[The following material is copied verbatim (the italics are in the original) from the essay entitled "Reflections and Reminiscences", in an essay collection by the late Prof. Fred Hoyle, published in book form under the general title Encounter with the Future (New York: Simon and Schuster, 1968). This particular excerpt spans pages 81, 82, and 83. It is quoted here as an authoritative broad corroboration - Hoyle had authority as one of the leading twentieth-century astrophysicists - of the position I broadly take in "Part D" of my own earnest-if-minor essay "Is Science Doomed?". I must not exaggerate by suggesting, without qualification, that Prof. Hoyle's passage is (a) a corroboration of my specific argument against a neomediaeval, neoscholastic, deference to scientific authority, and contrariwise (b) a corroboration of my specific argument for acquiring a grasp of the subtler mathematical side of scientific method through independent labour on conceptually deep questions. (My own example, in "Part D", is a sequence of questions anchored in "Faraday's Paradox" or "Faraday's Puzzle", seeming at a superficial level to refute a Maxwell equation.) The exaggerated phrasing is what I had used in my comment on this page up to UTC=20160622T1725Z. All the same, Prof. Hoyle explains things along the same broad and general lines as I do, stressing the importance of discovering the maths for oneself. I do speculate that if he were alive to be queried, he might prove willing develop his here-quoted ideas further, addressing my own distinction between the first two-thirds of the scientific terrain (for which, I argue in "Part D", routine problem-set assignments suffice) and the mission-critical final third (for which, I argue in "Part D", we have to attempt something like original work, even when within the confines of the classics). - I assume here, subject to correction by Simon and Schuster or their assignees, that my excerpt, being a short extract from Prof. Hoyle's long essay, satisfies the "fair use" provisions of copyright law.]
/.../ where a child is keen to learn, present methods seem woefully and even shockingly inadequate. For the keener student an absolute minimum of formal teaching should be prescribed. I will take mathematics as an example, because mathematics is supposed to be the hardest of all subjects to learn. I suspect it would be possible, given sufficient incentive, to design a complete set of examples, starting with the first or second grade, and ending at a stiff university standard. Perhaps a hundred thousand examples would be needed to cover the whole range. The correct procedure I am sure is to learn by doing, not by being told what to do. /.../
/.../ The need is for someone clever enough to build our best textbooks into an appropriate set of problems. Would children work under these conditions? To me at least, working a problem successfully, looking at the answer and checking that I am right, has always been vastly more interesting than formal lessons. I suspect this is a natural human reaction, and that the same is true for everybody. The reason why mathematics is too difficult for the majority of people is that the examples given to the student at the end of a class are too complicated, unless what has been said in the class has been thoroughly understood. Since the majority of students do not, and cannot, understand by hearing instead of doing, the normal experience is to find oneself completely incapable of solving the required exercises. This would have been obviated if the first exercises had been extremely easy and had been followed by slightly less easy ones, and so on.
All these things apply a fortiori at university level. /.../