"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition."

“It is very possible for a teacher to be fluent in speech, earnest in manner, happy in his choice of illustration, and to be a very inefficient teacher, nevertheless. We are often apt to think it enough if we deliver a good lesson, and to forget that, after all, its value depends upon the degree in which it is really received and appropriated by the children. Now, in order to secure that what we teach shall really enter their minds, and be duly fixed and comprehended there, it is above all things necessary that we should be able to use effectively the important instrument of instruction to which our attention is now to be drawn.”

“What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary 268 training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking—consecutively, closely, logically—can be effectually taught.”

“I have undertaken to say a few words to you on the "Art of Questioning". It is a subject of great importance to all of you who desire to become good teachers ; for, in truth, the success and efficiency of our teaching depend more on the skill and judgment with which we put questions than on any other single circumstance.”