develop. This is an instance of a true tap root.
(b) The plumule is beginning to form the stem.
(c) The cotyledons are gradually getting smaller, for the seedling is feeding on them.
These points should be emphasised by means of the blackboard, the pupils themselves drawing the seedlings as exactly as possible, always naming each part.
2. Seedlings of Sunflower.—These the pupils should describe as far as possible by themselves. They should notice from the green colour and absence of soil on the cotyledons that they are above ground, and that there is a portion of the seedling between the cotyledons and the beginning of the root; this the teacher tells them is called the hypocotyledonary portion of the stem, and the pupils ought to be able from previous lessons to explain the word, or even to make it up for themselves.
3. Seedlings of Maize.—Here the pupils will be able to describe by themselves the endosperm and the primary root, provided that only one root has shown itself. If the lateral roots have begun to develop, the teacher must explain which are lateral and which primary, and point out the difference between the primary root of this seedling and that of the bean and sunflower. It should be noticed that there is only one cotyledon, and here the point to emphasise is, that the bean and sunflower live on the food contained in, or made by, the cotyledons; the maize on the food present in the endosperm.
The seedlings three weeks old should then be compared with those already observed, the differences in length of radicle and plumule being noted.
The observation of these seedlings will naturally suggest the subject of growth. What is growth? By judicious questioning the teacher will show that it is impossible to define it, except by its manifestations in plants and animals; it is associated with the taking in of food; then by comparing the growth of a building or rock with that of a plant and animal, it will be possible to give some idea of growth by accretion as distinct from growth by assimilation; thus the mystery of growth will be gradually approached, the teacher pointing out that growth is only possible where there is life. This should be illustrated in every possible way, e.g., growth of the body, of the mind, of a school, a nation, etc.
Lastly, the effect of environment on growth will be illustrated by the seedlings grown in different media.
The home work in connection with this lesson should consist of: (1) Descriptions of seedlings; instead of maize, wheat may be given; nasturtium instead of bean; these the teacher must have ready for distribution; a drawing of each should be insisted on, with parts named; (2) Short notes on the conditions of growth and its essential nature.
The children should also be invited to grow seedlings for themselves; these should be exhibited in subsequent lessons.