AMSI Calls for a Halt of the Mathematics Curriculum Review

The Australian Mathematical Sciences Institute has finalised and released its submission for ACARA’s consultation on their draft mathematics curriculum. For eccentric reasons, we haven’t properly read AMSI’s submission. (Seriously.) We understand that it is a good and strong statement. The second and key paragraph is

In early April, AMSI, together with some of its key partners, released a joint statement on the proposed new curriculum “Why maths must change”. AMSI initially endorsed the revised draft curriculum in our joint statement. However, there is now an opportunity to comment on the draft curriculum, and we have revised our position, following extensive consultation with representatives of our member organisations. Many members expressed concern, and indeed alarm, at numerous proposed changes. AMSI and its members believe that the new curriculum should be delayed, and we ask ACARA to halt the current review process.

Continue reading “AMSI Calls for a Halt of the Mathematics Curriculum Review”

What Are the Arguments FOR the Draft Mathematics Curriculum?

This one is a companion to our problem-solving treasure hunt, and again amounts to a competition. We have written roughly ten million words on what is wrong with the draft mathematics curriculum. And plenty of people, including a number of big shots, have signed the open letter calling for the draft curriculum to be withdrawn. But where are the arguments for the draft curriculum? There is undoubtedly support for the draft curriculum. In particular, we are aware of a decent amount of snark directed towards the open letter and this blog. What we are unaware of is any substantive arguments in favour of the draft mathematics curriculum. The only articles of which we are aware, we posted on here and here. The first article came out before the draft curriculum and doesn’t amount to a substantive defense of anything. The second article was written in direct response to the open letter, and is so weak as to warrant no response beyond the comments already posted. And, apart from these two articles we are aware of nothing. No blog posts. No tweets. No anything. Just an arrogant and vacuous dismissal of the draft’s critics.* And now to the competition:

What is the strongest argument FOR the draft mathematics curriculum?

To be clear, what we’re asking for are very specific examples of good things within the draft curriculum, examples of content and/or elaborations that are genuine plusses. So, for example, claiming “the focus on mathematising” as a good won’t win a prize. First of all because the suggestion is really stupid, and secondly because such a generalist statement provides no specific evidence of how the mathematising is good. If you really want to argue that the mathematising is a plus then the argument must be based around very specific examples. Similar to our problem-solving competition, the intention here is not to imply or to prove that there is nothing of value in the draft curriculum. Rather, the competition is intended to imply and to prove that there is very little of value in the draft curriculum. Your job is to try to prove us wrong. Answer in the comments below. The provider of the most convincing evidence will win a signed copy of the number one best-selling** A Dingo Ate My Math Book.   *) If anyone is aware of any article/post/tweet/anything in support of the draft curriculum, which also contains at least a hint of evidence, please let us know and we will seek to address it. **) In Polster and Ross households.  

Update (29/07/21)

We’ve finally ended this. The winner is really nobody, but we’ve awarded it to John Friend. See here for details.  

One Week to Email Submissions on the Draft Curriculum

Submissions on ACARA’s draft mathematics curriculum close next week, on July 8, And, note, you do not have to use ACARA’s sheep-herding submission form. You can email your comments to ACARA, via the yellow “Email submissions and comments” button, near the bottom of ACARA’s consultation page. (We could include the email link here, but somehow that feels incorrect.)

Should you submit something? Yes, you should, for the same reason that you should vote against ScoMoFo in the next election. The point isn’t that your action is likely to change anything; the point is that it feels good. So, if it feels good to simply submit the open letter, then do that. But you should submit something.

Not convinced? Then maybe the following will help convince you. ACARA’s consultation page encourages feedback with the following line:

The online survey includes open fields to allow you to provide general comments about what you think we have improved and what you think needs further improvement.

So, either 1) what they’ve improved, or 2) what needs further improvement. On the off chance you believe something might fall into a third category, perhaps you might want to let ACARA know about it.

Does the Draft Mathematics Curriculum Contain Any Problem-Solving?

We’ve written about this before, and the point is obvious. But, it’s apparently not sufficiently obvious for some wilfully blind mathematicians. So, let’s go again. Plus, there’s a prize for the best comment.*

ACARA is playing people with a cute syllogism.

  • Problem-solving is good.
  • The draft curriculum contains lots of problem-solving.
  • Therefore the draft curriculum is good.

Yep, the syllogism is flawed from the get go. But in this post we want to focus on the second line, and we ask:

Does the draft mathematics curriculum contain any problem-solving?

Certainly the draft curriculum contains a hell of a lot of something. As we’ve noted, the draft refers to “investigating” or some variation of the word 298 times. And, students get to “explore” and the like 236 times, and they “model” or whatever 264 times. That’s a baker’s ton of inquiring and real-worlding, which some people, including some really clueless mathematicians, regard as a good thing. Ignoring such cluelessness, what about genuine mathematical problem-solving?

The draft curriculum refers to “problem(s)” to “solve” 154 times. But what do they mean? When, if ever, is the draft referring to a clearly defined mathematical problem that has a clearly defined answer, and which is to be solved with a choice of clearly defined mathematical techniques? To the extent that there are any such “problems”, do they rise above the level of a trivial exercise or computation? In the case of such trivial “problems”, is the label “problem-solving” more than a veil-thin disguise for the mandating of inquiry-learning?

In brief, is there more than a token amount of the draft’s “problem-solving” that is not either real-world “exploring/modelling/investigating” or routine exercises/skills to be taught in a ridiculously inappropriate inquiry manner?

Perhaps genuine mathematical problem-solving is there, and we are honestly curious to see what people have found or can find. But, we’ve found essentially nothing.

And so, to the competition. Find the best example of genuine, mathematical problem-solving in the draft curriculum. Answer in the comments below. The most convincing example will win a signed copy of the number one best-selling** A Dingo Ate My Math Book.

 

*) Yes, yes. we have those other competitions we still haven’t finalised. We will soon, we promise. As soon as we’re out of this ACARA swamp, we’ll be taking significant time out to catch up on our massive tidying backlog.

**) In Polster and Ross households.

 

Update (29/07/21)

We’ve finally ended this. The winner is, hilariously, Glen. See here for details.

 

 

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

 

What if You Hate this Blog

Having put out a few fires, I will return to posting on the draft curriculum. (The open letter can still be signed, here.) But, first, a meta-post on the draft.

It was brought to my attention that a Professor who might have otherwise contemplated signing the open letter did not even consider signing, because the open letter is seen to be “associated with” my ACARA page. The Professor decided that they could not “endorse” the style of criticism that I (and perhaps some commenters) provide there. I am sure the Professor is far from alone. So, how to respond?

Dear Professor, and Others,

I will try to make this simple.

(1) If you agree with the open letter then maybe you should just sign the letter. If not, not. Why is this hard?

(2) The open letter is not mine. I was involved in its production, but it is not my letter. It is no one’s letter. It is a letter stating a point of view, and with a request for ACARA and the ACARA Board to withdraw the draft curriculum. In particular, the letter is not hosted on this blog, and the idea that signing the letter somehow amounts to an endorsement of me or my blog is absolutely absurd. Signing the letter is an endorsement of the letter. That’s what “endorse” means. See point (1).

(3) I have provided the ACARA page (and the draft curriculum page) as assistance, in the unlikely event that someone couldn’t make their way through ACARA’s documentation. You are of course free to ignore my page entirely, and to use other sources. The fact that there are no other sources may be a bit of a hurdle, but I’m afraid that is your problem to solve. In any case, it is up to you to decide how to evaluate ACARA’s draft curriculum, and to act accordingly. See point (1).

(4) I can understand why you may find this blog, and me, distasteful, or worse. I can understand there are good and popular arguments, even just in my own self-interest, for why I should write in a different style. I believe I can defend myself and my blog, but this is not the place to do it. It is not the place to do it, because my blog is not the issue here. The issue here is the draft curriculum and the open letter. See point (1).

See, it’s not really that hard, is it?

Kind Regards, Marty

p. s. See point (1).

 

Education Fires Back Again

There is another contribution from the Education community:

How to do the sums for an excellent maths curriculum

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.)

UPDATE (10/06/21)

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.

UPDATE (10/06/21)

As indicated by SRK, there is now (in effect) a response from John Sweller.

Maths Ed Fires Back

Today in The Conversation there is an article firing back at the open letter to ACARA:

The proposed new maths curriculum doesn’t dumb down content. It actually demands more of students

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.

 

UPDATE (28/6/21)

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?

Open Letter to ACARA and the ACARA Board

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.

UPDATE (08/07/21)

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.

Shallow conceptualisation

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.

Sincerely,

The ACARA Page

Honestly, it wasn’t our intention to write three hundred posts on ACARA and their appalling draft mathematics curriculum. But, we did. Given that we did, it seems worthwhile having a pinned metapost, so that anybody who wants to can find their way through the jungle. (There’s probably a better way to do this, with a separate blog page or whatever, but we can’t be bothered figuring that out right now.)

So, here we are: the complete works, roughly in reverse chronological order, and laid out as clearly as we can think to do it. It includes older posts and articles, on the current mathematics curriculum (which also sucks) and NAPLAN (which also also sucks).

Continue reading “The ACARA Page”