Note.—This is not the place for saying much that might be said concerning that which renders early instruction in mathematics unnecessarily difficult. But it may be remarked in brief that some of these difficulties arise from the terminology, some from the teacher’s accustomed point of view, and some from the multiplication of varying requirements.
(1) The phraseology used forms an obstacle, even to the easiest steps in fractions. The fraction ⅔, for example, is read two-thirds, and, accordingly, ⅔ × ⅘, two-thirds times four-fifths, instead of, multiplication by two and by four, and division by three and by five. The fact is overlooked that the third part of a whole includes the concept of this whole, which cannot be a multiplier, but only a multiplicand. This difficulty the pupils stumble over. The same applies to the mysterious word square root, employed instead of the expression: one of the two equal factors of a product. Matters grow even worse later on when they hear of roots of equations.
(2) Still more might be said in criticism of the erroneous view according to which numbers are recorded as sums of units. This is true as little as that sums are products; two does not mean two things, but doubling, no matter whether that which is doubled is one or many. The concept of a dozen chairs is not made up of 12 percepts of single chairs; it comprises only two mental products,—the general concept chair and the undivided multiplication by 12. The concept one hundred men likewise contains only two concepts,—the general concept man and the undivided number 100. So, also, in such expression as six foot, seven pound, in which language assists correct apprehension by the use of the singular. Number concepts remain imperfect so long as they are identified with series of numbers and recourse is had to successive counting.
(3) In arithmetical problems the difficulty attaching to the apprehension of the things dealt with is confounded with that of the solution itself. Principal and interest and time, velocity and distance and time, etc., are matters which must be familiar to the pupils, and hence must have been previously explained, long before use can be made of them for practice. The pupil to whom arithmetical concepts still give trouble should be given concrete examples so familiar to him that out of them he can create over again the mathematical notion and not be compelled to apply it to them.
[253.] The measuring of lines, angles, and arcs (for which many children’s games, constructive in tendency, may present the first occasion) leads over to observation exercises dealing with both planes and spheres. Skill in this direction having been attained, frequent application must be made of it, or else, like every other acquirement, it will be lost again. Every plan of a building, every map every astronomical chart, may afford opportunities for practice.
These observation exercises are to be organized in such a manner that upon the completion of mensuration the way is fully prepared for trigonometry, provided that besides the work in plain geometry, algebra has been carried as far as equations of the second degree.
Extended discussions as to the place and value of the ratio idea in elementary arithmetic are found in “The Psychology of Number,” by McLellan & Dewey,[27] and in “The New Arithmetic,” by W. W. Speer.[28] The former work advocates early practice in measuring with changeable units, claiming that the child should early acquire the idea of number as the expression of the relation that a measured somewhat bears to a chosen measurer, and making counting a special case of measuring. Mr. Speer makes the ratio idea still more prominent by furnishing the school with numerous sets of blocks of various sizes and shapes with which to drill the pupils into instantaneous recognition of number as the ratio between two quantities. For an extended examination of these principles the reader may well consult Dr. David Eugene Smith’s able treatise on the teaching of elementary mathematics.[29]
Note.—It is now nearly forty years since the author wrote a little book on the plan of Pestalozzi’s A, B, C, of observation, and he has often had it used by teachers since. Numerous suggestions have been given by others under the title, “Study of Forms.” The main thing is training the eye in gauging distances and angles, and combining such exercises with very simple computations. The aim is not merely to secure keenness of observation for objects of sense, but, preëminently, to awaken geometrical imagination and to connect arithmetical thinking with it. Indeed, exercises of this sort constitute the necessary, although commonly neglected, preparation for mathematics. The helps made use of must be concrete objects. Various things have been tried and cast aside again; most convenient for the first steps are triangles made from thin hard-wood boards. Of these only seventeen pairs are needed, all of them right-angled triangles with one side equal. To find these triangles, draw a circle with a radius of four inches, and trace the tangents and secants at 5°, 10°, 15°, 20°, etc., to 85°. The numerous combinations that can be made will easily suggest themselves. The tangents and secants must be actually measured by the pupils; from 45° on, the corresponding figures, at first not carried out beyond tenths, should be noted, and, after some repetition, learned by heart. On this basis very easy arithmetical examples may be devised for the immediate purpose of gaining the lasting attention of the pupils to matters so simple. Observations relating to the sphere require a more complicated apparatus, namely, three movable great circles of a globe. It would be well to have such means at hand in teaching spherical trigonometry. Needless to say, of course, observation exercises do not take the place of geometry, still less of trigonometry, but prepare the ground for these sciences. When the pupil reaches plain geometry, the wooden triangles are put aside, and observation is subordinated to geometrical construction. Meanwhile arithmetic is passing beyond exercises that deal merely with proportions, to powers, roots, and logarithms. In fact, without the concept of the square root, not even the Pythagorean Theorem can be fully grasped.
“Herbart’s A, B, C, of Sense Perception,” together with a number of minor educational works, has been translated into English.[30] It abounds in shrewd observations and ingenious devices, yet as a whole it represents one of those side excursions, which, though delightful to genius, is not especially useful to the world. To drill children into the habit of resolving a landscape into a series of triangles, may indeed be possible, but like any other schematization of the universe, is too artificial to be desirable. Nevertheless, a limited use of the devices mentioned in this section might tend to quicken an otherwise torpid mind.
[27] McLellan & Dewey, “The Psychology of Number,” International Education Series, D. Appleton & Co., New York, 1895.