using examples such as , and to illustrate the necessity that for any non-zero natural number 𝑛, (new AC9M8N02)
Some commenters were understandably puzzled by a side point: ACARA’s employment of the expression “non-zero natural number”. In this post, we’ll dispel any lingering lack of puzzlement.
The underlying question is whether the set of natural numbers begins with 1 or with 0. We have always been in the 1 Camp, but there is not much sense in any strong commitment: both camps are sufficiently populous that there is little point claiming or attempting to establish some orthodox position. This unfortunate truth is reflected in ACARA’s glossary (Word, idiots):
The set or depending on whether counting is started at 0 or 1. The elements of are also called the counting numbers, used to count the number of elements in finite sets.
The wording is clumsy, and not quite grammatical, but it gets the point across. Which ACARA then forgets, immediately.
What to make, for instance, of the elaboration above? There is some sense to the wording, in that it is consistent with either use of “natural number”. Nonetheless, the wording is jarring for those in the 1 Camp, and any confusion could have been avoided by using instead the phrase “positive integer”. In any case, there is worse.
The curriculum contains twenty-two content descriptors and elaborations that employ the phrase “natural number(s)”. For half of these items it is unclear and unimportant whether 0 is considered to be a natural number. In three items, including the above elaboration, it is made explicit that 0 is (or at least may be) taken to be a natural number. The remaining eight items are well illustrated by the following elaboration:
solving problems involving lowest common multiples and greatest common divisors (highest common factors) for pairs of natural numbers by comparing their prime factorisation (AC9M7N02)
Here, it is actively confusing to consider “natural numbers” to include 0, but ACARA is silent. There are eight such items.
It is annoying that “natural number” does not have a universally accepted meaning. This is not ACARA’s fault. It is ACARA’s responsibility, however, to address the double-meaning issue properly, not simply with a token and ignored glossary entry. ACARA could have done this in various ways.
The brutal approach would have been to avoid entirely the use of “natural number”, and simply refer to “positive integers” or “non-negative integers” as appropriate. That’s handcuffingly censorious, however, particularly for the early years, before negatives have been introduced. Notably, the curriculum first refers to “integers” beginning in Year 6; this feels sensible, even if the terminology is only intended for primary teachers. Still, once “integers” have been introduced, the term could have been put to better use.
A second approach would have been for ACARA to have bitten the bullet, to have defined “natural number” for the purposes of the Australian Curriculum. You know, as might be done by a national body empowered to guide understanding and national practice. But of course, as ACARA’s lead propagandist David de Carvalho has noted, the Curriculum’s “potential variability” is “a reflection of one of the strengths”.
A third approach would have been to have left “natural number” as ambiguous, but to use the expression with care, and to clarify when required. ACARA’s quarter-hearted attempt is exemplified by the index law elaboration, above.
But of course ACARA went primarily with a fourth approach: write a bunch of stuff without any sense of or concern for internal consistency or external usefulness. Just to hammer the fish once more, here is a further glossary entry, one of many similar:
An integer that is divisible by 2. The even natural numbers are