Tag: Geometry

ACARA Crash 13: The Establishment Blues

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June 28, 2021

7:27 pm

24 Comments on ACARA Crash 13: The Establishment Blues

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(We had thought about destroying another song, but decided against it. Still, people should stop to the listen to the great Rodriguez.)

The following are content-elaboration combos from Year 6 and Year 7 Measurement.

CONTENT (Year 6 Measurement)

establish the formula for the area of a rectangle and use to solve practical problems

ELABORATIONS

solving problems involving the comparison of lengths and areas using appropriate units

investigating the connection between perimeter and area for fixed area or fixed perimeter, for example, in situations involving determining the maximum area enclosed by a specific length of fencing or the minimum amount of fencing required to enclose a specific area

investigating the relationship between the area of a parallelogram and the area of a rectangle by rearranging a parallelogram to form a rectangle of the same area and explaining why all parallelograms on the same base and of the same height will have the same area

CONTENT (Year 7 Measurement)

establish the formulas for areas of triangles and parallelograms, using their relationship to rectangles and use these to solve practical problems using appropriate units

ELABORATIONS

exploring the spatial relationship between rectangles and different types of triangles to establish that the area of a triangle is half the area of an appropriate rectangle

using dynamic geometry software to demonstrate how the sliding of the vertex of a triangle at a fixed altitude opposite a side leaves the area of the triangle unchanged (invariant)

using established formulas to solve practical problems involving the area of triangles, parallelograms and rectangles, for example, estimating the cost of materials needed to make shade sails based on a price per metre

CONTENT (Year 7 Measurement)

establish the formula for the volume of a prism. Use formulas and appropriate units to solve problems involving the volume of prisms including rectangular and triangular prisms

ELABORATIONS

packing a rectangular prism, with whole-number side lengths, with unit cubes and showing that the volume is the same as would be found by multiplying the edge lengths or by multiplying the height by the area of the base

developing the connection between the area of the parallel cross section (base), the height and volume of a rectangular or triangular prism to other prisms

connecting the footprint and the number of floors to model the space taken up by a building

representing threefold whole-number products as volumes, for example, to represent the associative property of multiplication

using dynamic geometry software and prediction to develop the formula for the volume of prisms

exploring the relationship between volume and capacity of different sized nets used by Aboriginal and Torres Strait Islander Peoples to catch different sized fish

exploring Aboriginal and Torres Strait Islander Peoples’ water resource management and the relationship between volume and capacity

 

WitCH 61: Wheel of Misfortune

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May 5, 2021

1:42 pm

23 Comments on WitCH 61: Wheel of Misfortune

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We decided ACARA’s psychedelic circle, which we discussed briefly here, was worthy of its own post. So, apart from the potential to trigger an epileptic fit, what is wrong with ACARA’s wheel?

ACARA labels their wheel as depicting

[the] relationship between the six strands and three core concept organisers

The six strands make up the inner wheel, and are explained here (pages 4-5). The three “core concept organisers” are the three colours on the wheel rim, with the categories and subcategories as pictured below, and which are explained here (pages 5-8).

Being Carvalho With the Truth

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April 30, 2021

12:21 am

7 Comments on Being Carvalho With the Truth

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There are bigger, stupider fish to fry, but this also needs to be done, and it’s now or never.

Earlier this week, David de Carvalho, the CEO of ACARA, gave a speech at The Age Schools Summit. flagging the Daft Australian Mathematics Curriculum (the spelling is correct). Our most recent WitCH is based on an excerpt of de Carvalho’s speech, but the entire speech — also condensed to an op-ed — is worthy of scrutiny. And so, here we are.

Below are parts of de Carvalho’s speech that relate in some manner to the mathematics review, followed by our, mostly redundant, comments. What de Carvalho says is clear, and clearly ridiculous.

The data that we get from NAPLAN is important, but it is not a measure of overall school quality, and we need to remember that education is a complex multi-dimensional process where improvement does not rely on the rationalistic analysis of data and the application of associated managerial techniques.

“Rationalistic” is now a boogie-man word? That’s where we are? And once again, we have this idiotic selection: NAPLAN or touchy-feely. A pox on them both.

For the thousandth time, NAPLAN is garbage. Numeracy is garbage. And, while we’re at it, PISA is garbage. If there were decent national tests of arithmetic and mathematics, these would be a godsend; the subsequent “rationalistic analysis” would do more than anything else imaginable to lift Australian mathematics education. But, of course, courtesy of two very poxy houses, there’s not a snowflake’s chance in Hell of this occurring.

This is because the essential nature of education is that it involves a sense of historic continuity and conversation between generations, between teachers and their students, where a learner is engaged in the process of becoming a well-rounded human being …

This conversation between the generations is the basis of the curriculum.

An A+ for irony. If there is one central theme to the now decades of destruction to Australian education, it is an utter contempt for the past, the active denigration and wilful forgetting of Australia’s one-time educational excellence.

Up to now, we have been working with 18 teacher and curriculum reference groups established to support the review, made up of 360 teachers and curriculum authority representatives from across Australia, as well as consulting with our peak national subject bodies and key academics. I’ve also had discussions with the staff from 24 primary schools across the country.

That’s a lot of people. What could go wrong?

But on Thursday, we begin public consultations on the proposed revisions to the Foundation to Year 10 Australian Curriculum.

No, you aren’t. You are permitting people to fill in a survey, which, presumably, will never see the light of day. In no manner does this sham hold ACARA publicly accountable to publicly available opinion.

I’ve heard some stakeholders say that we should be “taking a chainsaw to the curriculum”, but chainsaws are not particularly subtle, and can leave an awful mess behind.

What is currently wrong with the Australian Curriculum is also not subtle. We’d go for dynamite, but a chainsaw would do in a pinch.

We need to avoid perpetuating a false dichotomy between factual knowledge and capabilities such as collaboration and critical thinking.

Yes, …

You can’t engage critically and creatively on a topic if you lack the relevant background knowledge …

Go on, …

So the ability to recall facts from memory is not necessarily evidence of having genuine understanding.

A very cute straw man, but please continue.

A student might, for example, memorise the formula for calculating the length of the hypotenuse if given the length of the other two sides of a right-angle triangle, but do they understand why that formula, known as Pythagoras’ theorem, works every time? The process of discovering that for themselves, with the assistance of the teacher, is what makes learning exciting. And it’s what make teaching exciting. Seeing the look of excitement on the face of the student when they experience that “aha!” moment.

Ah, so that’s what you’re getting at. You nitwit.

This example is so monumentally, multidimensionally stupid, it has its own post.

Some learning areas have only required some tidying – but others have required more focus. Maths for example has required greater improvement and updating.

If we look at our PISA and our TIMSS results, Australian students are not bad at knowing the “what”, but we are not as strong at “why” of mathematics, …

This is, to use the technical term, a fucking lie. It is an absurd, almost meaningless dichotomy. And you forgot the “how”, you idiot. But in any case Australian students absolutely suck at the “what”, and the “how”, and ACARA damn well knows it.

This is THE lie. This pretence, that Australian maths students are doing just fine on the basic knowledge and skills, is why it is absolutely impossible for ACARA to revise the mathematics curriculum in a meaningful manner. This is why, without even needing to read it, you know the Draft Curriculum is a farce.

… this joint statement from five of the leading maths and science organisations in the country is an indication of the level of interest out there about the curriculum.

Yep, “five of the leading maths and science organisations” can endorse this poisonous twaddle. That’s where we are.

Again – knowledge and capabilities being acquired together.

Mate, it doesn’t work that way. More often than not, and particularly in primary school, your precious “capabilities” are just getting in the damn way. You can wish it all you want, but learning mathematics just doesn’t work that way.

It should be noted that the current Australian Curriculum in Mathematics already includes four Proficiency Strands: understanding, fluency, problem-solving and reasoning. The issue has been that these proficiencies have not been incorporated into the Content Descriptions, which is what teachers focus on.

So the major change we have made in the proposed revisions to Maths is to make these proficiencies more visible by incorporating them into the Content Descriptions.

In other words, you are changing the Curriculum content, and you are pretending that these changes are simply an innocent act of tidying up. Do you imagine anyone believes you?

It is also important to note that in proposing these revisions, ACARA is not making any recommendations about pedagogical approaches. 

So that Pythagoras nonsense back there was just for the Hell of it? There’s a technical term for this. What was it again?

I expect we will see a stirring of the passions.

So you’re ok with passions. How do you feel about white-hot fury?

No doubt some will argue the proposed revisions don’t go far enough, while others will say they go too far, …

Of course it never crossed your mind that people might say, accurately, that you’re going in entirely the wrong fucking direction.

This discussion and civil debate is a good thing.

It is not a discussion, it is a fait accompli, and there is no benefit here in being civil. What is called for is complete and utter contempt.

WitCH 60: Pythagorean Construction

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April 28, 2021

11:35 pm

77 Comments on WitCH 60: Pythagorean Construction

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Yesterday, David de Carvalho, the CEO of ACARA, gave a speech at The Age Schools Summit. de Carvalho used his speech to set the stage for ACARA’s imminent launch of the draft of the revised Australian Curriculum. (See here.)

We shall take a more careful look at de Carvalho’s speech in the near future. For now, we’ll settle with a WitCH, an excerpt from de Carvalho’s speech:

“So the ability to recall facts from memory is not necessarily evidence of having genuine understanding.

A student might, for example, memorise the formula for calculating the length of the hypotenuse if given the length of the other two sides of a right-angle triangle, but do they understand why that formula, known as Pythagoras’ theorem, works every time? The process of discovering that for themselves, with the assistance of the teacher, is what makes learning exciting. And it’s what make teaching exciting. Seeing the look of excitement on the face of the student when they experience that ‘aha!’ moment.”

And when we understand a topic, it is easier to recall the facts because they are no longer just random bits of information but are organised into intelligible ideas. Not only do we know where the dots are, but we know why they are there and how to join them.”

UPDATE (29/04/21)

Just a reminder, this is a WitCH. So, what specifically is crap about de Carvalho’s Pythagoras suggestion?

To guide the discussion, below is (arguably) the simplest, algebra-free proof of Pythagoras, which, undeniably, all students should see. Where does/should this proof fit in with the teaching of Pythagoras, and where does de Carvalho’s suggestion fit in with either of these?

UPDATE (03/05/21)

It is extremely helpful of De Carvalho to have selected such a fundamentally flawed example. If he had chosen more judiciously it would be more work to counter, to make the case here against inquiry-based learning. De Carvalho having chosen almost an anti-example, however, makes clear that ACARA’s inquiry push is nothing like a reasoned best-choice approach in given situations, and much more a religious fundamentalism: inquiry is, simply by being inquiry, the preferred method.

De Carvalho contrasts the WHAT of Pythagoras — the equation — with the WHY, the proof (or proof-like evidence for) that equation. Putting aside for the moment De Carvalho’s atrocious suggestion for getting to the WHY, we’ll first note that De Carvalho has failed to ask two fundamental questions:

  • HOW does Pythagoras’s theorem work?
  • WHY2 do we teach Pythagoras’s theorem?

The first question is about the HOW of the mechanics of dealing with the equation \boldsymbol{a^2 + b^2 = c^2}: manipulating for the unknown, taking roots and so on. This HOW is not glamorous, and not intrinsically difficult, but it is fundamental. Of course the application of Pythagoras’s theorem, including the mechanics, is in the draft curriculum (beginning in Year 8), but De Carvalho’s casual airbrushing of the HOW is a tell.

(As  a side note, we’re pleased to see that the single stupidest line in the Australian Curriculum is still there, in the Pythagoras elaborations.)

And now, the second question: WHY2 teach Pythagoras’s theorem? There are two strong answers to this question, both of which, in different ways, demonstrate that De Carvalho has no understanding of his example.

The first reason, the WHAT to teach Pythagoras is because it is so important: it is the fundamental formula for distance in Cartesian (and Euclidean) geometry.

The second reason, to teach the WHY of Pythagoras’s theorem is because it is a historical icon and because it is so beautiful. The proof illustrated above is gorgeous, it can easily be learned by a primary school student, and it should be learned, by all students, as one learns a poem. (One can of course extend this to be a third reason, to teach other proofs and the nature of geometric proof.)

Here is why these reasons show De Carvalho’s example to be so empty:

  • The two reasons for teaching Pythagoras are almost totally disconnected.
  • Hunting for a proof is the absolute worst way to appreciate the beauty of Pythagoras.

Arguing for the WHY, De Carvalho notes,

And when we understand a topic, it is easier to recall the facts …

And of course, as a general point, De Carvalho is correct. In regards to Pythagoras, however, De Carvalho is simply wrong. Pythagoras is one of the easiest equations to learn, and students simply don’t need the WHY to know WHAT it is and HOW it works. Secondly, the WHY doesn’t help with the recall whatsoever. Pythagoras is a theorem about areas, and its application in school is always to distances. The idea that a longer proof about areas will help students recall and understand the use of a simple formula about distances is utterly ridiculous.

As for appreciating the beauty of Pythagoras: if you are given a beautiful poem then you simply teach the poem. It is absurd to think that De Carvalho’s “discovering that for themselves” — which, anyway, will almost certainly be faked — will give students any proper appreciation of Pythagoras. All it can do, and inevitably what it will do, is obscure the simple beauty.

There are zillions of examples of the WHY being critical to understanding the WHAT, and there are even examples where the WHY should replace the WHAT altogether. There are examples where a limited form of inquiry is worthwhile in discovering the WHY and the WHAT. Pythagoras, however, is none of none of these. Pythagoras is only an example of ACARA’s constructivist dogmatism, and of De Carvalho’s ignorance.

WitCH 9: A Distant Hope

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February 15, 2019

2:23 pm

4 Comments on WitCH 9: A Distant Hope

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This WitCH (as is the accompanying PoSWW) is an exercise and solution from Cambridge’s Mathematical Methods Units 1 and 2, and is courtesy of the Evil Mathologer. (A reminder that WitCH 2, WitCH3, Witch 7 and WitCH 8 are still open for business.)

Update

As Number 8 and Potii pointed out, notation of the form AB is amtriguous, referring in turn to the line through A and B, the segment from A to B and the distance from A to B. (This lazy lack of definition appears to be systemic in the textbook.) And, as Potii pointed out, there’s nothing stopping A being the same point as C.

And, the typesetting sucks.

And, “therefore” dots suck.

PoSWW 3: Not the Right Angle

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February 15, 2019

2:23 pm

4 Comments on PoSWW 3: Not the Right Angle

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This PoSWW (as is the accompanying WitCH) is from Cambridge’s Mathematical Methods Units 1 and 2. and is courtesy of the Evil Mathologer. (A reminder, we continue to post on Cambridge not because their texts are worse than others, but simply because their badness is what we get to see. We welcome all emails with any suggestions for PoSWWs or WitCHes.)

We will update this PoSWW, below, after people have had a chance to comment.

Update

Similar to Witch 6, the above proof is self-indulgent crap, and obviously so. It is obviously not intended to be read by anyone.

One can argue how much detail should be given in such a proof, particularly in a subject and for a curriculum that systemically trashes the concept of proof. But it is difficult to see why the diagram below, coupled with the obvious equations and an easy word, wouldn’t suffice.

 

Obtuse Triangles

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June 25, 2017

1:00 pm

9 Comments on Obtuse Triangles

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Whatever the merits of undertaking a line by line critique of the Australian Curriculum, it would take a long time, it would be boring and it would probably overshadow the large, systemic problems. (Also, no one in power would take any notice, though that has never really slowed us down.) Still, the details should not be ignored, and we’ll consider here one of the gems of Homer Simpson cluelessness.

In 2010, Burkard Polster and I wrote an Age newspaper column about a draft of the Australian Curriculum. We focused on one line of the draft, an “elaboration” of Pythagoras’s Theorem:

recognising that right-angled triangle calculations may generate results that can be integral, fractional or irrational numbers known as surds

Though much can be said about this line, the most important thing to say is that it is wrong. Seven years later, the line is still in the Australian Curriculum, essentially unaltered, and it is still wrong.

OK, perhaps the line isn’t wrong. Depending upon one’s reading, it could instead be meaningless. Or trivial. But that’s it: wrong and meaningless and trivial are the only options.

The weird grammar and punctuation is standard for the Australian Curriculum. It takes a special lack of effort, however, to produce phrases such as “right-angled triangle calculations” and “generate results”. Any student who offered up such vague nonsense in an essay would know to expect big red strokes and a lousy grade. Still, we can take a guess at the intended meaning.

Pythagoras’s Theorem can naturally be introduced with 3-4-5 triangles and the like, with integer sidelengths. How does one then obtain irrational numbers? Well, “triangle calculations” on the triangle below can definitely “generate” irrational “results”:

Yeah, yeah, \pi is not a “surd”.  But of course we can replace each \pi by √7 or 1/7 or whatever, and get sidelengths of any type we want. These are hardly “triangle calculations”, however, and it makes the elaboration utterly trivial: fractions “generate” fractions, and irrationals “generate” irrationals. Well, um, wow.

We assume that the point of the elaboration is that if two sides of a right-angled triangle are integral then the third side “generated” need not be. So, the Curriculum writers presumably had in mind 1-1-√2 triangles and the like, where integers unavoidably lead us into the world of irrationals. Fair enough. But how, then, can we similarly obtain the promised (non-integral) fractional sidelengths? The answer is that we cannot.

It is of course notable that two sides of a right-angled triangle can be integral with the third side irrational. It is also notable, however, that two integral sides cannot result in the third side being a non-integral fraction. This is not difficult to prove, and makes a nice little exercise; the reader is invited to give a proof in the comments. The reader may also wish to forward their proof to ACARA, the producers of the Australian Curriculum.

How does such nonsense make it into a national curriculum? How does it then remain there, effectively unaltered, for seven years? True, our 2010 column wasn’t on the front of the New York Times. But still, in seven years did no one at ACARA ever get word of our criticism? Did no one else ever question the elaboration to anyone at ACARA?

But perhaps ACARA did become aware of our or others’ criticism, reread the elaboration, and decided “Yep, it’s just what we want”. It’s a depressing thought, but this seems as likely an explanation as any.