New Cur 11: Circling Reason

Just for a change, this post will be about a good aspect of new Curriculum. Just kidding. Sort of.

The following is an elaboration and associated content descriptor from Year 8 Measurement:

solve problems involving the circumference and area of a circle using formulas and appropriate units (AC9M8M03)

deducing that the area of a circle is between 2 radius squares and 4 radius squares, and using 3 × radius2 as a rough estimate for the area of a circle 

There are two ways one might react to this elaboration. First, one might justifiably have no idea what is the meaning or intent of the elaboration, and then conclude that the curriculum was written by idiots. Or, one could recognise that the elaboration is at least attempting something good but that the attempt was an abject failure, and then conclude that the curriculum was written by idiots. All roads lead to Rome.

For those who have not bothered to decipher the elaboration, the intent is to teach some understanding of the area of circles, and thus also of π, by inscribing and circumscribing squares, as pictured:

If the radius of the circle is R then the outer, purple square has side length 2R and so has area 4R2. Next, by Pythagoras, the inner, red square has side length √2R. So, the red square has area 2R2. These areas are the “two radius squares” and “four radius squares” in the elaboration, and the area of the circle is clearly trapped between them. This is a nice thing to do.

There are significant problems with the elaboration, however, quite apart from its incredibly clumsy phrasing. Firstly and of least importance, the elaboration has essentially nothing to do with the parent content: we are not in any sense solving a problem; we are not using the formula for the area of a circle; and the instruction to use “appropriate” units is distracting and irrelevant. The second problem is that the elaboration fails to state the proper purpose of such calculations, using the trapping of the area of a circle to also trap the value of π; in this instance, π has been trapped between 2 and 4. (For this reason, the exercise is probably better done, at least to begin with, on a unit circle.) The elaboration’s “rough estimate” of 3 is also not attached to the actual trapping. Thirdly, although the application of Pythagoras’ theorem is very nice, it is also not properly possible: Pythagoras only gets introduced later, with content descriptor AC9M8M06.

The general problem is that entire framing of the activity as an exercise in “measurement” rather than abstract geometry buries the much deeper ideas that might be touched upon. The elaboration, properly presented, is a good Archimedes-like exercise in trapping π. But it could also go further, considering other polygons and considering also circumference and perimeters. The elaboration thus extended could lead to a discussion of a more formal conceptualisation of π, and at least some good hints for why “π” is the same for every circle. (We gave it a shot earlier this year, with a strong Year 8 Extension class: the notes are here.) But this is ACARA we’re discussing: it suffices to criticise the elaboration on its own terms.

Finally, it is worth noting that Archimedes is not mentioned in this elaboration, or anywhere in the curriculum. The standard absence of history and humans. There are, however, other elaborations attached to the same content descriptor, and three doors down we have,

exploring traditional weaving designs by First Nations Australians and investigating the significance and use of circles 

So, ACARA’s presentation of circle geometry is not entirely devoid of tradition or culture.

UPDATE (10/12/22)

Here is a diagram illustrating that Pythagoras can be avoided, as noted in the comment by Franz.

12 Replies to “New Cur 11: Circling Reason”

  1. If you rotate the inner square by 45^\circ, you can replace Pythagoras by the simple “doubling-the-square”-story. Just make sure you don’t accidentally mention Socrates or Meno.

    1. Thanks very much, Franz. I should have included that. Perhaps I short-changed the curriculum writers, and that’s what they had in mind. The argument and the dialogue between the unmentionables can be found here.

      1. Nice try, Marty. But I think you’re suffering from a form of pareidolia.

        (On the other hand, maybe I should hand back my prize so that you can give it to yourself …)

  2. Marty, I think that your parenthetical remark (“For this reason, the exercise is probably better done, at least to begin with, on a unit circle.”) is arguable because I think that a potential nugget in this problem (in my humble view) is precisely the realization that the problem, along with its method of solving, carries an invariance of scale. The scale (expressed by the radius of the circle) is carried in the factor denoted “radius-squares” (so it enters as a square). It would appear to me that it is not entirely out of reach of students even at a moderate level to understand that that factor (with the square) is not the main fruit of the effort put in here, but rather what is in front: that is, 2, 4, and \pi. (It may take a rather skilled teacher to do this well, of course.)

    I also like this problem for its invariance under rotation of the “inner” and the “outer” approximating square (independently). That is, rotation of either, in whatever way you like, does not change the value of the approximation (!). For the “inner” square, Franz used precisely that invariance. This highlights another message, albeit one that may be a few shoe sizes too large at this level: if I have invariance in a problem, I can use that “version” (here, rotation) that makes my life the easiest (here, we can even dispense with knowing Pythagoras’ Theorem by choosing a “good” rotation).

    1. If only ACARA understood what it was writing about. Then maybe we’d get elaborations that were insightful, clear and useful. I doubt any of this (including what Marty posted) was even remotely thought of or considered by ACARA. Which is what happens when you have a curriculum written by work experience kids rather than people who have a deep understanding of mathematics. (Same can be said for the Mathematics Study Design).

      1. Possibly I’m being overly generous, since it didn’t occur to me that ACARA might have (or should have) been considering Franz’s proto-Pythagorean argument. But I think, even with known idiots, one should be very careful to distinguish the argument from the arguer, even when their argument is unclear, even when the arguer sells themselves badly. The curriculum writers were thinking something with that elaboration, and something intrinsically good to think. Yes, it was expressed horribly and, yes, it is unclear exactly, or even approximately, what they intended. But I wouldn’t assume the worst-case reality here.

        It is very easy to take anything from ACARA, or any of the clown outfits, and assume their argument/policy/whatever is screwed. You’ll usually be correct. But it is lazy to do so, and it encourages tribal unthink.

        This is exactly what happened with Alan Tudge, and his attack of ACARA’s draft curriculum. Tudge is an untrustworthy hypocritical asshole, and a lot of his curriculum attack was self-serving culture war garbage. But, fundamentally, he was correct. He was almost entirely correct on the maths and the English fundamentals, and he was substantially correct on the humanities. But he was knee-jerk dismissed out of hand by pretty much everyone outside of the Liberal strongholds.

        1. OK, I see your point (I think). Even a steaming pile of garbage might have something resembling a small speck of gold in it, and we shouldn’t dismiss that small speck as garbage just because it’s found in the steaming pile.

          1. Yes. There’s probably not a pony in there, and it’s probably foolish to look for a pony in there, but if you seem something vaguely pony-shaped then consider it on its smelly merits.

    2. Thanks, Christian. Of course I agree that the invariance with R is important. I’m just suggesting that as a first step it is easier for students to think of pi as getting trapped between (e.g.) 2 and 4, rather than trapping the area of a circle of radius R. Yes, this presupposes “pi” makes sense, but that is where students will be coming from. The essential idea and computations are the same, and I think it’s better to leave the scaling and more subtle notions until after the essential work. But, I’m open-minded on it. My view is it’s best to leave out R at the start, but I can see the argument for leaving it in.

  3. The numbering of the content descriptors is totally unrelated to the order they are taught. There is no reason to think that Pythagoras will be taught after the area of a circle since they are both in year 8.

    1. Thanks, MW, but I don’t get what you’re saying. If it is intended that Pythagoras be used in teaching the above elaboration, and probably anyway, then that is a clear error in the content numbering of Year 8 Measurement. Very busy teachers should be able to presume that the curriculum numbering reflects some logical ordering.

      I’m not really fussed here. The “Pythagoras is later” issue is the least of my concerns with this content. And I understand that smart teachers will reorder the curriculum material, push the stats off to infinity and so on. (Smarter teachers will ignore the curriculum entirely.) But if the curriculum is in error then the curriculum is in error. That matters.

  4. More on tradition and culture. In ancient Egypt, the formula for calculating the area of a circle was (\frac{8}{9}d)^2 where d is the diameter. This is not a bad approximation. However, what struck me when I first encountered this is, that the area was expressed in terms of d rather than the radius r, and, from a practical point of view it is easier to measure d than r. The Egyptians were practical people. (R.J. Gillings, “Mathematics in the time of the Pharaohs”, Dover.)

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