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?
**WHY**^{2} do we teach Pythagoras’s theorem?

The first question is about the HOW of the mechanics of dealing with the equation : 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: WHY^{2} 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.