ACARA Crash 13: The Establishment Blues

(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

 

24 Replies to “ACARA Crash 13: The Establishment Blues”

  1. It looks to me like the Year 6 content is missing the elaboration “solving problems involving the *calculation* of lengths and areas using appropriate units”. Also, the elaboration involving parallelograms is unrelated to the content in question (maybe it’s supposed to go in the Year 7 syllabus instead?).

    For Year 7, isn’t it easier to see (especially in the case of an obtuse-angled triangle where the obtuse angle is on one end of the base under consideration) that the area of a triangle is half that of the parallelogram? It seems like teachers are going to have a harder time linking the area of that sort of triangle to the area of any sort of rectangle.

    With regards to prisms, four of the elaborations can be boiled down to “understanding the derivation of the formula for the volume of a prism, including special cases where the base is a rectangle or triangle”. I’m not sure I understand why all four of these methods need to be used to understand this; surely just one or two are enough to give students these understanding? (Does this indicate that these elaborations are not to be taken as syllabus material, but rather ideas for how the teachers may teach the content?) Further, there’s nothing in here about solving problems involving volumes of prisms.

    I’ll admit I’m not from Australia, so I don’t know much about the Aboriginal and Torres Strait Islander Peoples, but quickly Googling suggests to me that very few of these fishing nets are in the shape of rectangular or triangular prisms (at best, a few of them could be described as “roughly cylindrical or conical”, but cylinders and cones don’t seem to be mentioned in this syllabus), so it’s not immediately clear to me what sort of exploration activity is appropriate here. Maybe someone who’s more informed about this can comment better on this. (I do have to wonder what they mean by “capacity” in the context of water resource management other than “volume”; surely they’re the same thing when talking about liquids?)

    1. Thanks very much, edder. I’ll look carefully at your comments later today, although I’ll note quickly that you don’t quite get to my fundamental annoyance with this material.

      Briefly, on capacity, it seems that Australians have had a weird “capacity ≠ volume” fetish for decades. I really, really don’t get it. Plus, of course, who really cares about the capacity/volume/whatever of a net, roughly polyhedral or otherwise, and Aboriginal or other?

      1. I also do not get the thing with volume and capacity. It’s a nonsense. It’s a thing to mark kids wrong for.
        How do you rearrange a parallelogram? That reminds me of putting the wagon in a circle. I guess they mean cutting a parallelogram into pieces (cleverly) and rearranging the pieces.
        Rereading this stuff, gee the use of English is bad. these sentence fragments are hateable
        Picky example: “establish the formulas for areas of triangles and parallelograms, using their relationship to rectangles and use these …” “their” refers to triangles and rectangles”, but “these” refers to formulae.
        And the Aboriginal and Torres Strait Island net stuff – puh-lease. Does stuff like this impress or interest anyone in the Aboriginal and Torres Strait Island community? I bet a barra that it doesn’t.

    2. Thanks, edder. Re your Year 6 comment, yes it seems common in the draft to leave out reference to the practice such basic skills. And, note the “use to solve practical problems” in the content description. The only implication that I can see is that the practising of the basic skills is to be undertaken through these “practical problems”.

      And, yes, the parallelogram thing is weird. In itself, going from rectangles to parallelograms is a fine elaboration. But it hardly fits with the subsequent Year 7 content, which *doesn’t* provide any elaboration on finding the area of parallelograms.

      With finding the area of triangles, I guess it’s the trade-off between a simpler construction, and a sometimes harder construction but to create a simpler figure. I’m not the best judge, although it’s not a great strain to do both, and both could easily be mentioned as elaborations.

      With the prisms, I think the fundamental issue is, as commenters have indicated below, they simply haven’t settled on a clear and useful definition of volume. So, they kind of got stuck with a bunch of semi-definitions.

  2. The running flaw throughout all of this is the tendency to want to explain / present simple concepts by referring to more complicated concepts.

    The use of “establish…” for the first and third content descriptors seems weird. What sort of justification do they have in mind here – appealing to disjoint unions of congruent squares / cubes? I guess one could define a unit square / cube and then “establish” the formula for the area of a rectangle (or volume of rectangular prism) with integer side lengths by counting squares (cubes), but then what about all the other rectangles (prisms)? Perhaps “establish” here just means “make it seem plausible”. I wonder if at this level it’s just best to take LW and LWH as primitive.

    As for the second content descriptor, while I support doing a quick demonstration of the formulas for areas of parallelograms and triangles, I agree with Eddie that the descriptor presents the idea poorly.

    Other stuff which seems weird / bonkers:

    “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 ” – the isoperimetric inequality in primary school maths? I guess there’s an implied restriction to rectangles here, but even so…

    “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) ” – this seems backwards. If you didn’t already accept that area = 0.5bh, why would you think that moving a vertex around without changing the altitude does not change the area?

    “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 ” & “connecting the footprint and the number of floors to model the space taken up by a building” – if this were intentional parody, it’d be a little bit funny.

    “representing threefold whole-number products as volumes, for example, to represent the associative property of multiplication” – another instance where this seems backwards. Surely associativity of multiplication is more intuitive than any counterpart property of calculating volumes of prisms.

    1. Thanks, SRK. As with edder’s comments, I’ll look closely later. But you caught on the specific irritant for me: the use of “establish”. I don’t know what the word is intended to mean, and I very much doubt that the writers have any consistent meaning in mind, beyond “declare as a truth to be used”.

    2. Hi, SRK. Further on your comment.

      Yes, the only real alternative is to have LW and LWH as definitions. Of course it’s fine to motivate these definitions by reference to unit cubes filling integer-sided shapes, and to use this early on as a quasi-definition. Just as early on multiplication “is” repeated addition. But, it can’t stay that way.

      The “dynamic geometry” thing, for both triangles and prisms, is utter lunacy. “Ooh, it does get in!“.

      I have more sympathy with the associativity (and commutativity) thing. Yes, you cannot magically obtain associativity/commutativity from volume/area. But there has to be lots of sleight of hand with introducing and reinforcing number axioms in school. I’m fine with using geometric intuition to reinforce algebraic fundamentals.

  3. I would want to interpret “establish” as meaning the students actually need to know how explain or derive (sort of almost “prove”) the mathematical statement somehow. So for the area of a rectangle, they’d have to have some sense of what area means. They could tile the rectangle with the unit squares and count them, or else think about dilating shapes and what effect that should have on area in order to get the general rectangle formula.

    None of the elaborations in the first section touch on any of the considerations I would have in teaching students to establish the area of a rectangle. I think there are subtleties to elaborate on. Is there a chance they put the key ideas somewhere else and these are mismatched elaborations?

    1. I think they are unnecessarily vague about what they consider axiomatic. Why not just spell it out?

      ETA: I looked to try to answer my own question. It seems like the meaning of area as how many tiles you can fit is explored in Year 4, then in Year 5 they do lots of problems, before then ‘establishing’ the area of a rectangle in Year 6, so I guess they’re leaving it to the teachers to connect the dots and sequence things logically. It makes me wonder what it means to have a curriculum.

      1. Spoiler Alert: as a teacher I cannot remember the last time I consulted the actual curriculum documentation for anything other than VCE Units 3&4. It was probably more than a decade ago.

        Actually, the last time I read the VCE Units 3&4 curriculum guide was when the new draft came out…

        In between times, I just read the exam papers and examiners “reports”. They are far more enlightening than anything else.

        And before anyone asks “how do you decide what to teach at Year X?” simple: talk to your students, get a sense of what they already know and go from there. Working with numbers, working with algebra, a bit of geometry, dabble in some probability and/or basic statistics and make the rest up as you need to. If you really want, grab a few 1970’s (or earlier, preferably) textbooks as a source of practice problems that aren’t stupidly engineered “applications” and your lessons will almost plan themselves…

        1. Thanks, RF. Of course no one who can avoid it reads the current curriculum, and I predict that people will gouge out their eyes rather than read the draft curriculum.

          But there’s a triple-barrelled problem with your strategy. First, the clunkier the curriculum, the clunkier the textbooks, which means it will be more difficult to obtain a clear picture from the obvious sources. Secondly, you happen to be old enough and to have sufficiently good mathematical resources (human and written), that you know well enough what mathematics is, how that maths is the basis of decent lessons, and thus you will typically be able to swim over whatever nonsense is thrown at you and teach well; most younger teachers will be screwed. Thirdly, I get the sense that this is all much harder for primary teachers. who almost certainly will not have textbooks, will tend to be much less sure of what the maths really is and what really needs to be emphasised (arithmetic), and so really, really need proper guidance.

        2. In my short experience in teaching mathematics in a few schools, I have found that there are several classes in any subject (except Specialist Mathematics). To keep all the teachers in step, the text book is followed closely. Tests and assignments are generated by the publishers from the text book. The question as to what to teach to the students does not arise; just follow the text. Examiners’ reports are useful in giving advice – provided that you agree with the examiners.

          1. Thanks, Terry. But you should also note that the textbooks are pretty universally awful. A non-awful curriculum would at least create the possibility of a non-awful text.

      2. Thanks, wst. Yes, it is a mess. As I replied to SRK above, it is fine and reasonable to much around with unit squares or unit cubes as an introduction. But at some point you have to get from integer intuition to proper definition, and the draft makes it clear as mud how and when to do this. The fundamental problem is the use of “establishment” in a hopelessly vague manner.

  4. WST beat me to it – how does one “establish” a formula when the formula is the definition???

    1. A bloody good question. Again, one can use “establish” in a don’t-care-about-the-origins declaration-of-fact-to-be-used. But “establishing” the area of a rectangle and the area of a triangle are fundamentally different establishments.

      1. Having finished my MTeach recently, I wonder if they are just picking and choosing from tables of verbs that are meant to correspond to different levels of student learning: recognise, understand, solve, analyse, create, evaluate, establish? “Establish” is presumably more advanced than whatever they did in Year 5, but they’re not using it in any way anyone would recognise as meaning anything… Perhaps they just mean “they totally know the area of a rectangle for sure now.”

        I feel like they need to be stopped because they’re using words in a way only them and their friends can understand, and then they use the ability to make sense of them as their definition of pedagogical expertise. So, if I don’t understand, they can say I just don’t have the required expertise. It’s demoralising.

        ETA: it was recommended to us that we google for such tables of verbs in order to complete our assignments, and we were graded on whether our verbs were in the right row. I thought it was just a weird uni assignment thing, but maybe they actually believe that stuff?

        1. So, wst, when you suggest “they need to be stopped”, do you think perhaps organisations such as AMSI and AMT and AAS and AustMS should, just maybe, get off their fat asses and say something public to counter this crap? Or, do you think signing on to ridiculous public statements and submitting mealy mouthed nonsense to ACARA might suffice from organisations purportedly representing mathematicians?

          True, there is some sense to the latter strategy. Leaving the proper defense of Australian mathematics education up to a blacklisted blogger and a Ballarat physics guy is clearly a brilliant plan.

          1. Getting a public statement out there to counter this crap would be a great start from any of these groups.

            I also don’t see them doing it, for different reasons.

            AMSI won’t make such a statement and, even if they did, the majority of teachers who teach mathematics classes (sorry, I cannot refer to them all as mathematics teachers) would probably say: “who or what is AMSI?” For AAS, raise that from at least 50% to at least 90%.

            AMT does great work in their rather limited space, and doesn’t seem to have any great desire to get involved in curriculum matters. A lot of their work is based on things outside the curriculum (OK, so a lot of schools do teach the stuff they test in their various problem-solving challenges, but this is in spite of not because of the national curriculum).

            MAV, apart from being a state-based association has pretty much declared their hand in recent years and to publicly make a statement against ACARA would be… out of character.

            So, I guess I’m thankful to the Ballarat physics guy and a few fellow travelers.

            And, WST, well done on completing your degree. Now for the fun part: dealing with the VIT (and I don’t mean “fun” in the standard sense of the word).

            1. No offence, but the point of such a public statement by mathematician organisations is not to impress teachers. And yes, a critical public statement by a maths ed organisation such as MAV or AAMT or MERGA would also be valuable. And inconceivable. These organisations are the disease, not the cure.

              1. None taken.

                I just remember MAV running a session (after the supposed deadline had closed) for comment on the draft VCE curriculum (although I may not be remembering correctly because the whole thing, as usual, was a bit underwhelming)

                1. Your memory is fine. Feedback after the deadline still counts? Another underwhelming tokenistic effort from the MAV.

  5. “estimating the cost of materials needed to make shade sails based on a price per metre”

    Surely ACARA means price per square metre”

    It’s a war of attrition finding every error in this document. The above error captures the total literacy fail of the entire document. The document reads as if written by a group of primary school students (no offence intended to such students).

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