Strike it out—add it up

Abstract

If you were to ask a random sample of students from your school if they liked mathematics, what do you suppose their responses would be? Almost certainly most of them would respond negatively and some would do so with considerable vigour. But what of the others? Some students do like mathematics and we might well ask why this is so? Do you suppose it is because these students realise that knowledge of mathematics is a necessary survival skill in today’s increasingly technological society? Surely the answer is no. Do you suppose it is because they realise that mathematics has many useful real life applications or that lack of knowledge of mathematics will effectively prevent them for entering college in most disciplines? Again the answer is almost surely no. Certainly mathematics is relevant and we should teach our students that this is so and we should teach them useful and appropriate applications as well. At the same time, it is my experience that student attitudes towards mathematics are much like they are towards exercise—in general, they do not exercise because it contributes to good health, but because exercising is usually done in the form of a game or a competition that they find attractive in and of itself. Similarly, what turns students on to the study of mathematics and provides the needed motivation for learning are the remarkable patterns and surprising results with which our subject is so replete. Successful teachers realise this fact and utilise it to maintain a high level of interest and enthusiasm which enables them at the same time to teach ideas which, while useful and necessary, may be more prosaic.

The Strong Law of Small Numbers

“…6). 17,18,19, and 20 are examples of Moessner's process, which does indeed produce the square, cubes, fourth powers and factorials. Moessner's paper [20] is followed by a proof by Perron.…”

“…Moessner's paper [20] is followed by a proof by Perron. Subsequent generalizations are due to Paasche [22]: see [19] for a more recent exposition. 21.…”

Moessner’s Theorem: An Exercise in Coinductive Reasoning in Coq

“…As a result, our formalization corresponds well to the paper by Niqui and Rutten, and is very compact. -Lastly, in the process of formalizing Niqui and Rutten's proof, we uncovered a simple proof of Long [10,11] and Salié's [20] generalization. Though (once done) the generalization is not at all complicated, it was surprising to us that the extended version is just a corollary of the original Moessner's Theorem, and that the the bisimulation did not have to be modified.…”

“…Long [10,11] and Salié [20] generalized Moessner's result to apply to the situation in which the initial sequence is not the sequence of successive integers (1, 2, 3, . .…”

“…The first proof was given shortly thereafter by Perron [17] (who, curiously, was the editor of the journal where the conjecture was submitted). The theorem was then the subject of several popular accounts and generalizations [4,8,11,20,14,15,16].…”