On Teaching (-1849)

"If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?" (Saint Augustine of Hippo, "De Magistro", cca. 5th century)

"For in such a case, the teacher has a double job: the first to erase the [effects of] previous faulty instruction, the second to give the student true and correct training." (John of Salisbury, "Metalogicon", 1159)

"Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

"To know how to suggest is the great art of teaching. To attain it we must be able to guess what will interest." (Henri-Frédéric Amiel, 1864)

"The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1830)

"There is no teaching until the pupil is brought into the same state or principle in which you are; a transfusion takes place; he is you, and you are he; then is a teaching; and by no unfriendly chance or bad company can he ever lose the benefit." (Ralph W Emerson, "Essays", 1841)