To teach well, the teacher must have a background of^mar-ginal5' information and understanding. Such a background is not to be regarded primarily as a reserve to be drawn upon readily should the occasion arise, although this secondary role is not without significance. Fundamentally the reasons for insisting that the teacher shall "know far more than she expects to teach" concern the subtler qualities of her instruction its seriousness, its depth, its resourcefulness. This additional insight and appreciation will have to do with the history and evolution of arithmetical symbols, processes and concepts; with the philosophy, logic and mathematics of arithmetic; with the cultural and humanistic bearings of arithmetical knowledge; with the psychology and sociology of learning arithmetic; and with the socio-economic significance of arithmetic skills and understandings.From one point of view, arithmetic is a very old subject. Presumably, primitive man had learned to count long before history recorded the arithmetical achievements of the pre-BabyIonian and Egyptian cultures, and long before the Greeks apologetically busied themselves with the art of computing (logislica) while cultivating with greater zeal the art of numbers (arithmetika). From another point of view, however, our number system, and the methods of computation as we know them today, are relatively modern almost as young as "modern science." We need only recall that our present notation for decimal fractions did not come into general use until about 1600 or later, and that the counting board had begun to disappear only half a century earlier. The logical foundations of arithmetic were not established until the close of the 19th century; that is, not until then was it shown that propositions of arithmetic, as a formal body of doctrine concerning numbers and the operations by which numbers may be combined, are deducible from a few assumptions.

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A Suggestive if not unique interpretation of the history of mathematics is that proposed by Oswald Spengler in the first volume of his Decline of the West, in the chapter entitled "The Meaning of Numbers." This explanation, which might be termed a morpltological interpretation, is based on the thesis that every outstanding culture is (or was) an organic structure; as such, every culture has its own mathematic, which is a morphological and distinctive characterization of that culture. Accordingly, the so-called growth of mathematics throughout the ages has not been a continuous, homogeneous and cumulative development. Furthermore, the particular "mathematic" of any given culture is inherently a necessary part of that culture, an expression of its world-being, just as its literature, music, art and morals are also an intrinsic part of that culture. In short, the mathematic of a culture is the culmination of the symbolic expression of the soul and spirit of that culture.

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