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.