Mastering the fundamental ideas of a field involves not only the grasping of general principles, but also the development of an attitude toward learning and inquiry, toward guessing and hunches, toward the possibility of solving problems on one’s own. … To instill such attitudes by teaching requires something more than the mere presentation of fundamental ideas.  Just what it takes to bring off such teaching is something on which a great deal of research is needed, but it would seem that an important ingredient is a sense of excitement about discovery—discovery of regularities of previously unrecognized relations and similarities between ideas, with a resulting sense of self-
confidence in one’s abilities.  Various people who have worked on curricula in science and mathematics have urged that it is possible to present the fundamental structure of a discipline in such a way as to preserve some of the exciting sequences that lead a student to discover for himself. It is particularly the Committee on School Mathematics and the Arithmetic Project of the University of Illinois that have emphasized the importance of discovery as an aid to teaching.  They have been active in devising methods that permit a student to discover for himself the generalization that lies behind a particular mathematical operation, and they contrast this approach with the “method of assertion and proof” in which the generalization is first stated by the teacher and the class asked to proceed through the proof.  It has been pointed out by the Illinois group that the method of discovery would be too time-consuming for presenting all of what a student must cover in mathematics.  The proper balance between the two is anything but plain*…

Should students be encouraged to guess, in the interest of learning eventually how to make intelligent conjectures?  Possibly there are certain kinds of situations where guessing is desirable and where it may facilitate the development of intuitive thinking to some reasonable degree.  There may, indeed, be a kind of guessing that requires careful cultivation.  Yet, in many classes in school, guessing is heavily penalized and is associated somehow with laziness. … What we should teach students to recognize, probably, is when the cost of not guessing is too high, as well as when guessing itself is too costly.  We tend to do the latter much better than the former. … Perhaps when the student sees the consequences of error as too grave and the consequences of success as too chancy, he will freeze into analytic [rather than intuitive] procedures even though they may not be appropriate.  On these grounds, one may wonder whether the present system of rewards and punishments as seen by pupils in school actually tends to inhibit the use of intuitive thinking.  The assignment of grades in school typically emphasizes the acquisition of factual knowledge, primarily because that is what is most easily evaluated; moreover, it tends to emphasize straightforward examination that can be graded as “correct.”…  [T]he pedagogic problems in fostering such a gift [intuition] are severe and should not be overlooked in our eagerness to take the problem into the laboratory.  For one thing, the intuitive method, as we have noted, often produces the wrong answer.  It requires a sensitive teacher to distinguish an intuitive mistake—an interestingly wrong leap—from a stupid or ignorant mistake, and it requires a teacher who can give approval and correction simultaneously to the intuitive student.  To know a subject so thoroughly that he can go easily beyond the textbook is a great deal to ask of a high school teacher.  Indeed, it must happen occasionally that a student is not only more intelligent than his teacher but better informed, and develops intuitive ways of approaching problems that he cannot explain and that the teacher is simply unable to follow or re-create for himself.  It is impossible for the teacher properly to reward or correct such students, and it may very well be that it is precisely our more gifted students who suffer such unrewarded effort.  So along with any program for developing methods of cultivating and measuring the occurrence of intuitive thinking, there must go some practical consideration of the classroom problems and the limitations on our capacity for encouraging such skills in our students.
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*Professor Ralph A. Raimi of the University of Rochester observes, "forty years later:  The proper balance has not yet been found.  Perhaps it does not exist."