A

*function*is a rule that maps each item in a domain (the starting set) to one and only one member of a codomain (the finishing set).So, for example, the rule 'add 1' with domain and codomain the set of natural numbers, {1, 2, 3, 4, 5, ...}, which can be written as

*n*➝*n*+ 1, is a function. For each natural number there is one and only one natural number obtained by adding 1 to it (for example, 42 ➝ 43). This is the rule whereby we generate the 'next' number when counting on in 1s. Since 'nextness' is a fundamental property of the natural numbers, I would regard this as a 'core arithmetical function'. The same goes for 'subtract 1' (*n*➝*n −*1), which is the basis of the process of counting back in 1s and the concept of the 'previous' number. Significantly we need to include zero in the codomain in this case, since 1 ➝ 0.So, to 'master core arithmetical functions' must include being able to count on and back in 1s and being able to state the next and the previous natural numbers for any given whole number.

Then, since 10 is the base of our number system, we must also give prominence to the functions

*n*➝*n +*10 and*n*➝*n −*10 (for*n*≥ 10), and*n*➝*n +*100 and*n*➝*n −*100 (for*n*≥ 100). So, to master core arithmetical functions must include being able to count on and back in 10s and in 100s, starting with a given natural number. For example, we would expect a primary school child to start at 57 and count on in 10s (57, 67, 77, 87, 97, 107, and so on) or to count back in 10s (57, 47, 37, 27, 17, 7).Well, that's a start. More mastery of core arithmetical functions next week – if I have time. It's a busy week for me leading up to the Sunday before Christmas. With my musical and Christian hats on, I have a choir to knock into shape for our carol service on Sunday evening (19th December), so we can make a joyful noise together in celebration of the birth of Christ.

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