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.

 

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).

 

Open Letter

Open Letter to ACARA and the ACARA Board (02/06/21)

 

The Draft Curriculum

This is mainly the current, ACARA Crash series, on specific aspects of the draft curriculum.

 

Education Fires Back Again (10/06/21 – article from education academics)

Maths Ed Fires Back (09/06/21 – a response to the open letter)

ACARA Crash 12: Let X = X (02/06/21 – algebra in Year 7 Algebra)

ACARA Crash 11: Pulped Fractions (01/06/21 – fraction arithmetic in Year 7)

ACARA Crash 10: Dividing is Conquered (29/05/21 – division in Year 5 and Year 6)

ACARA Crash 9: Their Sorrows Will Multiply (28/05/21 – multiplication in Year 5 and Year 6)

You Got a Problem With That? (27/05/21 – problem-solving)

ACARA Crash 8 – Multiple Contusions (25/05/21 – multiplication tables in Year 4 Algebra)

ACARA Crash 7 – Spread Sheeet (24/05/21 – primes in Year 6 Number)

ACARA Crash 6 – Crossed Words (23/05/21 – word-hunting)

ACARA Crash 5 – Completing the Squander (22/05/21 – quadratics in Year 10 Optional)

ACARA Crash 4 – The Null Fact Law (21/05/21 – quadratics in Year 9 Algebra)

ACARA Crash 3 – Fool’s Gold (16/05/21 – golden ratio in Year 8 Number)

ACARA Crash 2 – Shell Game (15/05/21 – Fibonacci numbers in Year 6 Algebra)

ACARA Crash 1 – The Very Beginning (12/05/21 – counting in Foundation Number)

ACARA Crash 0 – It Was a Dark and Stormy Curriculum (18/05/21 – introductory material in the draft)

WitCH 61: Wheel of Misfortune (05/05/21 – Core Concepts and Organisers)

How Do You Solve a Problem Like ACARA (03/05/21 – problem-solving)

The ACARA Mathematics Draft is Out (summary page of draft documents)

 

Warm Up for the Draft Curriculum

Posts on public commentary, just prior to or with the draft’s release.

De Carvalho, AMSI and that Other Singapore (30/04/21 – comments by ACARA’s CEO and AMSI’s DIrector)

Being Carvalho With the Truth (30/04/21 – speech by ACARA’s CEO)

WitCH 60: Pythagorean Construction (28/04/21 – Pythagoras from ACARA CEO’s speech)

Leading By Example (15/04/21 – comments by AMSI’s Director and others)

Why Mathematics Education Must Change (12/04/21 – statement by AMSI, AAS and others)

ACARA is Confronted With the Big Ideas (17/03/21 – leaked review documents)

 

Curriculum Review

Posts on ACARA’s review documents, leading up to the draft.

Australia v Singapore (28/04/21 – ACARA’s curriculum comparison)

The Key to ACARA’s Universe (27/04/21 – ACARA’s Key Findings from curricula comparisons)

Massing Evidence (20/04/21 – more on the Literature Review)

ACARA’s Illiterature Review (11/04/21 – ACARA’s Literature Review)

 

The Current Australian Curriculum

Obtuse Triangles (25/06/2017 – Pythagoras in Year 9)

A Zillion and One Things to Talk About (18/06/2012 – statistics)

Irrational Thoughts (03/05/2010 – irrational numbers)

The Times Tables They Are A Changin’ (22/04/2010 – multiplication tables in draft curriculum)

New Draft Curriculum a Feeble Tool, Calculated to Bore (04/30/2010 – draft curriculum)

Summing Up a Failure (23/02/2009 – prelude to draft curriculum)

 

NAPLAN

The NAPLAN Numeracy Test Test (19/03/19 – numeracy)

NAPLAN’s Latest Last Legs (13/03/2019 – public criticism of NAPLAN)

We was Robbed (07/10/2018 – former ACARA CEO)

NAPLAN’s Numeracy Test (24/05/2018 – FOI application)

NAPLAN’s Numerological Numeracy (15/08/2017 – NAPLAN data)

NAPLAN’s Mathematical Nonsense, and What it Means for Rural Peru (13/07/21 – NAPLAN question)

Accentuate the Negative (27/05/2017 – NAPLAN problem)

NAPLAN in Kafkaland (12/05/2014 – FOI request)

NAPLAN, numeracy and nonsense (13/05/2013 – NAPLAN problems)

The best laid NAPLAN (09/05/2011 – numeracy)

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.

ACARA Crash 12: Let X = X

(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.
Right again.

Let X=X.

You know, that it’s for you.
It’s a blue sky curriculum.
Parasites are out tonight.
Let X=X.

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?

Let X=X.

I got this postcard.
And it read, it said: Dear Amigo – Dear Partner.
Listen, uh – I just want to say thanks.
So…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.
XXXXOOOO.

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.

LEVEL DESCRIPTION 

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.

CONTENT 

create algebraic expressions using constants, variables, operations and brackets. Interpret and factorise these expressions, applying the associative, commutative, identity and distributive laws as applicable

ELABORATIONS

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

OMISSION

Solving simple linear equations

JUSTIFICATION

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

 

ACARA Crash 11: Pulped Fractions

We’re still crazy-nuts with work, so, for today, it’s just another fish. This one is from Year 7 Number. and appears to be the sum of fraction arithmetic in Year 7.

LEVEL DESCRIPTION 

As students engage in learning mathematics in Year 7 they … develop their understanding of integer and rational number systems and their fluency with mental calculation, written algorithms, and digital tools and routinely consider the reasonableness of results in context

ACHIEVEMENT STANDARD 

By the end of Year 7, students use all four operations in calculations involving positive fractions and decimals, using the properties of number systems and choosing the computational approach. … They determine equivalent representations of rational numbers and choose from fraction, decimal and percentage forms to assist in computations. They solve problems involving rational numbers, percentages and ratios and explain their choice of representation of rational numbers and results when they model situations, including those in financial contexts.

CONTENT 

determine equivalent fraction, decimal and percentage representations of rational numbers. Locate and represent positive and negative fractions, decimals and mixed numbers on a number line

ELABORATIONS

investigating equivalence of fractions using common multiples and a fraction wall, diagrams or a number line to show that a fraction such as \color{blue}\boldsymbol{\frac23} is equivalent to \color{blue}\boldsymbol{\frac46} and \color{blue}\boldsymbol{\frac69} and therefore \color{blue}\boldsymbol{\frac23 < \frac56}

expressing a fraction in simplest form using common divisors

applying and explaining the equivalence between fraction, decimal and percentage representations of rational numbers, for example, \color{blue}\boldsymbol{16\%, 0.16, \frac{16}{100}} and \color{blue}\boldsymbol{\frac4{25}}, using manipulatives, number lines or diagrams

representing positive and negative fractions and mixed numbers on various intervals of the real number line, for example, from -1 to 1, -10 to 10 and number lines that are not symmetrical about zero or without graduations marked

investigating equivalence in fractions, decimals and percentage forms in the patterns used in the weaving designs of Aboriginal and Torres Strait Islander Peoples

CONTENT

carry out the four operations with fractions and decimals and solve problems involving rational numbers and percentages, choosing representations that are suited to the context and enable efficient computational strategies

ELABORATIONS 

exploring addition and subtraction problems involving fractions and decimals, for example, using rectangular arrays with dimensions equal to the denominators, algebra tiles, digital tools or informal jottings

choosing an appropriate numerical representation for a problem so that efficient computations can be made, such as \color{blue}\boldsymbol{12.5\%, \frac{1}{8}, 0.125} or \color{blue}\boldsymbol{\frac{25}{1000}}

developing efficient strategies with appropriate use of the commutative and associative properties, place value, patterning, multiplication facts to solve multiplication and division problems involving fractions and decimals, for example, using the commutative property to calculate \color{blue}\boldsymbol{\frac23} of \color{blue}\boldsymbol{\frac12} giving \color{blue}\boldsymbol{\frac12} of \color{blue}\boldsymbol{\frac23 = \frac13}

exploring multiplicative (multiplication and division) problems involving fractions and decimals such as fraction walls, rectangular arrays, algebra tiles, calculators or informal jottings

developing efficient strategies with appropriate use of the commutative and associative properties, regrouping or partitioning to solve additive (addition and subtraction) problems involving fractions and decimals

calculating solutions to problems using the representation that makes computations efficient such as 12.5% of 96  is more efficiently calculated as \color{blue}\boldsymbol{\frac18} of 96, including contexts such as, comparing land-use by calculating the total local municipal area set aside for parkland or manufacturing and retail, the amount of protein in daily food intake across several days, or increases/decreases in energy accounts each account cycle

using the digits 0 to 9 as many times as you want to find a value that is 50% of one number and 75% of another using two-digit numbers

CONTENT

model situations (including financial contexts) and solve problems using rational numbers and percentages and digital tools as appropriate. Interpret results in terms of the situation

ELABORATIONS

calculating mentally or with calculator using rational numbers and percentages to find a proportion of a given quantity, for example, 0.2 of total pocket money is spent on bus fares, 55% of Year 7 students attended the end of term function, 23% of the school population voted yes to a change of school uniform

calculating mentally or with calculator using rational numbers and percentages to find a proportion of a given quantity, for example, 0.2 of total pocket money is spent on bus fares,  of Year 7 students attended the end of term function,  of the school population voted yes to a change of school uniform

interpreting tax tables to determine income tax at various levels of income, including overall percentage of income allocated to tax

using modelling contexts to investigate proportion such as proportion of canteen total sales happening on Monday and Friday, proportion of bottle cost to recycling refund, proportion of school site that is green space; interpreting and communicating answers in terms of the context of the situation

expressing profit and loss as a percentage of cost or selling price, comparing the difference

investigating the methods used in retail stores to express discounts, for example, investigating advertising brochures to explore the ways discounts are expressed

investigating the proportion of land mass/area of Aboriginal Peoples’ traditional grain belt compared with Australia’s current grain belt

investigating the nutritional value of grains traditionally cultivated by Aboriginal Peoples in proportion to the grains currently cultivated by Australia’s farmers

WitCH 62: Video Killed the Proficiency Star

It’s not fun, but we gotta do it.

Yesterday, we wrote about Ofsted’s review of mathematics education. We wrote that it is a great review, and it is a great review. It is perfect in its reactionaryism. But, alas, there’s also a video.

Below is Hannah Stoten, one of the review authors and future kidnap victim, launching the review. An excellent review, and a well-spoken author. What could go wrong?

 

UPDATE (01/06/21)

Well, that’s not good:

Nonetheless, we’re willing to assume that Hannah is innocent here, and plans to kidnap her and bring her to Australia are proceeding apace.

RatS 12: Sit Up and Think of England

Good news. We’re giving ACARA, and our readers, the night off. No painful reading tonight; just painful reality.

Ofsted is the UK’s ACARA-ish organisation, although the “ish” hides the fact that Ofsted appears to be competent. Last week, Oftsed published a review into mathematics education. We’re sure we are missing something, because the review appears to be important, clear and correct.

Reportedly the work of Hannah Stoten,* the document lays out in a clear and methodical manner what a mathematics education entails, and thus the nature of a proper mathematics curriculum. Here is how the review gets going:

How the review classifies mathematics curriculum content

For this review, we have classified mathematical curriculum content into declarative, procedural and conditional knowledge.

Declarative knowledge is static in nature and consists of facts, formulae, concepts, principles and rules.

All content in this category can be prefaced with the sentence stem ‘I know that’.

Procedural knowledge is recalled as a sequence of steps. The category includes methods, algorithms and procedures: everything from long division, ways of setting out calculations in workbooks to the familiar step-by-step approaches to solving quadratic equations.

All content in this category can be prefaced by the sentence stem ‘I know how’.

Conditional knowledge gives pupils the ability to reason and solve problems. Useful combinations of declarative and procedural knowledge are transformed into strategies when pupils learn to match the problem types that they can be used for.

All content in this category can be prefaced by the sentence stem ‘I know when’.

When pupils learn and use declarative, procedural and conditional knowledge, their knowledge of relationships between concepts develops over time. This knowledge is classified within the ‘type 2’ sub-category of content (see table below). For example, recognition of the deep mathematical structures of problems and their connection to core strategies is the type 2 form of conditional knowledge.

Summary table of content categories considered in the review:

Category Type 1 Type 2
Declarative "I know that" Facts and formulae Relationship between facts (conceptual understanding)
Procedural "I know how" Methods Relationship between facts, procedures and missing facts (principles/mechanisms)
Conditional "I know when" Strategies Relationship between information, strategies and missing information (reasoning)

Is this perfect? Of course not. It would be easy to nitpick over borderline calls. But as a basic guide to building and analysing a curriculum, it is beautifully simple and clear. As guides should be. And as ACARA’s Wheel of Death most definitely is not.

There’s plenty more we could quote. Like the whole damn thing. But we’ll restrain ourselves, and give just a few more. Here’s a note on “core knowledge”:

Foundational knowledge, particularly proficiency in number, gives pupils the ability to progress through the curriculum at increasing rates later on. The path of learning that begins with a diligent focus on core declarative and procedural knowledge is not a straight line, therefore, but a curve. This is a function of the curriculum’s intelligent design. For example, in countries where pupils do well, pupils are able to attempt more advanced aspects of multiplication and division in Year 4 if they have been given more time on basic arithmetic in Year 1. This may explain why successful curriculum approaches tend to emphasise core knowledge early on.

So, arithmetic skills are kind of important, especially early on. Who would have guessed?

Here’s an early comment on problem-solving:

Problem-solving requires pupils to hold a line of thought. It is not easy to learn, rehearse or experience if the facts and methods that form part of a strategy for solving a problem type are unfamiliar and take up too much working memory. For example, pupils are unlikely to be able to solve an area word problem that requires them to multiply 2 lengths with different units of measurement if they do not realise that the question asks them to use a strategy to find an area. They are also unlikely to be successful if they do not know many number bonds, unit measurement facts, conversion formula or an efficient method of multiplication to automaticity. Therefore, the initial focus of any sequence of learning should be that pupils are familiar with the facts and methods that will form the strategies taught and applied later in the topic sequence.

What’s this? Give the kids the knowledge and skills and techniques before having them embark on problem-solving? Are these people nuts?

One last one, on “positive attitude”:

Pupils are more likely to develop a positive attitude towards mathematics if they are successful in it, especially if they are aware of their success. However, teachers should be wary of the temptation to invert this causal pathway by, for example, substituting fun games into lessons as a way of fostering enjoyment and motivation. This is because using games as a learning activity can lead to less learning rather than more.

Some pupils become anxious about mathematics. It is not the nature of the subject but failure to acquire knowledge that is at the root of the anxiety pathway. The origins of this anxiety may have even been present at the start of a pupil’s academic journey. However, if teachers ensure that anxious pupils acquire core mathematical knowledge and start to experience success, those pupils will begin to associate the subject with enjoyment and motivation.

It’s hard to believe, but they seem to be suggesting that to get a kid to like mathematics you should get them good at it, rather than pretending they’re good at it. Crazy, crazy stuff.

Read the whole damn thing. We haven’t read it all yet, and yeah, we’ll probably find something in there that annoys us (because we’re that type). But we haven’t found it yet. It is a great, great document.

UPDATE (31/05/21)

A couple of colleagues, Simon the Likeable and The Hot Dog Man, indicated that they were puzzled by the Review. They both wondered if perhaps what Hannah Stoten is saying isn’t really simple. Indeed, they are correct. In a nutshell, this is Stoten’s message:

The last 50+ years have been a complete screw up. Forget about them, and start again.

 

*) (Update 31/05/21) And Steve Wren. Efforts are already underway to kidnap them both. No one tell them.

ACARA Crash 10: Dividing is Conquered

This Crash is a companion to, and overlaps with, the previous Crash, on multiplication. It is from Year 5 and Year 6 Number. and is, as near as we can tell, the sum of the instruction on techniques of division for F-6.

ACHIEVEMENT STANDARD (YEAR 5)

They apply knowledge of multiplication facts and efficient strategies to … divide by single-digit numbers, interpreting any remainder in the context of the problem.

CONTENT (YEAR 5)

choose efficient strategies to represent and solve division problems, using basic facts, place value, the inverse relationship between multiplication and division and digital tools where appropriate. Interpret any remainder according to the context and express results as a mixed fraction or decimal

ELABORATIONS

developing and choosing efficient strategies and using appropriate digital technologies to solve multiplicative problems involving multiplication of large numbers by one- and two-digit numbers

solving multiplication problems such as 253 x 4 using a doubling strategy, for example, 253 + 253 = 506, 506 + 506 = 1012

solving multiplication problems like 15 x 16 by thinking of factors of both numbers, 15 = 3 x 5, 16 = 2 x 8; rearranging the factors to make the calculation easier, 5 x 2 = 10, 3 x 8 = 24, 10 x 24 = 240

using an array model to show place value partitioning to solve multiplication, such as 324 x 8, thinking 300 x 8 = 2400, 20 x 8 = 160, 4 x 8 = 32 then adding the parts, 2400 + 160 + 32 = 2592; connecting the parts of the array to a standard written algorithm

investigating the use of digital tools to solve multiplicative situations managed by First Nations Ranger Groups and other groups to care for Country/Place including population growth of native and feral animals such as comparing rabbits or cane toads with platypus or koalas, or the monitoring of water volume usage in communities

LEVEL DESCRIPTION (YEAR 6)

use all four arithmetic operations with natural numbers of any size

ACHIEVEMENT STANDARD (YEAR 6)

Students apply knowledge of place value, multiplication and addition facts to operate with decimals.

CONTENT (YEAR 6)

apply knowledge of place value and multiplication facts to multiply and divide decimals by natural numbers using efficient strategies and appropriate digital tools. Use estimation and rounding to check the reasonableness of answers

ELABORATIONS

applying place value knowledge such as the value of numbers is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a natural number by a decimal of at least tenths

applying and explaining estimation strategies to multiplicative (multiplication and division) situations involving a natural number that is multiplied or divided by a decimal to at least tenths before calculating answers or when the situation requires just an estimation

deciding to use a calculator in situations that explore multiplication and division of natural numbers being multiplied or divided by a decimal including beyond hundredths

explaining the effect of multiplying or dividing a decimal by 10, 100, 1000… in terms of place value and not the decimal point shifting

ACARA Crash 9: Their Sorrows Shall Be Multiplied

We still have no time for the deep analysis of this shallow nonsense. So, we’ll just continue with the fish.

Below are two content-elaborations combos, from Year 5 and Year 6 Number. As near as we can tell, that’s about the sum of the instruction on techniques of multiplication for F-6.

ACHIEVEMENT STANDARD (YEAR 5)

They apply knowledge of multiplication facts and efficient strategies to multiply large numbers by one-digit and two-digit numbers

CONTENT (YEAR 5)

choose efficient strategies to represent and solve problems involving multiplication of large numbers by one-digit or two-digit numbers using basic facts, place value, properties of operations and digital tools where appropriate, explaining the reasonableness of the answer

ELABORATIONS

interpreting and solving everyday division problems such as, ‘How many buses are needed if there are 436 passengers, and each bus carries 50 people?’, deciding whether to round up or down in order to accommodate the remainder

solving division problems mentally like 72 divided by 9, 72 ÷ 9, by thinking, ‘how many 9 makes 72’, ? x 9 = 72 or ‘share 72 equally 9 ways’

investigating the use of digital technologies to solve multiplicative situations managed by First Nations Ranger Groups and other groups to care for Country/Place including population growth of native and feral animals such as comparing rabbits or cane toads with platypus or koalas, or the monitoring of water volume usage in communities

LEVEL DESCRIPTION (YEAR 6)

use all four arithmetic operations with natural numbers of any size

ACHIEVEMENT STANDARD (YEAR 6)

Students apply knowledge of place value, multiplication and addition facts to operate with decimals.

CONTENT (YEAR 6)

apply knowledge of place value and multiplication facts to multiply and divide decimals by natural numbers using efficient strategies and appropriate digital tools. Use estimation and rounding to check the reasonableness of answers

ELABORATIONS

applying place value knowledge such as the value of numbers is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a natural number by a decimal of at least tenths

applying and explaining estimation strategies to multiplicative (multiplication and division) situations involving a natural number that is multiplied or divided by a decimal to at least tenths before calculating answers or when the situation requires just an estimation

deciding to use a calculator in situations that explore multiplication and division of natural numbers being multiplied or divided by a decimal including beyond hundredths

explaining the effect of multiplying or dividing a decimal by 10, 100, 1000… in terms of place value and not the decimal point shifting

UPDATE (29/50/21)

we’ve just discovered some multiplication techniques tucked inside some division elaborations, as indicated in this companion Crash. The two Crashes should be considered together (and should have been just one Crash, dammit.)

You Got a Problem With That?

We’ll take a day off from bashing the draft curriculum, in order to bash the draft curriculum. This one’s not a Crash post, but it gets to the disfigured heart of the draft.

Yesterday, a good friend and colleague, let’s call him Mr. Big, threw a book at us. By Alexandre Borovik and Tony Gardiner, the book is called The Essence of Mathematics Through Elementary Problems. The book is free to download, and it is beautiful.

There is much to say about this book. It is, unsurprisingly, a collection of problems and solutions. By “elementary”, the authors mean, in the main, in the domain of secondary school mathematics. Note that “elementary” does not equate to “easy”, although there are easy problems as well.

The problems have been chosen with great care. As the authors write, the problems are included for two reasons:

    • they constitute good mathematics
    • they embody in a distilled form the quintessential spirit of elementary mathematics

As indicated by the the Table of Contents, the problems in The Essence of Mathematics are also arranged very carefully, by topic and in a roughly increasing level of conceptual depth, and the book includes interesting and insightful commentary. Their twenty problems and solutions embodying 3 – 1 = 2 is a beautiful illustration.

The Essence of Mathematics also contains an incredibly important message. Here is the very first problem in the book:

1(a)   Compute for yourself, and learn by heart, the times tables up to 9 × 9.

Regular readers will know exactly where we’re going with this. Chapter 1 of Essential Mathematics is titled Mental Skills, it includes simple written skills as well, and the message is obvious. As the authors write,

The chapter is largely devoted to underlining the need for mastery of a repertoire of instantly available techniques, that can be used mentally, quickly, and flexibly to analyse less familiar problems at sight.

In particular, on their first problem,

Multiplication tables are important for many reasons. They allow us to appreciate directly, at first hand, the efficiency of our miraculous place value system – in which representing any number, and implementing any operation, are reduced to a combined mastery of

(i) the arithmetical behaviour of the ten digits 0–9, and

(ii) the index laws for powers of 10.

Fluency in mental and written arithmetic then leaves the mind free to notice, and to appreciate, the deeper patterns and structures which may be lurking just beneath the surface.

What does all this have to do with ACARA’s draft curriculum? Alas, nothing whatsoever.

The draft curriculum is the antithesis of Essence. The “problems” and “investigations” and “models” in the draft curriculum are anything but well-chosen, being typically sloppy and ill-defined, with no clear direction or purpose. The draft curriculum also displays nothing but contempt for the prior mastery of basic facts and skills required for problem-solving, or anything.

Essence is not a textbook, but the authors clearly see a large role for problem-solving in mathematics education, and, with genuine modesty, they can imagine their book as a natural supplement to a good curriculum. Such a role can mean slow and open-ended learning, or at least open-ended teaching:

Learning mathematics is a long game; and teachers and students need the freedom to digress, to look ahead, and to build slowly over time. 

The value of such digressions and explorations, however, does not negate the primary goal of mathematics education:

Teachers at each stage must be free to recognise that their primary responsibility is not just to improve their students’ performance on the next test, but to establish a firm platform on which subsequent stages can build.

The effect [of political pressures] has been to downgrade the more important challenges which every student should face: namely

    • of developing a robust mastery of new, forward-looking techniques (such as fractions, proportion, and algebra), and
    • of integrating the single steps students have at their disposal into larger, systematic schemes, so that they can begin to tackle and solve simple multi-step problems.

Building systematic schemes out of the mastery of techniques. Or, there’s the alternative:

A didactical and pedagogical framework that is consistent with the essence, and the educational value of elementary mathematics cannot be rooted in false alternatives to mathematics (such as numeracy, or mathematical literacy).

There is problem-solving, and there is “problem-solving”. ACARA is shovelling the latter.

 

UPDATE (28/05/21)

Mrs. Big, AKA Mrs. Uncle Jezza, has given the draft curriculum a very good whack in the comments, below. As part of that, she has noted an excellent quotation that begins the Preface of Essential. The quotation is by Richard Courant and Herbert Robbins, and is from the Preface of their classic What is Mathematics?

“Understanding mathematics cannot be transmitted by painless entertainment … actual contact with the content of living mathematics is necessary. The present book … is not a concession to the dangerous tendency toward dodging all exertion.”

While we’re here, we’ll include another great quote, from the About section of Essential, by John von Neumann:

“Young man, in mathematics you don’t understand things. You just get used to them.”

Understanding is a fine goal, but it can also be a dangerously distracting goal. ACARA’s “deep understanding” is an absurdity.