2: Arithmetic

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A child of the new generation Refused to learn multiplication He said, “Don’t conclude That I’m stupid,, or rude. I am simply without motivation.”

Joel Henry Hildebrand (1881–1983)

Many important aspects of serious mathematics have their roots in the world of arithmetic. This is a world everyone can enjoy and master. In this chapter we re-visit, or maybe meet for the first time, key aspects of arithmetic that are often overlooked – ending with an introduction to the basic result on the distribution of primes.

The place of arithmetic in elementary mathematics can only be understood if one realises that, from upper primary school onwards, mathematics should no longer focus on more and more complicated calculations. Rather it moves beyond a set of procedures for grinding out answers, and should become a structural laboratory, where we gain insight into simple phenomena, and where we begin to appreciate how calculation can be managed, or tamed. The focus on structure leads in the main to matters which can be best expressed algebraically. This chapter concentrates mainly on structural aspects of number that are strictly arithmetical (e.g. related to numerals and place value), or where the relevant structural approach is “pre-algebraic” – with occasional forays into the world of algebra.

We repeat the observation that the “essence of mathematics” in the title is mostly left implicit in the problems. And while there is some discussion of this “essence” in the text between the problems, most of the relevant observations that we make are to be found in the solutions, or in the Notes which follow many of the solutions.