On Teaching (1875-1899)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

"[…] not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices." (Ernst Mach, "The Science of Mechanics", 1883)

"The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers; […] The modem, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things." (Truman H Safford, "Mathematical Teaching and Its Modern Methods", 1886)

"Thoroughly to teach another is the best way to learn for yourself." (Tyron Edwards, "A Dictionary of Thoughts", 1891)

"And as the number of combinations that can be made on the chess-board, is so great that probably no two games exactly alike were ever played; so no two games which the student plays with nature to wrest from her hidden truths, which were worth playing at all, ever made use of quite the same methods in quite the same way." (Alfred Marshall, "Principles of Economics", 1890)

"One of the most baneful delusions by which the minds, not only of students, but even of many teachers of mathematics in our classical colleges, have been afflicted is that mathematics can be mastered by the favored few, but lies beyond the grasp and power of the ordinary mind." (Florian Cajori, "The Teaching and History of Mathematics in the United State", 1890)

"The true aim of the teacher must be to impart an appreciation of method and not a knowledge of facts." (Karl Pearson, "The Grammar of Science", 1892)

"The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store." (Florian Cajori, "A History of Mathematics", 1893)