Ueber den Mathematischen Unterricht an den [höheren] Schulen; Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 11, p. 131.
[518]. It is above all the duty of the methodical text-book to adapt itself to the pupil’s power of comprehension, only challenging his higher efforts with the increasing development of his imagination, his logical power and the ability of abstraction. This indeed constitutes a test of the art of teaching, it is here where pedagogic tact becomes manifest. In reference to the axioms, caution is necessary. It should be pointed out comparatively early, in how far the mathematical body differs from the material body. Furthermore, since mathematical bodies are really portions of space, this space is to be conceived as mathematical space and to be clearly distinguished from real or physical space. Gradually the student will become conscious that the portion of the real space which lies beyond the visible stellar universe is not cognizable through the senses, that we know nothing of its properties and consequently have no basis for judgments concerning it. Mathematical space, on the other hand, may be subjected to conditions, for instance, we may condition its properties at infinity, and these conditions constitute the axioms, say the Euclidean axioms. But every student will require years before the conviction of the truth of this last statement will force itself upon him.—Holzmüller, Gustav.
Methodisches Lehrbuch der Elementar-Mathematik (Leipzig, 1904), Teil 1, Vorwort, pp. 4-5.
[519]. Like almost every subject of human interest, this one [mathematics] is just as easy or as difficult as we choose to make it. A lifetime may be spent by a philosopher in discussing the truth of the simplest axiom. The simplest fact as to our existence may fill us with such wonder that our minds will remain overwhelmed with wonder all the time. A Scotch ploughman makes a working religion out of a system which appalls a mental philosopher. Some boys of ten years of age study the methods of the differential calculus; other much cleverer boys working at mathematics to the age of nineteen have a difficulty in comprehending the fundamental ideas of the calculus.—Perry, John.
The Teaching of Mathematics (London, 1902), pp. 19-20.
[520]. Poor teaching leads to the inevitable idea that the subject [mathematics] is only adapted to peculiar minds, when it is the one universal science and the one whose four ground-rules are taught us almost in infancy and reappear in the motions of the universe.—Safford, T. H.
Mathematical Teaching (Boston, 1907), p. 19.
[521]. The number of mathematical students ... would be much augmented if those who hold the highest rank in science would condescend to give more effective assistance in clearing the elements of the difficulties which they present.—De Morgan, A.
Study and Difficulties of Mathematics (Chicago, 1902), Preface.