As was our previous post, this one concerns a very small but very telling detail of the new mathematics curriculum. A minor perversion of the curriculum is the renaming of the study of geometry as “Space”. This stupidity was noted by AMSI last year, in their submission on the draft curriculum:

*We believe that the use of ‘Space,’ for the title of a content strand will be confusing to schoolchildren and indeed, teachers, who are likely to associate the term with astronomy. We think that the strand title ‘Geometry’ is mathematically appropriate. We reiterate the importance of accuracy in mathematical language.*

AMSI could work on their comma placement, but working on their emphases is probably more important. Yes, ACARA’s choice of ‘Space’ as a strand title was stupid, but it was also stupid for AMSI to whine about this when there were many whale-sized fish to fry. Of course, ACARA was more than happy to waste precious time and attention by batting back this trivia (at 2.3):

*The [ACARA] team has proposed the term “Space” as a broader characterisation of the field. It should be noted that the National Statement on Mathematics for Australian Schools (1991) used Space as a strand name. Subsequently the term was used in mathematics curricula in this country for two decades, without any apparent confusion with astronomy including NAPLAN Minimum standards – numeracy.*

ACARA was substantially correct, at least when playing on the terms that AMSI chose. “Space” is just a name, and a stupid strand by any other name would be just as stupid. It is ironic that ACARA chose to cite the National Statement from 1990 (not 1991), since this Statement was universally condemned for its mathematical illiteracy. Nonetheless, a name is just a name.

That doesn’t explain why ACARA bothered to change the name. The real problem with “Space”, as we have argued (at 2.3), is that it is not simply a new, alternative name for geometry. On the contrary, it is clear that “Space” was intentionally chosen by ACARA in order to signal a real-world emphasis directly at odds with the proper, abstract study of geometry. The name “Space” is poisonous precisely because it is not just another name. Here, we want to add one more detail to this name nonsense, to demonstrate ACARA’s ideological obsessiveness.

ACARA’s content descriptor codes are long. For example, one such code is

AC9M5N03.

The “5N03” is the helpful bit, indicating that we are looking at the 3rd Number descriptor at the Year 5 level. The other characters, indicating that we’re looking at Version 9 of the Mathematics component of the Australian Curriculum, feels a bit clumsily overkill, but bureaucrats like that kind of thing. So, sure, if they don’t mind their codes being eight characters in length then let them have eight characters.

Or nine characters.

As it happens, “Statistics” and “Shape” begin with the same letter. Which means we cannot employ a code such as AC9M5S03, since this would be ambiguous. Instead, we must use

AC9M5SP03

and

AC9M5ST03.

Which is untidy and very easy to misread. And which was entirely unnecessary: if ACARA had done nothing, had just stuck with the name “Geometry”, the coding would have simply worked.

But that’s ACARA: never let common sense get in the way of ideological obsession.

This really annoys me. The discipline is geometry, not space. There is no space in the mathematics classification codes, which covers all of mathematics. It’s perverse. And yes I agree that it is probably done only to impose some completely unnecessary “real world”-ness. Ugh.

Yes, it’s definitely a stupid name but the name is also a stupid distraction. Your last sentence is the important point.

For me the most distressing aspect, regardless of the title, is the complete absence of the word “prove” in the Year 7 curriculum. Geometric facts are just stated , “demonstrated” or “investigated with technology” but never actually proved, or more interestingly, explained as axioms. In my experience teaching Year 7 in all types of schools with children of all levels of readiness for high school mathematics, students absolutely embrace their first experience of geometric proof, when the process is well explained and they are given the chance to engage with it. Issues with formal language can easily be overcome by starting with coloured diagrams (Oliver Bryne’s 1857 Euclid’s Elements is a terrific resource). Young people totally understand the challenge of “but how to do you know that is true ..?” as many have younger siblings who continually ask “why?” and won’t accept answers without asking “why?” again and again :-). Students are fascinated by the process of deductive reasoning based on axioms – a special gift they get only in mathematics class. And it’s a short, very exciting, step to ask “What if we changed the axioms?” and look at some other types of geometry and their surprising applications. The failure of the Year 7 curriculum to engage and inspire even the most rudimentary experience of proof is a real missed opportunity.

Thanks very much, ENZ. You are absolutely correct, both in the lack of proof and in the general lack of logical structure. The predominance of fuzzy words is telling. It is a huge failing of the new (and old) AC.

The proofs thing is on my Cur To Do List, twice. To be upfront, I’m flat out gunning for non-ACARA villainy right now, and so am just keeping things ticking by whacking the low-hanging ACARA fruit. There is a number of bigger and more important targets on my ACARA list, but they will take more time and care.

Interestingly, for me at least, I seem to be an outlier and don’t care that much for axiomatic geometry. I didn’t care much for it as a kid, and I don’t see it as that important now. But I know lots of smart guys do, so I assume I am wrong. But, however and wherever it is done, an appreciation of the fundamental structure of defining and assuming and proving must be nurtured. It is not.

The reason I like teaching axiomatic geometry is that it is a an accessible and fun entry point to rigorous proof and reasoning. It can be nice when paired with an introduction to formal logic. Problems with numbers and sets are also fine of course, but there is just something intrinsically appealing to drawing pictures and distilling the right words from a geometric argument.

As I said, smart people disagree with me.

Glen,

Would I be correct in assuming axiomatic geometry was successfully collated by Euclid in his elements

around 300 BCE? Presumably not all his own ideas though?

I have inherited a translated copy of this text and find the wording somewhat complicated but logically fascinating

as a large set of axioms are built from a few postulates

Regards

Steve R

Hi Steve, I actually don’t know for sure when various things happened and to whom they should be credited. I think it may still be a matter of research with math historians and I’m ignorant of it.

I do like the content though! Discussions are dropping the parallel postulate are fun. Especially if you’ve been able to get up to the sum of angles in a triangle, because you can instead suppose that parallel lines meet at infinity and end up in a different geometry. I hope this kind of knowledge becomes more popular to teach because I’d hate for it to be completely supplanted by the more modern approaches.

Glen, Yes; you are correct in making this assumption. Heath’s three volume translation, published by Dover, contains many notes and comments. (Do you realise that the proof of Proposition 1 is incorrect?)

A clever on-line version is here:

http://aleph0.clarku.edu/~djoyce/elements/elements.html

A modern translation, with the Greek, is here:

Click to access Elements.pdf

I used this last version when I taught students in PNG about Euclid’s Elements.

BTW, I am warming to “space” instead of “geometry”.

In his book “Elementary mathematics from an advanced standpoint”, Felix Kilne emphasises the importance of capturing the interest of the students, especially in the early years. “A more abstract presentation will be possible only in the upper classes” (p. 3). It’s a good read – and still in print thanks to Dover.

Click to access Felix%20Klein%20-%20Elementary%20Mathematics%20From%20an%20Advanced%20Standpoint_%20Arithmetic%2C%20Algebra%2C%20Analysis%20%281945%29.pdf

What a terrific read. Thank you!

Thanks, Terry. Very interesting. I’ve never properly looked at Kilne (otherwise known as Klein), but clearly worth the effort. i’m also curious about Weber-Wellstein.

Maybe the curriculum writers are harking back to the days of Newton, Leibniz, and Kant when space and time were regarded as fundamental concepts.

They’re harking back further than that. There’s something Medieval about ACARA’s “It’s the vibe” framing.

Well, it was a long shot.

Is ‘Space’ really more “real-world” then geometry? The mathematics of Earth-Measurement seems more concrete than just a place for things to be.

Hi, Simon.

Do you mean is the curriculum strand of Space more real-world than a traditional presentation of geometry? Yes.

I don’t understand your second sentence.

I was addressing your line “it is clear that “Space” was intentionally chosen by ACARA in order to signal a real-world emphasis directly at odds with the proper, abstract study of geometry”

But if you at the words:

Space comes from / means “room, area, distance, stretch of time” – it then got applied to “outer space” as there is so much space out there. But it doesn’t really imply anything other room.

Geometry literally means measuring land – and the mathematics associated with measurement of shapes on that land.

To me, “geometry” seems much more concrete and practical than the amorphous “space”.

Refs: https://www.etymonline.com/word/space, https://www.etymonline.com/word/geometry

Hi, Simon. I don’t think the origins of a word, or of a discipline, necessarily tell you much about the current nature of the discipline. “Geometry” doesn’t now mean measuring the Earth, or necessarily measuring much of anything. Particularly when there is a whole other strand called “Measurement”.

The fact remains, ACARA chose to change the name from “Geometry” to “Space” for a reason. What reason? They chose to do that even though the “S” of “Space” conflicted with the “S” of “Statistics”. They stuck by their decision even though AMSI objected to the name.

ACARA really really wants the name “Space”. Why? What non-idiotic reason can there be?

I won’t even try to read the mind of ACARA in making this change. However…

Immanuel Kant (and others of that time) regarded mathematics as being based on the two basic notions of time and space. Time was measured in a linear fashion, which leads to the number line, which leads to arithmetic, which leads to algebra. The notion of space leads to Euclidean geometry.

So, from this very old-fashioned point of view, I am warming to the term “space”.

From a modern perspective, “space” is used in many parts of mathematics: e.g. metric space, topological space, vector space, Hilbert space, Banach space. All of these have a basis in geometry.

Not the point. A good point, and not the point.