This one’s from AMT‘s 2007 upper primary Australian Mathematics Competition. Yes, it’s a while ago, and we are not aware that such BODMAS nonsense has appeared since on the AMC, and of course such BODMAS nonsense is endemic elsewhere. But we hold, or at least held, AMT to a higher standard.
We’re working on a long ACARA post, which, hopefully, will be up in a day or so. In the meantime and as a bit of background for the coming post, readers may wish to have a wander through the Singaporean Primary Mathematics Syllabus.* (The syllabus begins with explanatory chapters, and the content description begins on page 34.)(Added 12/10/21 – The Secondary Syllabus 1-4 is here.)
Jen Deyzel is a retired primary school teacher, a “septuagenarian” (now octogenarian) with decades of experience. She came up in conversation with a parent, whose children are tutored by Jen. In 2017, Jen wrote an essay on the decline of Australia’s educational standards. Jen has kindly permitted us to reproduce her essay, below (and in PDF form here).
The Great Decline:
Why Australia’s Education Standards are Plummeting Fast
Each Year’s content in the draft curriculum begins with a Level description, and each of the thirteen Level description begins with the exact same sentences:
The Australian Curriculum: Mathematics focuses on the development of a deep knowledge and conceptual understanding of mathematical structures and fluency with procedures. Students learn through the approaches for working mathematically, including modelling, investigation, experimentation and problem solving, all underpinned by the different forms of mathematical reasoning. [emphasis added]
Yep, “an equal focus on building fluency“, no doubt about it.
Continuing to try to rid ourselves of ACARA irritants, the following are the “calculator” elaborations from Year 1 – Year 6 Number and Algebra (sic):
using the constant function on a calculator to add ten to single digit numbers, recording the numbers to make, show and explore the patterns in a 0 – 100 chart
with the use of a calculator, exploring skip-counting sequences that start from different numbers, discussing patterns
modeling skip counting sequences using the constant function on a calculator, while saying, reading and recording the numbers as they go
We began the Crash series by critiquing the draft curriculum’s approach to counting in Foundation. Our main concern was the painful verbosity and the real-world awfulness, but we also provided a cryptic hint of one specifically puzzling aspect. The draft curriculum’s content descriptor on counting is as follows:
“establish understanding of the language and processes of counting to quantify, compare, order and make correspondences between collections, initially to 20, and explain reasoning” (draft curriculum)
“explain reasoning”. Foundation kids.
OK, let’s not get distracted; we’ve already bashed this nonsense. Here, we’re interested in the accompanying elaborations. There are ten of them, which one would imagine incorporates any conceivable manner in which one might wish to elaborate on counting. One would be wrong.
The corresponding content descriptor in the current Mathematics Curriculum is as follows:
“Establish understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20, moving from any starting point” (current curriculum)
Notice how much more “cluttered” is the current descriptor… OK, OK stay focussed.
The current descriptor on counting has just (?) four elaborations, including the following two:
“identifying the number words in sequence, backwards and forwards, and reasoning with the number sequences, establishing the language on which subsequent counting experiences can be built” (current curriculum, emphasis added)
“developing fluency with forwards and backwards counting in meaningful contexts, including stories and rhymes” (current curriculum, emphasis added)
The point is, these elaborations also emphasise counting backwards, which seems an obvious idea to introduce and an obvious skill to master. And which is not even hinted at in any of the ten elaborations of the draft counting descriptor.
Why would the writers of the draft curriculum do that? Why would they consciously eliminate backward counting from Foundation? We’re genuinely perplexed. It is undoubtedly a stupid idea, but we cannot imagine the thought process that would lead to this stupid idea.
OK, we know what you’re thinking: it’s part of their dumbing down – maybe “dumbing forward” is a more apt expression – and they’ve thrown backward counting into Year 1. Well, no. In Year 1, students are introduce to the idea of skip-counting. And, yep, you know where this is going. So we’ll, um, skip to the end.
The current Curriculum has two elaborations of the skip-counting descriptor, one of which emphasises the straight, pure ability to count numbers backwards. And the draft curriculum? There are four elaborations on skip-counting, suggesting in turn the counting of counters in a jar, pencils, images of birds, and coins. Counting unadorned numbers? Forget it. And counting backwards? What, are you nuts?
OK, so eventually the draft curriculum seems, somehow, to get around to kids counting backwards, to look at “additive pattern sequences” and possibly to solve “subtraction problems”. The content descriptors are so unstructured and boneless, and the elaborations so vague and cluttered, it is difficult to tell. But how are the kids supposed to get there? Where is the necessary content description or elaboration:
Teach the little monsters to count backwards.
If it is there, somewhere in the draft curriculum, we honestly can’t see it. And if it is not there, that it is simply insane.
There is another contribution from the Education community:
This one does not directly address the open letter, although, given the framing and the links, it is difficult to not see the article as an intended rebuttal. Again, we know little of the authors, and we have not read the article with any attention. We’ll be interested in what commenters think. (Ball-not-man rules still apply.)
Glen has pointed out that the article is from April 21. So, it is definitely not in response to the open letter. However, the article came out soon after the ridiculous, pre-emptive strike statement from AMSI, AAS and others, and in its first sentence the article links to the reporting of this statement. Whatever merits it might have, the article is not an innocent reflection on educational method.
As indicated by SRK, there is now (in effect) a response from John Sweller.
Today in The Conversation there is an article firing back at the open letter to ACARA:
We haven’t read the letter, and we don’t know the authors, or of the authors. We’ll try to read the article and comment on the article soon, modulo home schooling and general exhaustion. For now, people can comment below (respectfully and on-topic and on-the-ball-not-the-man). We’ll be interested in what people think.
We’ve finally had the time to read this article. The comments below suffice, and we’re not going to waste our or others’ time with a detailed critique. .
Seriously, that’s the sum of the defense of the draft curriculum? That’s all they got?
(With apologies to the brilliant Laurie Anderson. Sane people should skip straight to today’s fish, below.)
I met this guy – and he looked like he might have been a math trick jerk at the hell brink.
Which, in fact, he turned out to be.
And I said: Oh boy.
You know, that it’s for you.
It’s a blue sky curriculum.
Parasites are out tonight.
You know, I could write a book.
And this book would be thick enough to stun an ox.
Cause I can see the future and it’s a place – about a thousand miles from here.
Where it’s brighter.
Linger on over here.
Got the time?
I got this postcard.
And it read, it said: Dear Amigo – Dear Partner.
Listen, uh – I just want to say thanks.
Thanks for all your patience.
Thanks for introducing me to the chaff.
Thanks for showing me the feedbag.
Thanks for going all out.
Thanks for showing me your amiss, barmy life and uh
Thanks for letting me be part of your caste.
Hug and kisses.
Oh yeah, P.S. I – feel – feel like – I am – in a burning building – and I gotta go.
Cause I – I feel – feel like – I am – in a burning building – and I gotta go.
OK, yes, we’re a little punch drunk. And drunk drunk. Deal with it.
Today’s fish is Year 7 Algebra. We have restricted ourselves to the content-elaboration combo dealing with abstract algebraic expressions. We have also included an omission from the current curriculum, together with the offical justification for that omission.
As students engage in learning mathematics in Year 7 they … explore the use of algebraic expressions and formulas using conventions, notations, symbols and pronumerals as well as natural language.
create algebraic expressions using constants, variables, operations and brackets. Interpret and factorise these expressions, applying the associative, commutative, identity and distributive laws as applicable
generalising arithmetic expressions to algebraic expressions involving constants, variables, operations and brackets, for example, 7 + 7+ 7 = 3 × 7 and 𝑥 + 𝑥 + 𝑥 = 3 × 𝑥 and this is also written concisely as 3𝑥 with implied multiplication
applying the associative, commutative and distributive laws to algebraic expressions involving positive and negative constants, variables, operations and brackets to solve equations from situations involving linear relationships
exploring how cultural expressions of Aboriginal and Torres Strait Islander Peoples such as storytelling communicate mathematical relationships which can be represented as mathematical expressions
exploring the concept of variable as something that can change in value the relationships between variables, and investigating its application to processes on-Country/Place including changes in the seasons
Solving simple linear equations
Focus in Year 7 is familiarity with variables and relationships. Solving linear equations is covered in Year 8 when students are better prepared to deal with the connections between numerical, graphical and symbolic forms of relationships.
I – feel – feel like – I am – in a burning building
The following is an open letter to David de Carvalho, CEO of ACARA, and to the ACARA Board. regarding the draft mathematics curriculum. The home of the letter is here, you can sign up here, and the list of current signatories is here.
Disclosure: The letter was not my idea, and it is not my letter, but I had a hand in bringing the letter to fruition. As to why I think the letter is important, see here.
The sign-up page for the open letter has now been closed. Greg Ashman, the boss of the open letter, will submit (or has already submitted) the letter and the list of signatories today, to ACARA and to the ACARA Board.
Greg Ashman deserves a huge thank you from the Australian maths ed world. Greg instigated the open letter, and managed it through, and it simply wouldn’t have existed without him. There are also a number of other, anonymous or semi-anonymous, people who deserve a very big thanks. Some for their persistent and incredibly irritating hammering on various drafts, and some for helping to sign up various Mr. Bigs.
And, a very big thanks to the hundreds of people who signed on to this strong and public declaration.
Open letter to Mr. David de Carvalho, CEO of ACARA, and the ACARA Board
On 29 April 2021, the Australian Curriculum, Assessment and Reporting Authority (ACARA) released its draft revisions to the Australian Mathematics Curriculum, with a consultation period ending on 8 July 2021. We are a group of mathematicians, mathematics educators, educational psychologists, parents and members of the public who take an active interest in mathematics education and in the curriculum. We agree that the Mathematics Curriculum desperately requires reform; it is repetitious, disconnected, unambitious and is lacking in critical elements. We are pleased that efforts to reform the curriculum are underway. We are profoundly concerned, however, with the structure of the current draft and with many of the proposed changes within.
The primary source of our concerns is the proposal to replace the four Proficiencies in the current Curriculum with the draft’s thirteen “Core Concepts”, grouped under three “Core Concept Organisers”. The Proficiencies – understanding, fluency, reasoning and problem-solving – are well-understood and provide a clear structure for teaching mathematics. In contrast, the Core Concepts are often poorly defined and overlapping, vary massively in scope and breadth, and their groupings into Core Concept Organisers, including the faddish “Mathematising”, are a mostly arbitrary and at times contradictory categorisation. The critical element of “thinking and reasoning”, for example, has somehow been reduced to just another concept among thirteen, sharing equal value with wordy descriptions of simple ideas. The end effect is a framework of little practical value as a guiding structure.
The Core Concepts are confused and confusing, but it is clear that they represent a push toward a central role for “problem-solving” and inquiry-based learning. Solving problems is obviously a core aspect of mathematical practice, is an important goal for mathematics education, and is already listed as one of the four Proficiencies in the current curriculum. The issue with the draft curriculum is that its “inquiries” are unanchored by clear and specific content, by underlying knowledge and skills. Moreover, the “problems” suggested to be “solved” are mostly exploratory and open-ended, effectively unsolvable and of questionable pedagogical value, and with little or no indication of the specific desired learning outcome. Insufficient attention is given to carefully constrained problems facilitating the practicing and subsequent extension of already mastered skills. Making things worse, the inclusion of inquiry methods in the content descriptors results in the descriptors being almost useless as determiners of actual content. This obscures the key ideas and basic skills to be learned, which are the foundational elements essential for any effective mathematical practice, including for problem-solving.
The draft is not so much pushing problem-solving as it is pushing for learning through activities referred to as “solving problems”, but which are actually ill-defined explorations. We do not believe that a curriculum document should mandate a specific method of mathematics teaching, and it is especially concerning that the draft curriculum is extensively mandating learning through “exploring” and “problem-solving”. There is strong evidence to indicate that methods without a proper balance that includes the explicit teaching of mathematical concepts are less effective, in particular for younger students grappling with new concepts and basic skills. The content of the mathematics curriculum, even for the lower years, is the result of millennia of human endeavour across cultures around the world – it is neither fair nor realistic to expect students to retrace this journey with a few pointers and inquiries in a few hours per week.
The emphasis in the draft curriculum on open-ended inquiry, without the systematic building of coherent knowledge, creates further serious issues. Some indication of these issues is provided in the following paragraphs, but many, many more examples could be given.
The delaying and devaluing of fluency, of “the basics”
The draft curriculum includes some particularly concerning Content descriptors, and rearrangement of material. The learning of the multiplication tables, for example, is first addressed only in Year 4, where it is framed in terms of “patterns” and “strategies”, with no emphasis on mastery. Similarly, the solving of linear equations such as ax + b =c, a foundational skill for all secondary school mathematics, is pushed in the draft from Year 7 to Year 8. There is simply no valid argument for these, and many other, dilutions and delays. Indeed, the draft curriculum has squandered the opportunity to address some glaring problems with the timing and emphasis of content in the current Curriculum.
The loss of natural mathematical connections
Mathematics in the current Curriculum consists of three strands, but the draft has split these into six strands. The very natural Number-Algebra strand, for instance, has become separate strands of Number and Algebra. This is unwieldy, effectively requires a redefinition of “algebra” and, most damagingly, it severs the critical pedagogical link between these two disciplines. Similarly, the strands of Measurement-Geometry and Statistics-Probability have been split into Measurement, Space, Statistics and Probability, for no benefit or good purpose.
Notwithstanding ACARA’s repetitive claims to be promoting “deep understanding”, the draft’s overwhelming emphasis on investigation and modelling has resulted in many critical mathematical concepts being underplayed and, in certain cases, not even being named. In Algebra, for example, fundamental terms such as “null factor” and “polynomial” and “completing the square” rate not a single mention. To give an analogy, it is as if a curriculum on Politics failed to mention “sovereignty” or “citizenship” or “separation of powers”.
The devaluing of mathematics
The problem-solving, investigation and modelling that is advocated by the draft curriculum is very heavily weighted towards real-world contexts. Indeed, the definition of “Problem solving” provided in the draft Curriculum’s “Key considerations” section explicitly mentions solving problems relating to the “natural and created worlds”, and pointedly omits references to solving problems stemming from mathematics itself. This approach squanders an excellent opportunity for students to gain an appreciation of mathematics as a beautiful discipline, a discipline which can be its own goal. This devaluing of mathematics is starkly displayed in the description of, and in the very name of, the Space strand. Whereas Geometry is concerned fundamentally with the study of abstract objects and their properties, the Space content is heavily slanted towards the study of real-world contexts. Learning in genuine real-world contexts is much more difficult, because the real world is inevitably full of distractions that cloud the clear principle to be learned.
Mathematical errors and non sequiturs
Some errors in the draft are subtle, but many are not. There is no purpose, for example, in directing students to “investigate … Fibonacci patterns in shells”, since such patterns simply do not exist. Such errors and confusions would typically be caught during a proper review by mathematicians; their existence in the draft curriculum places into serious question the nature and the extent of ACARA’s consultation process.
Finally, we make two points about ACARA’s presentation and promotion of the draft curriculum.
Part of ACARA’s justification for the strong emphasis on problem-solving has been that the mathematics curriculum in Singapore, an education system that performs extremely well in the mathematics component of the Programme for International Student Assessment (PISA), places an emphasis on problem-solving. We seriously question whether the Singaporean sense of “problem-solving” bears even a remote resemblance to ACARA’s use of the term but, in any case, ACARA’s justification fails on its own terms. To begin, there are other education systems that also place a premium on problem-solving but that do not perform at anywhere near the level of Singapore in PISA mathematics. Further, whatever the role of problem-solving in the Singaporean curriculum, this curriculum is also very demanding in terms of fluency with basic skills; no comparable requirements exist in the current Australian Curriculum, and the draft curriculum only pushes to weaken these requirements. The further elimination and weakening of fundamental skills will contribute to the root cause of Australian students’ slipping in international comparisons: the students end up knowing less mathematics.
Secondly, an important aspect of ACARA’s review is that it was intended to be modest in scope, with a focus on “refining” and “decluttering”. The draft curriculum fails in both respects. The radical introduction of the Core Concepts structure and “Mathematising”, the separation into twice the number of strands, the multipurpose nature of the Content, is all the antithesis of modest. This new structure is, inevitably, much clunkier, with massively increased curriculum clutter. The draft curriculum is barely readable.
In brief, the draft curriculum is systemically flawed. It is unworkable, and it fails to capture or to promote the high standard of mathematical knowledge, appreciation and understanding that Australia’s schoolchildren deserve.
The Australian mathematics curriculum requires proper review. Such a review, however, must be undertaken without a pre-ordained outcome, and with the proper participation and consultation of discipline experts. Indeed, ACARA’s own terms of reference for the review specify that the content changes are to be made by subject matter experts, namely mathematicians. It is difficult to imagine that this was the case.
We urge ACARA to remove the current draft mathematics curriculum for consideration and to begin a proper and properly open review, in line with community expectations and with Australia’s needs.