Bibliography.For a list of standard textbooks issued prior to the present generation, consult the bibliography in Stamper, History of the Teaching of Geometry, New York, 1908.
Bibliography.For a list of standard textbooks issued prior to the present generation, consult the bibliography in Stamper, History of the Teaching of Geometry, New York, 1908.
From the standpoint of theory there is or need be no relation whatever between algebra and geometry. Algebra was originally the science of the equation, as its name[37]indicates. This means that it was the science of finding the value of an unknown quantity in a statement of equality. Later it came to mean much more than this, and Newton spoke of it as universal arithmetic, and wrote an algebra with this title. At present the term is applied to the elements of a science in which numbers are represented by letters and in which certain functions are studied, functions which it is not necessary to specify at this time. The work relates chiefly to functions involving the idea of number. In geometry, on the other hand, the work relates chiefly to form. Indeed, in pure geometry number plays practically no part, while in pure algebra form plays practically no part.
In 1687 the great French philosopher, Descartes, wishing to picture certain algebraic functions, wrote a work of about a hundred pages, entitled "La Géométrie," and in this he showed a correspondence between the numbers of algebra (which may be expressed by letters) and the concepts of geometry. This was the first great step in the analytic geometry that finally gave us the graph
in algebra. Since then there have been brought out from time to time other analogies between algebra and geometry, always to the advantage of each science. This has led to a desire on the part of some teachers to unite algebra and geometry into one science, having simply a class in mathematics without these special names.
It is well to consider the advantages and the disadvantages of such a plan, and to decide as to the rational attitude to be taken by teachers concerning the question at issue. On the side of advantages it is claimed that there is economy of time and of energy. If a pupil is studying formulas, let the formulas of geometry be studied; if he is taking up ratio and proportion; let him do so for algebra and geometry at the same time; if he is solving quadratics, let him apply them at once to certain propositions concerning secants; and if he is proving that (a+b)2equalsa2+ 2ab+b2, let him do so by algebra and by geometry simultaneously. It is claimed that not only is there economy in this arrangement, but that the pupil sees mathematics as a whole, and thus acquires more of a mastery than comes by our present "tandem arrangement."
On the side of disadvantages it may be asked if the same arguments would not lead us to teach Latin and Greek together, or Latin and French, or all three simultaneously? If pupils should decline nouns in all three languages at the same time, learn to count in all at the same time, and begin to translate in all simultaneously, would there not be an economy of time and effort, and would there not be developed a much broader view of language? Now the fusionist of algebra and geometry does not like this argument, and he says that the cases are not parallel, and he tries to tell why they are not.He demands that his opponent abandon argument by analogy and advance some positive reason why algebra and geometry should not be fused. Then his opponent says that it is not for him to advance any reason for what already exists, the teaching of the two separately; that he has only to refute the fusionist's arguments, and that he has done so. He asserts that algebra and geometry are as distinct as chemistry and biology; that they have a few common points, but not enough to require teaching them together. He claims that to begin Latin and Greek at the same time has always proved to be confusing, and that the same is true of algebra and geometry. He grants that unified knowledge is desirable, but he argues that when the fine arts of music and color work fuse, and when the natural sciences of chemistry and physics are taught in the same class, and when we follow the declension of a German noun by that of a French noun and a Latin noun, and when we teach drawing and penmanship together, then it is well to talk of mixing algebra and geometry.
It is well, before deciding such a question for ourselves (for evidently we cannot decide it for the world), to consider what has been the result of experience. Algebra and geometry were always taught together in early times, as were trigonometry and astronomy. The Ahmes papyrus contains both primitive algebra and primitive geometry. Euclid's "Elements" contains not only pure geometry, but also a geometric algebra and the theory of numbers. The early works of the Hindus often fused geometry and arithmetic, or geometry and algebra. Even the first great printed compendium of mathematics, the "Sūma" of Paciuolo (1494) contained all of the branches of mathematics. Much of this later attempt was not,however, an example of perfect fusion, but rather of assigning one set of chapters to algebra, another to geometry, and another to arithmetic. So fusion, more or less perfect, has been tried over long periods, and abandoned as each subject grew more complete in itself, with its own language and its peculiar symbols.
But it is asserted that fusion is being carried on successfully to-day by more than one enthusiastic teacher, and that this proves the contention that the plan is a good one. Books are cited to show that the arrangement is feasible, and classes are indicated where the work is progressing along this line.
What, then, is the conclusion? That is a question for the teacher to settle, but it is one upon which a writer on the teaching of mathematics should not fear to express his candid opinion.
It is a fact that the Greek and Latin fusion is a fair analogy. There are reasons for it, but there are many more against it, the chief one being the confusion of beginning two languages at once, and the learning simultaneously of two vocabularies that must be kept separate. It is also a fact that algebra and geometry are fully as distinct as physics and chemistry, or chemistry and biology. Life may be electricity, and a brief cessation of oxidization in the lungs brings death, but these facts are no reasons for fusing the sciences of physics, biology, and chemistry. Algebra is primarily a theory of certain elementary functions, a generalized arithmetic, while geometry is primarily a theory of form with a highly refined logic to be used in its mastery. They have a few things in common, as many other subjects have, but they have very many more features that are peculiar to the one or the other. The experience of the world has ledit away from a simultaneous treatment, and the contrary experience of a few enthusiastic teachers of to-day proves only their own powers to succeed with any method. It is easy to teach logarithms in the seventh school year, but it is not good policy to do so under present conditions. So the experience of the world is against the plan of strict fusion, and no arguments have as yet been advanced that are likely to change the world's view. No one has written a book combining algebra and geometry in this fashion that has helped the cause of fusion a particle; on the contrary, every such work that has appeared has damaged that cause by showing how unscientific a result has come from the labor of an enthusiastic supporter of the movement.
But there is one feature that has not been considered above, and that is a serious handicap to any effort at combining the two sciences in the high school, and this is the question of relative difficulty. It is sometimes said, in a doctrinaire fashion, that geometry is easier than algebra, since form is easier to grasp than function, and that therefore geometry should precede algebra. But every teacher of mathematics knows better than this. He knows that the simplest form is easier to grasp than the simplest function, but nevertheless that plane geometry, as we understand the term to-day, is much more difficult than elementary algebra for a pupil of fourteen. The child studies form in the kindergarten before he studies number, and this is sound educational policy. He studies form, in mensuration, throughout his course in arithmetic, and this, too, is good educational policy. This kind of geometry very properly precedes algebra. But the demonstrations of geometry, the study by pupils of fourteen years of a geometry that was written forcollege students and always studied by them until about fifty years ago,—that is by no means as easy as the study of a simple algebraic symbolism and its application to easy equations. If geometry is to be taught for the same reasons as at present, it cannot advantageously be taught earlier than now without much simplification, and it cannot successfully be fused with algebra save by some teacher who is willing to sacrifice an undue amount of energy to no really worthy purpose. When great mathematicians like Professor Klein speak of the fusion of all mathematics, they speak from the standpoint of advanced students, not for the teacher of elementary geometry.
It is therefore probable that simple mensuration will continue, as a part of arithmetic, to precede algebra, as at present; and that algebra into or through quadratics will precede geometry,[38]drawing upon the mensuration of arithmetic as may be needed; and that geometry will follow this part of algebra, using its principles as far as possible to assist in the demonstrations and to express and manipulate its formulas. Plane geometry, or else a year of plane and solid geometry, will probably, in this country, be followed by algebra, completing quadratics and studying progressions; and by solid geometry, or a supplementary course in plane and solid geometry, this work being elective in many, if not all, schools.[39]It is also probable that a general review of mathematics, where the fusion idea may be carried out, will prove to be a feature of the last year of the high school, and one
that will grow in popularity as time goes on. Such a plan will keep algebra and geometry separate, but it will allow each to use all of the other that has preceded it, and will encourage every effort in this direction. It will accomplish all that a more complete fusion really hopes to accomplish, and it will give encouragement to all who seek to modernize the spirit of each of these great branches of mathematics.
There is, however, a chance for fusion in two classes of school, neither of which is as yet well developed in this country. The first is the technical high school that is at present coming into some prominence. It is not probable even here that the best results can be secured by eliminating all mathematics save only what is applicable in the shop, but if this view should prevail for a time, there would be so little left of either algebra or geometry that each could readily be joined to the other. The actual amount of algebra needed by a foreman in a machine shop can be taught in about four lessons, and the geometry or mensuration that he needs can be taught in eight lessons at the most. The necessary trigonometry may take eight more, so that it is entirely feasible to unite these three subjects. The boy who takes such a course would know as much about mathematics as a child who had read ten pages in a primer would know about literature, but he would have enough for his immediate needs, even though he had no appreciation of mathematics as a science. If any one asks if this is not all that the school should give him, it might be well to ask if the school should give only the ability to read, without the knowledge of any good literature; if it should give only the ability to sing, without the knowledge of good music; if it should give only the ability to speak, without any training in the useof good language; and if it should give a knowledge of home geography, without any intimation that the world is round,—an atom in the unfathomable universe about us.
The second opportunity for fusion is possibly (for it is by no means certain) to be found in a type of school in which the only required courses are the initial ones. These schools have some strong advocates, it being claimed that every pupil should be introduced to the large branches of knowledge and then allowed to elect the ones in which he finds himself the most interested. Whether or not this is sound educational policy need not be discussed at this time; but if such a plan were developed, it might be well to offer a somewhat superficial (in the sense of abridged) course that should embody a little of algebra, a little of geometry, and a little of trigonometry. This would unconsciously become a bait for students, and the result would probably be some good teaching in the class in question. It is to be hoped that we may have some strong, well-considered textbooks upon this phase of the work.
As to the fusion of trigonometry and plane geometry little need be said, because the subject is in the doctrinaire stage. Trigonometry naturally follows the chapter on similar triangles, but to put it there means, in our crowded curriculum, to eliminate something from geometry. Which, then, is better,—to give up the latter portion of geometry, or part of it at least, or to give up trigonometry? Some advocates have entered a plea for two or three lessons in trigonometry at this point, and this is a feature that any teacher may introduce as a bit of interest, as is suggested inChapter XVI, just as he may give a popular talk to his class upon the fourth dimension or the non-Euclidean geometry. The lastingimpression upon the pupil will be exactly the same as that of four lessons in Sanskrit while he is studying Latin. He might remember each with pleasure, Latin being related, as it is, to Sanskrit, and trigonometry being an outcome of the theory of similar triangles. But that either of these departures from the regular sequence is of any serious mathematical or linguistic significance no one would feel like asserting. Each is allowable on the score of interest, but neither will add to the pupil's power in any essential feature.
Each of these subjects is better taught by itself, each using the other as far as possible and being followed by a review that shall make use of all. It is not improbable that we may in due time have high schools that give less extended courses in algebra and geometry, adding brief practical courses in trigonometry and the elements of the calculus; but even in such schools it is likely to be found that geometry is best taught by itself, making use of all the mathematics that has preceded it.
It will of course be understood that the fusion of algebra and geometry as here understood has nothing to do with the question of teaching the two subjects simultaneously, say two days in the week for one and three days for the other. This plan has many advocates, although on the whole it has not been well received in this country. But what is meant here is the actual fusing of algebra and geometry day after day,—a plan that has as yet met with only a sporadic success, but which may be developed for beginning classes in due time.
There are two difficult crises in the geometry course, both for the pupil and for the teacher. These crises are met at the beginning of the subject and at the beginning of solid geometry. Once a class has fairly got into Book I, if the interest in the subject can be maintained, there are only the incidental difficulties of logical advance throughout the plane geometry. When the pupil who has been seeing figures in one plane for a year attempts to visualize solids from a flat drawing, the second difficult place is reached. Teachers going over solid geometry from year to year often forget this difficulty, but most of them can easily place themselves in the pupil's position by looking at the working drawings of any artisan,—usually simple cases in the so-called descriptive geometry. They will then realize how difficult it is to visualize a solid from an unfamiliar kind of picture. The trouble is usually avoided by the help of a couple of pieces of heavy cardboard or box board, and a few knitting needles with which to represent lines in space. If these are judiciously used in class for a few days, until the figures are understood, the second crisis is easily passed. The continued use of such material, however, or the daily use of either models or photographs, weakens the pupil, even as a child is weakened by being kept too long in a perambulator. Such devices have their place; they are usefulwhen needed, but they are pernicious when unnecessary. Just as the mechanic must be able to make and to visualize his working drawings, so the student of solid geometry must be able to get on with pencil and paper, representing his solid figures in the flat.
But the introduction to plane geometry is not so easily disposed of. The pupil at that time is entering a field that is entirely unfamiliar. He is only fourteen or fifteen years of age, and his thoughts are distinctly not on geometry. Of logic he knows little and cares less. He is not interested in a subject of which he knows nothing, not even the meaning of its name. He asks, naturally and properly, what it all signifies, what possible use there is for studying geometry, and why he should have to prove what seems to him evident without proof. To pass him successfully through this stage has taxed the ingenuity of every real teacher from the time of Euclid to the present; and just as Euclid remarked to King Ptolemy, his patron, that there is no royal road to geometry, so we may affirm that there is no royal road to the teaching of geometry.
Nevertheless the experience of teachers counts for a great deal, and this experience has shown that, aside from the matter of technic in handling the class, certain suggestions are of value, and a few of these will now be set forth.
First, as to why geometry is studied, it is manifestly impossible successfully to explain to a boy of fourteen or fifteen the larger reasons for studying anything whatever. When we confess ourselves honestly we find that these reasons, whether in mathematics, the natural sciences, handwork, letters, the vocations, or the fine arts, are none too clear in our own minds, in spite of any pretentious language that we may use. It is therefore most satisfactory to anticipate the question at once, and to setthe pupils, for a few days, at using the compasses and ruler in the drawing of geometric designs and of the most common figures that they will use. This serves several purposes: it excites their interest, it guards against the slovenly figures that so often lead them to erroneous conclusions, it has a genuine value for the future artisan, and it shows that geometry is something besides mere theory. Whether the textbook provides for it or not, the teacher will find a few days of such work well spent, it being a simple matter to supplement the book in this respect. There was a time when some form of mechanical drawing was generally taught in the schools, but this has given place to more genuine art work, leaving it to the teacher of geometry to impart such knowledge of drawing as is a necessary preliminary to the regular study of the subject.
Such work in drawing should go so far, and only so far, as to arouse an interest in geometric form without becoming wearisome, and to familiarize the pupil with the use of the instruments. He should be counseled about making fine lines, about being careful in setting the point of his compasses on the exact center that he wishes to use, and about representing a point by a very fine dot, or, preferably at first, by two crossed lines. Unless these details are carefully considered, the pupil will soon find that the lines of his drawings do not fit together, and that the result is not pleasing to the eye. The figures here given are good ones upon which to begin, the dotted construction lines being erased after the work is completed. They may be constructed with the compasses and ruler alone, or the draftsman's T-square, triangle, and protractor may be used, although these latter instruments are not necessary. We shouldconstantly remember that there is a danger in the slavish use of instruments and of such helps as squared paper.
Just as Euclid rode roughshod over the growing intellects of boys and girls, so may instruments ride roughshod over their growing perceptions by interfering with natural and healthy intuitions, and making them the subject of laborious measurement.[40]
Just as Euclid rode roughshod over the growing intellects of boys and girls, so may instruments ride roughshod over their growing perceptions by interfering with natural and healthy intuitions, and making them the subject of laborious measurement.[40]
The pupil who cannot see the equality of vertical angles intuitively better than by the use of the protractor is abnormal. Nevertheless it is the pupil's interest that is at stake, together with his ability to use the instruments of daily life. If, therefore, he can readily be
supplied with draftsmen's materials, and is not compelled to use them in a foolish manner, so much the better. They will not hurt his geometry if the teacher does not interfere, and they will help his practical drawing; but for obvious reasons we cannot demand that the pupil purchase what is not really essential to his study of the subject. The most valuable single instrument of the three just mentioned is the protractor, and since a paper one costs only a few cents and is often helpful in the drawing of figures, it should be recommended to pupils.
There is also another line of work that often arouses a good deal of interest, namely, the simple field measures that can easily be made about the school grounds. Guarding against the ever-present danger of doing too much of such work, of doing work that has no interest for the pupil, of requiring it done in a way that seems unreal to a class, and of neglecting the essence of geometry by a line of work that involves no new principles,—such outdoor exercises in measurement have a positive value, and a plentiful supply of suggestions in this line is given in the subsequent chapters. The object is chiefly to furnish a motive for geometry, and for many pupils this is quite unnecessary. For some, however, and particularly for the energetic, restless boy, such work has been successfully offered by various teachers as an alternative to some of the book work. Because of this value a considerable amount of such work will be suggested for teachers who may care to use it, the textbook being manifestly not the place for occasional topics of this nature.
For the purposes of an introduction only a tape line need be purchased. Wooden pins and a plumb line caneasily be made. Even before he comes to the propositions in mensuration in geometry the pupil knows, from his arithmetic, how to find ordinary areas and volumes, and he may therefore be set at work to find the area of the school ground, or of a field, or of a city block. The following are among the simple exercises for a beginner:
1. Drive stakes at two corners,AandB, of the school grounds, putting a cross on top of each; or make the crosses on the sidewalk, so as to get two points between which to measure. Measure fromAtoBby holding the tape taut and level, dropping perpendiculars when necessary by means of the plumb line, as shown in the figure. Check the work by measuring fromBback toAin the same way. Pupils will find that their work should always be checked, and they will be surprised to see how the results will vary in such a simple measurement as this, unless very great care is taken. If they learn the lesson of accuracy thus early, they will have gained much.
2. Take two stakes,X,Y, in a field, preferably two or three hundred feet apart, always marked on top with crosses so as to have exact points from which to work. Let it then be required to stake out or "range" the line fromXtoYby placing stakes at specified distances. One boy stands atYand another atX, each with a plumb line. A third one takes a plumb line and stands atP, the observer atXmotioning to him to move his plumb line to the right or the left until it is exactly in line withXandY. A stake is then driven atP, and the pupil atXmoves on to the stakeP. ThenQis located in the same way, and thenR, and so on. The work is checked by ranging back fromYtoX. In some of the simple exercises suggested later it is necessary to range a line so that this work is useful in making measurements. The geometric principle involved is that two points determine a straight line.
3. To test a perpendicular or to draw one line perpendicular to another in a field, we may take a stout cord twelve feet long, having a knot at the end of every foot. If this is laid along four feet, the ends of this part being fixed, and it is stretched as here shown, so that the next vertex is five feet from one of these ends and three feet from the other end, a right angle will be formed. A right angle can also be run by making a simple instrument, such as is described in Chapter XV. Still another plan of drawing a line perpendicular to another lineAB, from a pointP, consists in swinging a tape fromP, cuttingABatXandY, and then bisectingXYby doubling the tape. This fixes the foot of the perpendicular.
4. It is now possible to find the area of a field of irregular shape by dividing it into triangles and trapezoids, as shown in the figure. Pupils know from their work in arithmetic how to find the area of a triangle or a trapezoid, so that the area of the field is easily found. The work may be checked by comparingthe results of different groups of pupils, or by drawing another diagonal and dividing the field into other triangles and trapezoids.
These are about as many types of field work as there is any advantage in undertaking for the purpose of securing the interest of pupils as a preliminary to the work in geometry. Whether any of it is necessary, and for what pupils it is necessary, and how much it should trespass upon the time of scientific geometry are matters that can be decided only by the teacher of a particular class.
A second difficulty of the pupil is seen in his attitude of mind towards proofs in general. He does not see why vertical angles should be proved equal when he knows that they are so by looking at the figure. This difficulty should also be anticipated by giving him some opportunity to know the weakness of his judgment, and for this purpose figures like the following should be placed before him. He should be asked which of these lines is longer,ABorXY. Two equal lines should then be arranged in the form of a letter T, as here shown, and he should be asked which is the longer,ABorCD. A figure that is very deceptive, particularly if drawn larger and with heavy cross lines, is this one in whichABandCDare really parallel, but do not seem to be so. Other interesting deceptions have to do with producing lines, as in these figures, where it is quite difficult in advance to tell whetherABandCDare in the same line, and similarly forWXandYZ. Equallydeceptive is this figure, in which it is difficult to tell which lineABwill lie along when produced. In the next figureABappears to be curved when in reality it is straight, andCDappears straight when in reality it is curved. The first of the following circles seems to be slightly flattened at the pointsP,Q,R,S, and in the second one the distanceBDseems greater than the distanceAC. There are many equally deceptive figures, and a few of them will convince the beginner that the proofs are necessary features of geometry.
It is interesting, in connection with the tendency to feel that a statement is apparent without proof, to recall an anecdote related by the French mathematician, Biot, concerning the great scientist, Laplace:
Once Laplace, having been asked about a certain point in his "Celestial Mechanics," spent nearly an hour in trying to recall the chain of reasoning which he had carelessly concealed by the words "It is easy to see."
Once Laplace, having been asked about a certain point in his "Celestial Mechanics," spent nearly an hour in trying to recall the chain of reasoning which he had carelessly concealed by the words "It is easy to see."
A third difficulty lies in the necessity for putting a considerable number of definitions at the beginning of geometry, in order to get a working vocabulary. Although practically all writers scatter the definitions as much aspossible, there must necessarily be some vocabulary at the beginning. In order to minimize the difficulty of remembering so many new terms, it is helpful to mingle with them a considerable number of exercises in which these terms are employed, so that they may become fixed in mind through actual use. Thus it is of value to have a class find the complements of 27°, 32° 20', 41° 32' 48", 26.75°, 33-1/3°, and 0°. It is true that into the pure geometry of Euclid the measuring of angles in degrees does not enter, but it has place in the practical applications, and it serves at this juncture to fix the meaning of a new term like "complement."
The teacher who thus anticipates the question as to the reason for studying geometry, the mental opposition to proving statements, and the forgetfulness of the meaning of common terms will find that much of the initial difficulty is avoided. If, now, great care is given to the first half dozen propositions, the pupil will be well on his way in geometry. As to these propositions, two plans of selection are employed. The first takes a few preliminary propositions, easily demonstrated, and seeks thus to introduce the pupil to the nature of a proof. This has the advantage of inspiring confidence and the disadvantage of appearing to prove the obvious. The second plan discards all such apparently obvious propositions as those about the equality of right angles, and the sum of two adjacent angles formed by one line meeting another, and begins at once on things that seem to the pupil as worth the proving. In this latter plan the introduction is usually made with the proposition concerning vertical angles, and the two simplest cases of congruent triangles.
Whichever plan of selection is taken, it is important to introduce a considerable number of one-step exercisesimmediately, that is, exercises that require only one significant step in the proof. In this way the pupil acquires confidence in his own powers, he finds that geometry is not mere memorizing, and he sees that each proposition makes him the master of a large field. To delay the exercises to the end of each book, or even to delay them for several lessons, is to sow seeds that will result in the attempt to master geometry by the sheer process of memorizing.
As to the nature of these exercises, however, the mistake must not be made of feeling that only those have any value that relate to football or the laying out of a tennis court. Such exercises are valuable, but such exercises alone are one-sided. Moreover, any one who examines the hundreds of suggested exercises that are constantly appearing in various journals, or who, in the preparation of teachers, looks through the thousands of exercises that come to him in the papers of his students, comes very soon to see how hollow is the pretense of most of them. As has already been said, there are relatively few propositions in geometry that have any practical applications, applications that are even honest in their pretense. The principle that the writer has so often laid down in other works, that whatever pretends to be practical should really be so, applies with much force to these exercises. When we can find the genuine application, if it is within reasonable grasp of the pupil, by all means let us use it. But to put before a class of girls some technicality of the steam engine that only a skilled mechanic would be expected to know is not education,—it is mere sham. There is a noble dignity to geometry, a dignity that a large majority of any class comes to appreciate when guided by an earnest teacher; but thebest way to destroy this dignity, to take away the appreciation of pure mathematics, and to furnish weaker candidates than now for advance in this field is to deceive our pupils and ourselves into believing that the ultimate purpose of mathematics is to measure things in a way in which no one else measures them or has ever measured them.
In the proof of the early propositions of plane geometry, and again at the beginning of solid geometry, there is a little advantage in using colored crayon to bring out more distinctly the equal parts of two figures, or the lines outside the plane, or to differentiate one plane from another. This device, however, like that of models in solid geometry, can easily be abused, and hence should be used sparingly, and only until the purpose is accomplished. The student of mathematics must learn to grasp the meaning of a figure drawn in black on white paper, or, more rarely, in white on a blackboard, and the sooner he is able to do this the better for him. The same thing may be said of the constructing of models for any considerable number of figures in solid geometry; enough work of this kind to enable a pupil clearly to visualize the solids is valuable, but thereafter the value is usually more than offset by the time consumed and the weakened power to grasp the meaning of a geometric drawing.
There is often a tendency on the part of teachers in their first years of work to overestimate the logical powers of their pupils and to introduce forms of reasoning and technical terms that experience has proved to be unsuited to one who is beginning geometry. Usually but little harm is done, because the enthusiasm of any teacher who would use this work would carry the pupils over the difficulties without much waste of energy on theirpart. In the long run, however, the attempt is usually abandoned as not worth the effort. Such a term as "contrapositive," such distinctions as that between the logical and the geometric converse, or between perfect and partial geometric conversion, and such pronounced formalism as the "syllogistic method,"—all these are happily unknown to most teachers and might profitably be unknown to all pupils. The modern American textbook in geometry does not begin to be as good a piece of logic as Euclid's "Elements," and yet it is to be observed that none of these terms is found in this classic work, so that they cannot be thought to be necessary to a logical treatment of the subject. We need the word "converse," and some reference to the law of converse is therefore permissible; the meaning of thereductio ad absurdum, of a necessary and sufficient condition, and of the terms "synthesis" and "analysis" may properly form part of the pupil's equipment because of their universal use; but any extended incursion into the domain of logic will be found unprofitable, and it is liable to be positively harmful to a beginner in geometry.
A word should be said as to the lettering of the figures in the early stages of geometry. In general, it is a great aid to the eye if this is carried out with some system, and the following suggestions are given as in accord with the best authors who have given any attention to the subject:
1. In general, letter a figure counterclockwise, for the reason that we read angles in this way in higher mathematics, and it is as easy to form this habit now as to form one that may have to be changed. Where two triangles are congruent, however, but have their sides arranged in opposite order, it is better to letter them sothat their corresponding parts appear in the same order, although this makes one read clockwise.
2. For the same reason, read angles counterclockwise. Thus ∠Ais read "BAC," the reflex angle on the outside of the triangle being read "CAB." Of course this is not vital, and many authors pay no attention to it; but it is convenient, and if the teacher habitually does it, the pupils will also tend to do it. It is helpful in trigonometry, and it saves confusion in the case of a reflex angle in a polygon. Designate an angle by a single letter if this can conveniently be done.
3. Designate the sides opposite anglesA,B,C, in a triangle, bya,b,c, and use these letters in writing proofs.
4. In the case of two congruent triangles use the lettersA,B,CandA',B',C', orX,Y,Z, instead of letters chosen at random, likeD,K,L. It is easier to follow a proof where some system is shown in lettering the figures. Some teachers insist that a pupil at the blackboard should not use the letters given in the textbook, hoping thereby to avoid memorizing. While the danger is probably exaggerated, it is easy to change with some system, usingP,Q,RandP',Q',R', for example.
5. Use small letters for lines, as above stated, and also place them within angles, it being easier to speak of and to see ∠mthan ∠DEF. The Germans have a convenient system that some American teachers follow to advantage, but that a textbook has no right to require. They use, as in the following figure,Afor the point,afor the opposite side, and the Greek letter α (alpha) for the angle. The learning of the first three Greek letters,alpha (α), beta (β), and gamma (γ), is not a hardship, and they are worth using, although Greek is so little known in this country to-day that the alphabet cannot be demanded of teachers who do not care to use it.
6. Also use small letters to represent numerical values. For example, writec= 2πrinstead ofC= 2πR. This is in accord with the usage in algebra to which the pupil is accustomed.
7. Use initial letters whenever convenient, as in the case ofafor area,bfor base,cfor circumference,dfor diameter,hfor height (altitude), and so on.
Many of these suggestions seem of slight importance in themselves, and some teachers will be disposed to object to any attempt at lettering a figure with any regard to system. If, however, they will notice how a class struggles to follow a demonstration given with reference to a figure on the blackboard, they will see how helpful it is to have some simple standards of lettering. It is hardly necessary to add that in demonstrating from a figure on a blackboard it is usually better to say "this line," or "the red line," than to say, without pointing to it, "the lineAB." It is by such simplicity of statement and by such efforts to help the class to follow demonstrations that pupils are led through many of the initial discouragements of the subject.
No definite rules can be given for the detailed conduct of a class in any subject. If it were possible to formulate such rules, all the personal magnetism of the teacher, all the enthusiasm, all the originality, all the spirit of the class, would depart, and we should have a dull, dry mechanism. There is no one best method of teaching geometry or anything else. The experience of the schools has shown that a few great principles stand out as generally accepted, but as to the carrying out of these principles there can be no definite rules.
Let us first consider the general question of the employment of time in a recitation in geometry. We might all agree on certain general principles, and yet no two teachers ever would or ever should divide the period even approximately in the same way. First, a class should have an opportunity to ask questions. A teacher here shows his power at its best, listening sympathetically to any good question, quickly seeing the essential point, and either answering it or restating it in such a way that the pupil can answer it for himself. Certain questions should be answered by the teacher; he is there for that purpose. Others can at once be put in such a light that the pupil can himself answer them. Others may better be answered by the class. Occasionally, but more rarely, a pupil may be told to "look that up forto-morrow," a plan that is commonly considered by students as a confession of weakness on the part of the teacher, as it probably is. Of course a class will waste time in questioning a weak teacher, but a strong one need have no fear on this account. Five minutes given at the opening of a recitation to brisk, pointed questions by the class, with the same credit given to a good question as to a good answer, will do a great deal to create a spirit of comradeship, of frankness, and of honesty, and will reveal to a sympathetic teacher the difficulties of a class much better than the same amount of time devoted to blackboard work. But there must be no dawdling, and the class must feel that it has only a limited time, say five minutes at the most, to get the help it needs.
Next in order of the division of the time may be the teacher's report on any papers that the class has handed in. It is impossible to tell how much of this paper work should be demanded. The local school conditions, the mental condition of the class, and the time at the disposal of the teacher are all factors in the case. In general, it may be said that enough of this kind of work is necessary to see that pupils are neat and accurate in setting down their demonstrations. On the other hand, paper work gives an opportunity for dishonesty, and it consumes a great deal of the teacher's time that might better be given to reading good books on the subject that he is teaching. If, however, any papers have been submitted, about five minutes may well be given to a rapid review of the failures and the successes. In general, it is good educational policy to speak of the errors and failures impersonally, but occasionally to mention by name any one who has done a piece of work that is worthy of special comment. Pupils may better be praised in publicand blamed in private. There is such a thing, however, as praising too much, when nothing worthy of note has been done, just as there is danger of blaming too much, resulting in mere "nagging."
The third division of the recitation period may profitably go to assigning the advance lesson. The class questions and the teacher's report on written work have shown the mental status of the pupils, so that the teacher now knows what he may expect for the next lesson. If he assigns his lesson at the beginning of the period, he does not have this information. If he waits to the end, he may be too hurried to give any "development" that the new lesson may require. There can be no rule as to how to assign a new lesson; it all depends upon what the lesson is, upon the mental state of the class, and not a little upon the idiosyncrasy of the teacher. The German educator, Herbart, laid down certain formal steps in developing a new lesson, and his successors have elaborated these somewhat as follows:
1.Aim.Always take a class into your confidence. Tell the members at the outset the goal. No one likes to be led blindfolded.
2.Preparation.A few brief questions to bring the class to think of what is to be considered.
3.Presentation of the new.Preferably this is done by questions, the answers leading the members of the class to discover the new truth for themselves.
4.Apperception.Calling attention to the fact that this new fact was known before, in part, and that it relates to a number of things already in the mind. The more the new can be tied up to the old the more tenaciously it will be held.
5.Generalization and application.
It is evident at once that a great deal of time may be wasted in always following such a plan, perhaps in ever following it consciously. But, on the other hand, probably every good teacher, whether he has heard of Herbart or not, naturally covers these points in substantially this order. For an inexperienced teacher it is helpful to be familiar with them, that he may call to mind the steps, arranged in a psychological sequence, that he would do well to follow. It must always be remembered that there is quite as much danger in "developing" too much as in taking the opposite extreme. A mechanical teacher may develop a new lesson where there is need for only a question or two or a mere suggestion. It should also be recognized that students need to learn to read mathematics for themselves, and that always to take away every difficulty by explanations given in advance is weakening to any one.
Therefore, in assigning the new lesson we may say "Take the next two pages," and thus discourage most of the class. On the other hand, we may spend an unnecessary amount of time and overdevelop the work of those same pages, and have the whole lesson lose all its zest. It is here that the genius of the teacher comes forth to find the happy mean.
The fourth division of the hour should be reached, in general, in about ten minutes. This includes the recitation proper. But as to the nature of this work no definite instructions should be attempted. To a good teacher they would be unnecessary, to a poor one they would be harmful. Part of the class may go to the board, and as they are working, the rest may be reciting. Those at the board should be limited as to time, for otherwise a premium is placed on mere dawdling. They should be soarranged as to prevent copying, but the teacher's eye is the best preventive of this annoying feature. Those at their seats may be called upon one at a time to demonstrate at the blackboard, the rest being called upon for quick responses, as occasion demands. The European plan of having small blackboards is in many respects better than ours, since pupils cannot so easily waste time. They have to work rapidly and talk rapidly, or else take their seats.
What should be put on the board, whether the figure alone, or the figure and the proof, depends upon the proposition. In general, there should be a certain number of figures put on the board for the sake of rapid work and as a basis for the proofs of the day. There should also be a certain amount of written work for the sake of commending or of criticizing adversely the proof used. There are some figures that are so complicated as to warrant being put upon sheets of paper and hung before the class. Thus there is no rule upon the subject, and the teacher must use his judgment according to the circumstances and the propositions.
If the early "originals" are one-step exercises, and a pupil is required to recite rapidly, a habit of quick expression is easily acquired that leads to close attention on the part of all the class. Students as a rule recite slower than they need to, from mere habit. Phlegmatic as we think the German is, and nervous as is the American temperament, a student in geometry in a German school will usually recite more quickly and with more vigor than one with us. Our extensive blackboards have something to do with this, allowing so many pupils to be working at the board that a teacher cannot attend to them all. The result is a habit of wasting the minutesthat can only be overcome by the teacher setting a definite but reasonable time limit, and holding the pupil responsible if the work is not done in the time specified. If this matter is taken in hand the first day, and special effort made in the early weeks of the year, much of the difficulty can be overcome.
As to the nature of the recitation to be expected from the pupil, no definite rule can be laid down, since it varies so much with the work of the day. In general, however, a pupil should state the theorem quickly, state exactly what is given and what is to be proved, with respect to the figure, and then give the proof. At first it is desirable that he should give the authorities in full, and later give only the essential part in a few words. It is better to avoid the expression "by previous proposition," for it soon comes to be abused, and of course the learning of section numbers in a book is a barbarism. It is only by continually stating the propositions used that a pupil comes to have well fixed in his memory the basal theorems of geometry, and without these he cannot make progress in his subsequent mathematics. In general, it is better to allow a pupil to finish his proof before asking him any questions, the constant interruptions indulged in by some teachers being the cause of no little confusion and hesitancy on the part of pupils. Sometimes it is well to have a figure drawn differently from the one in the book, or lettered differently, so as to make sure that the pupil has not memorized the proof, but in general such devices are unnecessary, for a teacher can easily discover whether the proof is thoroughly understood, either by the manner of the pupil or by some slight questioning. A good textbook has the figures systematically lettered in some helpful way that is easily followed by the classthat is listening to the recitation, and it is not advisable to abandon this for a random set of letters arranged in no proper order.
It is good educational policy for the teacher to commend at least as often as he finds fault when criticizing a recitation at the blackboard and when discussing the pupils' papers. Optimism, encouragement, sympathy, the genuine desire to help, the putting of one's self in the pupil's place, the doing to the pupil as the teacher would that he should do in return,—these are educational policies that make for better geometry as they make for better life.
The prime failure in teaching geometry lies unquestionably in the lack of interest on the part of the pupil, and this has been brought about by the ancient plan of simply reading and memorizing proofs. It is to get away from this that teachers resort to some such development of the lesson in advance, as has been suggested above. It is usually a good plan to give the easier propositions as exercises before they are reached in the text, where this can be done. An English writer has recently contributed this further idea: