WitCH 78: Seeing Through De Carvalho’s Prism

Happy April Fools’ Day everyone. Guess which gang of fools is gonna get fooled, and guess which gang of fools is gonna do the fooling.

There’s not a whole lot of point to this any more, but it’s niggling. So we’ll post again on ACARA‘s thoughtful and classy CEO, David de Carvalho.

We have already written on Mister Wisdom’s Keynote address to the Christian Schools National Policy Forum, and De Carvalho’s cute reference to ACARA’s manufactured Joint Statement as his sole example of the “public conversation”. We noted the key nonsense of his address, and most else is not worthy of comment, just the standard ACARA fluff dipped in De Carvalho’s wading pool philosophy. But, mirroring another of his gems, De Carvalho takes a moment to consider a mathematical example in pseudo-depth. We’ll make it a WitCH, so we don’t have to do any work and can get an early start on the vodka.

In his address, De Carvalho spouts the standard nonsense on ACARA’s supposed “refining, realigning and decluttering” of the Curriculum. Readers will recall that ACARA achieved this magnificent tidying by ramming the general capabilities – the “critical and creative thinking” and whatnot – into the curriculum itself, making the entire thing an impenetrable, inquiry-infested swamp.

To make his point, De Carvalho notes “the false dichotomy between factual knowledge and the ability to think creatively and critically”. Skipping over the questionable grammar, we then have De Carvalho distinguishing “the ability to recall facts” from “”knowledge”. It is at this point that we get to De Carvalho’s example, with “knowledge” somehow sliding into “genuine understanding”:

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 volume of a prism but do they understand why that formula works every time and when they should use it to solve some real-world problem? The process by which a student arrives at that point of understanding, with the assistance of the teacher, is what makes learning exciting.

De Carvalho has an unerring gift for creating the anti-example. What does De Carvalho mean? What are students supposed to be taught so that they “understand” the formula for the volume of a prism and its “use”?

To add context, below is the relevant content and associated elaborations from Year 7-9 of the (only visible) draft curriculum. Make what sense of it you will. Or, just give up and go straight for the vodka.


Apply computational thinking and digital tools to construct tables of values from formulas involving several variables, and systematically explore the effect of variation in one variable while assigning fixed values for other variables


experimenting with different sets of tables of values from formulas, for example, using volume of a rectangular prism = length × width × height, and specifying a fixed width and equal length and varying the height


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


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


choose and justify the appropriate metric units for solving problems involving perimeter, area, volume and capacity. Solve practical problems involving the volume and capacity of prisms and converting from one metric unit to another


solve problems involving the volume of right prisms and cylinders in practical contexts and explore their relationship to right pyramids and cones


investigating the volume and capacity of prisms and cylinders, to solve authentic problems

determining and describing how changes in the linear dimensions of a shape affect its surface area or volume, including proportional and non-proportional change

solving problems involving volume and capacity, for example, rain collection and storage, optimal packaging and production 

experimenting with various open prisms, pyramids, cylinders and cones to develop an understanding that pyramids and cones are derived from prisms and cylinders respectively and that their volumes are directly related by a constant factor of 1/3

8 Replies to “WitCH 78: Seeing Through De Carvalho’s Prism”

  1. Here is my take on the C in this WiTCH:

    Take a “real world example”, say, for example a school building, which I will assume is a prism.

    I could work out the volume by taking some measurements and then using the formula I have memorised.

    Or, I could fill the building with sand/water/vodka then (assuming I managed to completely fill the space and didn’t lose anything in doing so) find a way to measure the volume of filling used by carefully decanting into another vessel of known volume.

    Is this an “authentic problem”?

    I think not. Hence WiTCH.

    1. Thanks, RF. I think there’s plenty of C to go around, but you struck on a big bit. What are the “authentic problems” that require the volume of a prism? And, except maybe for the manufacturing of Toblerone, which are going to impress a kid?

    2. I have an issue with how the word “authentic” is used in education. To me, “authentic” means “genuine”; e.g. this diamond is authentic. So an “authentic problem” is a genuine problem. However, as often happens in the social sciences, perfectly good words with an established meaning are used differently, and dare I say pompously. The language of education is one of its greatest impediments.

      1. I think it’s worse than that. I think they say “authentic problem” because they know the problems are not at all authentic. It’s bluffing.

  2. It’s hard to know where to start with this egregious rubbish. But look for example at the last paragraph, about “experimenting” to somehow show “that pyramids and cones are derived from prisms and cylinders respectively”. They are, are they? Who knew? And how are year 9 schoolchildren expected to discover that – and with what sort of experimentation? This sounds like wishful thinking, poor (well, non-existent) pedagogy, hideous curriculum design, and bad mathematics. I have no trouble with some simple solid geometry in a curriculum. But it does the subject no favour to dress it up with pointless and meaningless “applications”, and with a vague hand-wave at “experimentation”.

    Also, why not call a rectangular prism a “box”? Geez …

    To finish, a quote from one of my favourite authors, the essayist William Hazlitt, from “On the ignorance of the learned”:

    “Any one who has passed through the regular gradations of a classical education, and is not made a fool by it, may consider himself as having had a very narrow escape.” Replace “classical” with “mathematical” and you have ACARA’s brave new world.

    1. Thanks, Alsadair. It’s always the same. They spout these motherhoodisms of “knowledge” and “understanding”, but when one looks at the details of what is proposed, it is nothing but a gas of absurdity.

      I’ll look up the Hazlitt quote/essay. I’m not sure of his objection to “classical education”, but your ACARA version is spot on.

      1. Hazlitt was not against classical learning as such, but of the woeful teaching of it in his day, which was basically rote learning helped along with enthusiastic whipping by teachers. He points out that surviving such an education does not, in fact, make one “educated”; but instead a pitiful fool who can spout facts without knowledge.

        You also might enjoy “The pains of education” by the 18th century satirist Charles Churchill: https://www.poetrynook.com/poem/pains-education

        Of course, the root problem with the entire curriculum is that ACARA is trying to develop a subject called “mathematics”, but without actually including any mathematics in it.

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