A month or so ago, we posted on Euclid et al, asking about the proper role of proof in a mathematics curriculum. The question and subsequent discussion was purely theoretical of course, since proof barely exists in the Australian Curriculum. Here’s the proof.
The following is the entire list of content descriptors and (optional) elaborations we could find that indicate in any half-way explicit manner students engaging with the proofs of theorems. (There may be a few others, hidden somewhere in the swamp.)*
YEAR 8
use Pythagoras’ theorem to solve problems involving the side lengths of right-angled triangles (AC9M8M06)
discussing and comparing different applications, demonstrations and proofs of Pythagoras’ theorem, from Egypt and Mesopotamia, Greece, India and China with other historical and contemporary applications and proofs
identify the conditions for congruence and similarity of triangles and explain the conditions for other sets of common shapes to be congruent or similar, including those formed by transformations (AC9M8SP01)
applying logical reasoning and tests for congruence and similarity, to problems and proofs involving plane shapes (AC9M8SP01)
establish properties of quadrilaterals using congruent triangles and angle properties, and solve related problems explaining reasoning (AC9M8SP02)
establishing the properties of squares, rectangles, parallelograms, rhombuses, trapeziums and kites using geometric properties and proof, such as the sum of the exterior angles of a polygon is equal to a complete turn or 360° (AC9M8SP02)
YEAR 9
solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles (AC9M9M03)
investigating theorems and conjectures involving triangles; for example, the triangle inequality and generalising links between the Pythagorean rule for right-angled triangles, and related inequalities for acute and obtuse triangles; determining the minimal sets of information for a triangle from which other measures can all be determined
design, test and refine algorithms involving a sequence of steps and decisions based on geometric constructions and theorems; discuss and evaluate refinements (AC9M9SP03)
developing an algorithm for an animation of a geometric construction, or a visual proof, evaluating the algorithm using test cases
YEAR 10
apply deductive reasoning to proofs involving shapes in the plane and use theorems to solve spatial problems (AC9M10SP01)
distinguishing between a practical demonstration and a proof; for example, demonstrating that triangles are congruent by placing them on top of each other, as compared to using congruence tests to establish that triangles are congruent
developing proofs involving congruent triangles and angle properties, communicating the proof using a sequence of logically connected statements
applying an understanding of relationships to deduce properties of geometric figures; for example, the base angles of an isosceles triangle are equal
investigating proofs of geometric theorems and using them to solve spatial problems; for example, applying logical reasoning and similarity to proofs and numerical exercises involving plane shapes; using visual proofs to justify solutions
using dynamic geometric software to investigate the shortest path that touches 3 sides of a rectangle, starting and finishing at the same point and proving that the path forms a parallelogram
OPTIONAL YEAR 10
exploring how deductive reasoning and diagrams are used in proving geometric theorems related to circles
*) The Curriculum contains many references to “demonstrate/ing” this or that. Whatever the intended meaning, however, it is clear that “demonstration” is generally being used to indicate some argument or illustration that may fall well short of a proof: see, for example, AC9M8M06 and AC9M10SP01, above. But, just to add a little more mud to the mire, the Curriculum glossary defines “proof” as “A mathematical argument that demonstrates whether a proposition is true” (emphasis added).
‘using dynamic geometric software to investigate the shortest path that touches 3 sides of a rectangle, starting and finishing at the same point and proving that the path forms a parallelogram’
Any idea what this means? I would have interpreted ‘the shortest path that touches 3 sides of a rectangle, starting and finishing at the same point’ to describe a triangle. Maybe they mean a path that starts on a particular side of the rectangle, touches the 3 other sides, before returning to that point? Maybe it’s a typo, and they meant to say 4 sides? Maybe I’m missing something?
Yeah, I couldn’t make sense of it either. Which also makes “proving” it a little tough.
Indeed. I doubt ACARA expects students to be ‘distinguishing between a practical demonstration and a proof’ in this one…
APS,
It’s far from clear but…
…I think they might mean start at point P
Within a square and prove shortest path
From P to touch 3 Sides and return to P
is a parallelogram
Eg. https://math.stackexchange.com/questions/4103108/find-the-shortest-path-from-a-point-inside-the-square-touching-its-3-sides
Steve R
Why do y’all use such fancy words? Does it make you feel smarter and better and richer? Does it cover up for something?
My brain turns off when I see that pompous edubloat standards writing. Instead of dressing stuff up with fancy words, try to actually think about what you do and what you skip and just say so. COME UNE IH CATE!
God yeah. Can’t pick a worst one but ‘exploring how deductive reasoning and diagrams are used in proving geometric theorems related to circles’ is up there. They mean ‘prove circle theorems’ (I hope), but they’ve added so much extra verbiage (‘exploring how deductive reasoning and diagrams are used’) that, not only have they made the sentence more confusing, but they have actually changed its literal meaning.
Taken literally, students will apparently focus on the nature of deductive reasoning, or the history of diagrams…
So, if I may take a puzzled eye to this… proof exists in the Australian curriculum, but only in Geometry.
Geometry has pretty much gone from all VCE Mathematics (OK there are lines and planes in 2D and 3D in Specialist as well as transformations in Methods and Specialist, but I struggle to call that Geometry) so… is there any need (either of the carrot or stick variety) for Years 8 to 10 teachers to cover these proofs in any great detail…?
You mean any need other than to avoid the wrath of the God of Mathematics? Not that I can see.
In SA Geometry is a single topic in Stage1 (year 11) Mathematics and is normally only done by students who are intending to study Specialist Maths in stage2 (year 12). It has been dropped from stage 2, and no Geometric proof occurs in stage2. Stage 2 Specialist does include proof by induction.
Good teachers will ignore the nonsensical BS in the AC and VC and attempt to teach their students actual mathematics.
1. Yes. 2. Easier said than done.
3. Agree, hence the qualifier ‘attempt’
I make an attempt with my Year 9 students with questions such as “Prove that the sum of two rational numbers is always a rational number.”