A few weeks ago, we posted on a klutzy index law elaboration in the new curriculum:

*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):

**natural numbers**

*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:

**even numbers**

*An integer that is divisible by 2. The even natural numbers are *

Idiots.

Based on my experience, it seems that most secondary students think that all numbers are positive integers.

I have nothing to add here except to say that I would have gone with the title ‘Unnatural Selection’ (maybe it was on your shortlist?) Perhaps the ACARA curriculum is in the running for a Darwin Award. We can only hope.

Happy and safe Christmas and New Year to you Marty and to all readers.

I’m not against anyone who defines zero as a natural number (well, some people who define 0 as a natural number I actively avoid, but that is for other reasons) but I will argue that starting the set of natural numbers at 1 just feels more… natural.

The set does come up a fair bit too, in number theory which, perhaps wrongly, I like to use to introduce ideas of , and related ideas to students who are yet to experience much in the way of proper algebra.

Perhaps the only saving grace here is that ACARA is not writing anything which is externally assessed (for now).

When there are several commonly accepted conventions/definitions, it is appropriate/necessary to declare which one you will use. Pick and stick. There is an unwillingness by our education ‘authorities’ to do this. ACARA clearly don’t want to to do it.

(I won’t turn this into a VCAA bash by noting its unwillingness to declare a convention for the pseudocode that it has poisoned the new Study Design with).

Except that you did.

At least ACARA has made an attempt at a definition…

VCAA’s “glossary” does not come close.

So, as much as I don’t like ACARA’s documents (attacking the written word, not the author) the fact that they have made an effort is worth something.

Perhaps.

I think we’ll have to define “attempt”.

Indeed, There are several commonly accepted conventions/definitions …

Ha! Love it!

But unlike some organisations which are best known by acronyms, I will admit the mistake.

As your lawyer, I advise that you admit nothing. Settle privately and make sure a confidentiality agreement is signed.

Thank you Marty, the clearest and most useful articulation of the matter I have seen in a long lifetime of Maths learning and teaching! And thank you for all else, best New Year’s wishes, Ian