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昨今の教育をめぐる話題に算数・数学の学力低下の問題がある。また．教師の指導カ不足が強く言われてもいる。ここでは，これから算数を教えることになる小学校の教師や中学校，高校の数学の教師を目指す学生の現状を，数学の内容に関する基礎知識の学生による意味理解について話題とし，いくつかの項目を挙げて説明していき，教師を目指す学生の教師への素養に向上をもたらす方策についても議論をしていきたい。

The method of casting out 9s has been used for centuries, perhaps even as long as a millenium, for checking computations with integers involving the four mathematical operations. According to Eves (1980, 160–68) it was used by Hindu-Arabic scholars in the Middle Ages. It appears to have been imbibed by Western culture as a part of the decimal system of representation of numbers. With the invention of the electronic calculator its practical value has diminished. However, it is still an intriguing application of modular arithmetic that can be generalized to arithmetic in other bases. This article explains how casting out 9s is done, examines some reasons for including it as a topic for exploration in the mathematics curriculum, and uses modular arithmetic to explore its mathematical basis and its generalizability to computations in bases other than ten. A method of detecting errors in the transmission of computer code with some affinities to its analogues is also explored.

The splendid computational check known as "casting out 9's" has fallen into disuse in our schools. This is hardly surprising in view of the current obsession, at the elementary level, with outmoded computational algorithms at the expense of any real mathematical understanding. Indeed it is our experience that most university students have no idea what the phrase "casting out 9's" means. Easily accessible discussions are rare in contemporary literature; however, they may be found in [1], [3], [4], and [6]. The history of the topic may be found in [5]. Before explaining the significance of casting out nines, let us establish the notation which will be used in this note.

L/iscovery can be an important process in elementary mathematics education. The unit described here encourages self-discovery and allows the student to occasionally make a discovery that is new even to the teacher. All that the unit requires is a pencil, some paper, a creative imagination, and a "Let's see what would happen if . . ." attitude toward the problem. It is based on the concept of the digit sums of numbers, specifically of sets of numbers that have a pattern linking them together. The topic of digit sums (or digital roots) is an old one, sometimes referred to as "casting out nines," that is experiencing a revival in many textbooks. The topic is simple enough for elementary school children to handle with ease (this article is based upon experiences with fourth-grade children), yet it is one that appears in advanced-level number theory. The ideas are more than mere curiosities.

CONSIDERING THE RECENT INTEREST in proof by nines, one might conclude this is a mid-twentieth century discovery. Probably we can concede it is a rediscovery. In University Arithmetic by Charles Davies, LL.D., A. S. Barnes & Co., New York, 1868, the process is explained and is used throughout the book. In Advanced Arithmetic, Teachers Edit ion, G. A. Wentworth, Ginn & Co., Boston, 1898, the check by nines is called for and no explanation given, indicating it was expected to be part of a teacher's store of arithmetic knowledge.

We Americans Hate Despots—those who want us to obey, not to think. Or do we? Do those of us who turn the knob on our TV know why it works? Are despots always human? May we become slaves to mechanical devices without realizing it? What is the price we pay for the privilege of getting results without the toil of thought.

Today, Perhaps More Than ever before, we at least pay lip service to teaching for understanding in arithmetic. Too often, however, this “ understanding” still implies mere knowledge of how to perform a mechanical process. Checking any one of the four processes of arithmetic by casting out of nines is a good illustration of this fact; though, occasionally some wide awake boy or girl in junior high school arithmetic challenges the teacher by asking such questions as: “Why does it work?” “Will it work with any other number?” These are good, intelligent questions, but questions which many teachers who feel rather secure in teaching most of the arithmetic find difficulty in answering. And yet, at least partial answers are understandable, interesting and challenging to pupils.

One of the most difficult pedagogical exercises in teaching college mathematics is finding ways of showing students how professional mathematicians really work. Many college-level texts written in the "theorem-proof-corollary-remark" style help convince students that mathematics is actually discovered in the same way. At least somewhere along the line, students should be exposed to, and preferably experience for themselves, the search for pattern, the tentative generalizing, the further search for confirming or counter-examples, and the hypothesizing of a result, which in actual fact comprises a large part of the professional mathematician's work. Ideally, they should then continue on and confirm the result by proving it, or seeing it proved, deductively. Various parts of the history of mathematics are well suited to illustrate the above process: See, for example, the brilliant discussion of the development and proof of Euler's theorem for polyhedra in Lakatos [1]. However, it is much more satisfying for the student to be able to go through the process himself. This paper describes an investigation that may be undertaken with college students for this purpose. The structure is interesting, yet the ultimate deductive proof is short and simple enough for most students to be comfortable with. Sloane [2] has multiplied the digits of a number together to obtain a second number, then multiplied the digits of this second to obtain a third, etc. He defines the (multiplicative) persistence of a number as the number of iterations of this process required to produce a single digit number from the original one. (This paper is concerned exclusively with positive, whole, base 10 numbers.) For example, 76 has a persistence of three because it requires three steps to reduce it to one digit: 76 -*42 -*8. Of course, changing multiplication to addition in the above produces the old operation of "casting out nines." A similar but perhaps more interesting operation may be defined as follows: Form a new number by squaring and then adding the digits of the old number. We will call this the "sum of square digits" or "SSD" operator, and denote it by -*. Thus, for example, 145 12 + 42 + 52, or 42.