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

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.)

ACARA Crash 8: Multiple Contusions

OK, roll out the barrel, grab the gun: it’s time for the fish. Somehow we thought this one would take work but, really, there’s nothing to say.

It has obviously occurred to ACARA that the benefits of their Glorious Revolution may not be readily apparent to us mathematical peasants. And, one of the things we peasants tend to worry about are the multiplication tables. It is therefore no great surprise that ACARA has addressed this issue in their FAQ:

When and where are the single-digit multiplication facts (timetables) covered in the proposed F–10 Australian Curriculum: Mathematics?

These are explicitly covered at Year 4 in both the achievement standard and content descriptions for the number strand. Work on developing knowledge of addition and multiplication facts and related subtraction and division facts, and fluency with these, takes place throughout the primary years through explicit reference to using number facts when operating, modelling and solving related problems.

Nothing spells sincerity like getting the name wrong.* It’s also very reassuring to hear the kids will be “developing knowledge of … multiplication facts”. It’d of course be plain foolish to grab something huge like 6 x 3 all at once. In Year 4. And, how again will the kids “develop” this knowledge? Oh yeah, “when operating, modelling and solving related problems”. It should work a treat.

That’s the sales pitch. That’s ACARA’s conscious attempt to reassure us peasants that everything’s fine with the “timetables”. How’s it working? Feeling good? Wanna feel worse?

What follows is the relevant part of the Year Achievement Standards, and the Content-Elaboration for “multiplication facts” in Year 4 Algebra.

ACHIEVEMENT STANDARD

By the end of Year 4, students … model situations, including financial contexts, and use … multiplication facts to … multiply and divide numbers efficiently. … They identify patterns in the multiplication facts and use their knowledge of these patterns in efficient strategies for mental calculations. 

CONTENT

recognise, recall and explain patterns in basic multiplication facts up to 10 x 10 and related division facts. Extend and apply these patterns to develop increasingly efficient mental strategies for computation with larger numbers

ELABORATIONS

using arrays on grid paper or created with blocks/counters to develop and explain patterns in the basic multiplication facts; using the arrays to explain the related division facts

using materials or diagrams to develop and record multiplication strategies such as skip counting, doubling, commutativity, and adding one more group to a known fact

using known multiplication facts for 2, 3, 5 and 10 to establish multiplication facts for 4, 6 ,7 ,8 and 9 in different ways, for example, using multiples of ten to establish the multiples 9 as ‘to multiply a number by 9 you multiply by 10 then take the number away’; 9 x 4 = 10 x 4 – 4 , 40 – 4 = 36 or using multiple of three as ‘to multiply a number by 9 you multiply by 3, and then multiply the result by 3 again’

using the materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication

using known multiplication facts up to 10 x 10 to establish related division facts

 

Alternatively, the kids could just learn the damn things. Starting in, oh, maybe Year 1? But what would we peasants know.

 

*) It has since been semi-corrected to “times-tables”.

ACARA Crash 7: Spread Sheeet

(In keeping with our culturally sensitive ways, the title should be read with a thick Mexican accent.)

We’re working on a non-WitCHlike Crash post, but no way will that be done tonight. Luckily, frequent commenter Glen has flagged some easily postable nonsense, and we can keep the Crash ball rolling.

This A-Crash consists of a Content-Elaboration combo for Year 6 Number:

CONTENT

identify and describe the properties of prime and composite numbers and use to solve problems and simplify calculations

ELABORATIONS

understanding that a prime number has two unique factors of one and itself and hence 1 is not a prime number

testing numbers by using division to distinguish between prime and composite numbers, recording the results on a number chart to identify any patterns

representing composite numbers as a product of their factors including prime factors when necessary and using this form to simplify calculations involving multiplication such as \color{blue}\boldsymbol{15 \times 16} as \color{blue}\boldsymbol{5 \times 3 \times 4 \times 4} which can be rearranged to simplify calculation to \color{blue}\boldsymbol{5 \times 4 \times 3 \times 4 =20 \times 12}

using spread sheets to list all of the numbers that have up to three factors using combinations of only the first three prime numbers, recognise any emerging patterns, making conjectures and experimenting with other combinations

understanding that if a number is divisible by a composite number then it is also divisible by the prime factors of that number, for example, 216 is divisible by 8 because the number represented by the last three digits is divisible by 8, and hence 216 is also divisible by 2 and 4, using this to generate algorithms to explore

 

UPDATE (25/05/21)

Thanks, everyone, so far. We’re going nuts with work, so a quick WitCHlike update while the window is open.

0) How can ACARA be so, so, so appallingly bad with their grammar and punctuation? We honestly don’t get it. Is the content descriptor accidentally missing a pronoun, and a comma, and a preposition, or do they genuinely like how it reads?

1) Yes, the free-floating and otherwise irritating “hence”, the fact that “prime” is undefined is appalling. So is using “1” and “one” in the same sentence to refer to the same thing. So is “two unique factors of one and itself and …”.

2) Possibly John’s guess on the second elaboration is correct. What would be focussed and useful is to take a 12 x 12 table of numbers and cross off the multiples (and circle 1). So, you get the kids to do the sieve of Eratosthenes thing, and emphasise the multiples as composites. You know, a clearly expressed investigation, with clear purposes.

3) This is Year 6, and so we’re not so concerned about “Fundamental theorem of arithmetic” not being mentioned here, although of course both existence and uniqueness of the prime factorisation should have been spelled out, even if only as something to “explore”. It’s way too important to be included as just a “by the way” part of a multiplication trick. As a side point, in regard to our previous Crash post, it is notable when and how “Fundamental theorem” first appears.

4) 15 x 16? Really?

5) We’re guessing the spread sheeet activity was intended to mean using each prime at most once. Given these people can’t write, however, it’s only a guess. But if so, that would be a reasonable exercise, IF you ditched the spread sheeet, and IF you repeated the exercise a few times with varying selections of primes. None of which will happen.

6) It is unbelievably stupid to introduce prime stuff in combination with divisibility tricks. The former is, well, fundamental, and the latter is a base ten game.

7) “The number represented by the last three digits”. Of what? Who talks this way? Who talks this way and expects to be understood?

8) What are the other digits of 216?

9) Even if there were other digits, a number ending in 216 is a really stupid choice to demonstrate divisibility by 8. These things matter.

ACARA Crash 6: Crossed Words

Lots of 'em

Word/Phrase Number of Occurrences Clarification

Not lots of 'em

Word/Phrase Number of Occurrences Clarification

 

Alright, kiddies, this is one that you can play at home. Just grab your handy copy of the Daft Australian Curriculum, and go word searching. For example, you might look up the word “aimless” and, strangely, nothing will occur. On the other hand, look up “effective”/”effectively” and, just as strangely, you will get plenty of hits.

So, go to it. Look up your favourite/anti-favourite mathematical words and phrases, and let us know the number of hits in the comments. We’ll keep track of the results in our handy dandy Lots and Not-Lots tables, above.

Just a few quick notes:

*) Different derivatives of the same root word or phrase should be grouped together.

*) We’ll add clarifying notes on usage of the word/phrase when it seems appropriate.

*) We won’t be checking your puzzling skills very carefully. We’ll simply put up the numbers, and it’s up to others to do the checking. Then, we’ll correct the totals when need be.

Happy hunting.

ACARA Crash 5: Completing the Squander

The previous A-Crash consisted of everything we could find in the Daft Curriculum on the algebraic treatment of polynomials and polynomial equations. This companion A-Crash consists of everything we could find in the Year 10 draft on the same material. 

CONTENT (Year 10)

expand and factorise expressions and apply exponent laws involving products, quotients and powers of variables. Apply to solve equations algebraically

ELABORATIONS

reviewing and connecting exponent laws of numerical expressions with positive and negative integer exponents to exponent laws involving variables

using the distributive law and the exponent laws to expand and factorise algebraic expressions

explaining the relationship between factorisation and expansion

applying knowledge of exponent laws to algebraic terms, and simplifying algebraic expressions using both positive and negative integral exponents to solve equations algebraically

CONTENT (Year 10 Optional Content)

numerical/tabular, graphical and algebraic representations of quadratic functions and their transformations in order to reason about the solutions of \color{blue}\boldsymbol{f(x) = k}

 ELABORATIONS

connecting the expanded and transformed representations

deriving and using the quadratic formula and discriminant to identify the roots of a quadratic function

identifying what can be known about the graph of a quadratic function by considering the coefficients and the discriminant to assist sketching by hand

solving equations and interpreting solutions graphically

recognising that irrational roots of quadratic equations of a single real variable occur in conjugate pairs

ACARA Crash 4: The Null Fact Law

Well, the plan to post each day lasted exactly one day.* We have an excuse,** but we won’t make excuses. We’ll try to do better.

This A-Crash consists of two Content-Elaboration combos for Year 9 Algebra.

CONTENT

expand and factorise algebraic expressions including simple quadratic expressions

ELABORATIONS

recognising the application of the distributive law to algebraic expressions

using manipulatives such as algebra tiles or an area model to expand or factorise algebraic expressions with readily identifiable binomial factors, for example, \color{blue}\boldsymbol{4x(x + 3) = 4x^2 +12x} or \color{blue}\boldsymbol{(x + 1)(x + 3) = x^2 + 4x + 3}

recognising the relationship between expansion and factorisation and identifying algebraic factors in algebraic expressions including the use of digital tools to systematically explore factorisation from \color{blue}\boldsymbol{x^2 + bx + c} where one of \color{blue}\boldsymbol{b} or \color{blue}\boldsymbol{c} is fixed and the other coefficient is systematically varied

exploring the connection between exponent form and expanded form for positive integer exponents using all of the exponent laws with constants and variables

applying the exponent laws to positive constants and variables using positive integer exponents 

investigating factorising non-monic trinomials using algebra tiles or strategies such as the area model or pattern recognition

CONTENT

graph simple non-linear relations using graphing software where appropriate and solve linear and quadratic equations involving a single variable graphically, numerically and algebraically using inverse operations and digital tools as appropriate

 ELABORATIONS

graphing quadratic and other non-linear functions using digital tools and comparing what is the same and what is different between these different functions and their respective graphs

using graphs to determine the solutions to linear and quadratic equations

representing and solving linear and quadratic equations algebraically using a sequence of inverse operations and comparing these to graphical solutions

graphing percentages of illumination of moon phases in relationship with Aboriginal and Torres Strait Islander Peoples’ understandings that describe the different phases of the moon

 

*) Luckily, 1 is a Fibonacci number.

**) “Burkard, please put down the whip.”

ACARA Crash 0: It Was a Dark and Stormy Curriculum

It was a dark and stormy curriculum; the jargon fell in torrents—except at occasional intervals, when it was checked by violent gusts of puffery which swept up the streets (for it is in Australia that our scene lies), rattling along the schooltops, and fiercely agitating the scanty flames of thought that struggled against the darkness.

Yeah, yeah, a mixed metaphor, or something, but it’s really late and we’re really tired. Anyway, the point is that the writing in the Daft Mathematics Curriculum sucks, and the Introduction really sucks. Like Bulwer-Lytton level of sucking. And, of course, embedded in the suckingness, there is the awfulness.

We’ll get back to hammering the Content-Elaborations as soon as possible, since that’s where the rotten meets the road. Someone, however, has to write something about the godawful Introduction. Which is much easier said than done. The damn thing is sixteen pages. Sixteen Bulwer-Lytton pages. By way of comparison, the introduction to the Current Curriculum is three short, to-the-point pages. (Such modification is what David de Carvalho likes to refer to as “refining” and “decluttering”.)

OK, sure, “mathematising” may be a brilliant new concept.* Nonetheless, someone has to be plenty pleased with themselves to believe that their shiny new toy warrants five times the introduction. And, yes, the ACARA writers are indeed pleased; the Introduction is dripping with smug satisfaction, declarations of the wonderfulness of their new scheme. It goes without saying that this continual self-congratulation really assists the overall flow.

Alright, time to hold our noses and dive in. We’ll take it section by section.**

 

RATIONALE (p 1)

 This section is mostly florid motherhooding: “deep learning” and “creative” and so forth. One sentence, however, is worth noting, as it is a portent: 

“Throughout schooling, actions such as posing questions, abstracting, recognising patterns, practising skills, modelling, investigating, experimenting, simulating, making and testing conjectures, play an important role in the growth of students’ mathematical knowledge and skills.”

This is explicitly advocating an experimental/”problem-solving” approach to learning mathematics. Yes “Practising skills” and “abstracting” are there (learning facts is not), but they are just two “actions” in a very long list. Moreover, it is simply false to claim that the rest of these “actions” can play more than a trivial role in “the growth of students’ mathematical knowledge and skills”. Unless, that is, ACARA reinterprets “skills” to include the skill of modelling and so forth. Which ACARA can do, but which then also means ACARA is playing a cup and balls trick with their terminology.

 

FUNDAMENTAL STRUCTURE (pp 2-5)

 Each year level of the Draft Curriculum contains

*) Year Level Description “overview of the learning”

*) Achievement Standards – “expected quality of learning”

*) Content items“essential knowledge, understanding and skills”

*) Elaborations on each Content item – “suggestions and illustrations”

The content and companion elaborations are organised under six “strands“:

*) Number, Algebra, Measurement, Space, Statistics, Probability.

This replaces the current structure of three double strands — Number-Algebra and so forth — and it is self-evidently ridiculous. It ignores and thus weakens the critical connection between number and algebra. It also means that to have “Algebra” in primary school, ACARA simply has to make stuff up; they have to redefine “algebra” to be pattern-hunting or whatnot.

This is then not simply a case of having the same stuff under different labels. Once algebra is separated from number, it discourages semi-algebraic approaches to arithmetic, and to arithmetic problems. It discourages taking natural conceptual steps from arithmetic to algebra, which can be done, and should be done, in primary school.

The numerous strands also makes it easier for ACARA to push the overhyped Statistics and, more generally, ACARA’s real-world fetishism. This comes out most clearly in the splitting of the current double strand of Measurement-Geometry into Measurement and Space.

Why “Space”? Why not Geometry? The description indicates exactly why:

Space develops ways of visualising, representing and working with the location, direction, shape, placement, proximity and transformation of objects at macro, local and micro size in natural and created worlds. It underpins the capacity to construct pictures, diagrams, maps, projections, models and graphic images that enable the manipulation and analysis of shapes and objects through actions and the senses. This includes notions such as continuity, curve, surface, region, boundary, object, dimension, connectedness, symmetry, direction, congruence and similarity in art, design, architecture, planning, transportation, construction and manufacturing, physics, engineering, chemistry, biology and medicine.

Bulwer-Lytton sits up in his grave, and tips his hat. 

The point of this, and the clear awfulness of this, is Geometry, the mathematical consideration of abstract objects, has been trivialised to a tiny element of real-world investigation. Space includes a ton of what would currently be thought of as coming under Measurement, effectively airbrushing Geometry out of existence. And, then, what is the Measurement strand? Well, it’s pretty much just measurement, just quantifying, which is a fine, correct use of the word. Except that as a strand of mathematics it’s pretty damn trivial.

 

STUFF OVERLYING THE SIX STRANDS — CORE CONCEPTS (pp 5-8)

This is the stuff underlying ACARA’s hideous wheel, and it is when things get truly appalling.

The Core Concepts are intended to replace the four “proficiencies” in the Current Curriculum:

*) Understanding, Fluency, Reasoning, and Problem-Solving.  (Current Curriculum)

The current proficiencies aren’t that helpful in practice, since at least the first three proficiencies are much more intermingled than is suggested.***  Still, the current proficiencies are fundamentally coherent. No longer …

The three “Core Concepts” are those blue arcs surrounding the six strands:

*) Mathematical Structures,  Mathematical Approaches,  Mathematising.

Even ignoring the garishness of “mathematising”, the entire thing is absurd. What can “mathematising” mean other than to deal with a “mathematical structure” with a “mathematical approach”. How is “mathematising” anything other than the verb form of the noun phrase “mathematical approaches”? Why is “abstraction” a structure, rather than abstracting as an approach? Why is “generalising” an “approach” rather than “generalisation” a structure? How is “thinking and reasoning” a separate approach? Are the other approaches unthinking and unreasoned? What does “manipulating mathematical objects” mean? Do the other approaches not involve manipulation of anything? Why bother asking any questions at all about something so self-evidently meaningless? Where’s our vodka?

In twenty years of investigating educational absurdity, this diagram and its description out-absurds anything else we’ve seen. By a mile.

 

STUFF OVERLYING MATHEMATICS — GENERAL CAPABILITIES (pp 8-10)

Everything in the (not just mathematics) Curriculum is supposed to promote the General Capabilities:

*) Numeracy, Literacy, Critical and Creative Thinking, Digital Literacy, Ethical Understanding, Personal and Social capability, Intercultural Understanding.  (Current and Draft Curriculum)

The Draft makes no mention of the last two general capabilities, which, given what comes next, is a little odd. Of course, whatever their intrinsic worth, the general capabilities can readily be used as an argument for real-world problem-solving and the like. Of course, that is exactly what is done.

Numeracy needs no comment, since it is already perverting everything, to the point where Arithmetic barely exists. Similarly, Digital Literacy is obvious: why think think when you can push a button and watch a movie? As for the others: Literacy is about communicating problems and real-world contexts; Critical and Creative Thinking is press-ganged into serving problem-solving; Ethical Understanding amounts to gathering and analysing data on whatever needs ethicising. 

 

MORE STUFF OVERLYING MATHEMATICS — CROSS-CURRICULUM PRIORITIES (pp 10-11)

Everything in the (not just mathematics) Curriculum is supposed to promote the Cross-Curriculum Priorities:

*) Aboriginal and Torres Strait Islanders, Asia, Sustainability (Current and Draft Curriculum)

The Draft ignores Asia, for God knows what reason. Sustainability is what you’d expect, the “modelling” of this or that. And, predictably, the description of Aboriginal and Torres Strait Islander mathematics is strained, embarrassing and plain silly:

[Aboriginal and Torres Strait Islander Peoples] tend to be systems thinkers who are adept at pattern and algebraic thinking, …

Pull the other one.

For example, within the probability and statistics strands, stochastic reasoning is developed through Aboriginal and Torres Strait Islander instructive games and toys.

Huh. They pulled the other one.

We really wish well-meaning clowns would cease this tendentious nonsense and instead focus on the stopping of aboriginals being beaten up by racist cops.

Just to be clear, the A and TSI description in the Introduction is ridiculous, but it is not half-way as ridiculous, nor a tenth-way as damaging, as ACARA’s Core Concepts nonsense. It’s easy to make fun of this stuff, and it should be made fun of, but it is not even close to the main game.

 

MATHEMATICS AND OTHER SUBJECTS (pp 11-12)

There is nothing exceptionally notable here. It is just another opportunity taken to push the real-world contexts of mathematics, exactly as was done with the General Capabilities.

 

MORE STUFF OVERLYING THE SIX STRANDS — KEY CONSIDERATIONS (pp 13-16)

This is the last section, and it is very weird. And very bad. It seems to be attempting to serve the same purpose as the Core Concepts, but with no proper connection to the Core Concepts nor, as far as we can see, to anything else in the Curriculum documentation. It’s as if the Core Concepts section didn’t exist, or someone realise/admitted that the Core Concepts section was meaningless.

in any case the Key Considerations are:

*) Understanding, Fluency, Reasoning, Problem-Solving, Experimentation, Investigation, Mathematical modelling, Computational Thinking, Computation algorithms and the use of digital tools of mathematics.

Note that the first four Key Considerations are exactly the four Proficiencies in the Current Curriculum — what the Core Concepts are meant to be replacing. But, then we have five more Considerations, all shoving us towards modelling, real-world contexts, computers and whatnot. The purpose of this is obvious, and it is bad.

There are minor changes in wording from the first three Proficiencies in the Current Curriculum to the corresponding Considerations. The new wording is generally worse, including an annoying amount of self-promotion, but is basically ok. The problem is with the rest of the Key Considerations.

The Proficiency on Problem-Solving is extensively reworded in the corresponding Consideration, with explicit linking to the next four (new) Considerations. Embedded in it is ACARA’s definition of Problem-Solving:

Students formulate and solve problems when they: apply mathematics to model and represent meaningful or unfamiliar situations; design investigations and plan their approaches; choose and apply their existing strategies to seek solutions; reflect upon and evaluate approaches; and verify that their answers are reasonable.

For those keeping track, this is definitely not Singapore

The last five Considerations are predictable and need no comment, except for Computational Thinking. This is described as follows:

The Australian Curriculum: Mathematics aims to develop students’ computational thinking through the application of its various components, including decomposition, abstraction, pattern recognition, modelling and simulation, algorithms and evaluation.

Framed as such, Computational Thinking is no different from standard aspects of Mathematical Thinking, except for the inclusion of “modelling and simulation” — which is jammed in even thought it doesn’t remotely fit — and “algorithms and evaluation”.

The point is then given away in the next line:

Computational thinking provides the strategic basis that underpins the central role of computation and algorithms in mathematics and their application to inquiry, modelling and problem solving in mathematics and other fields. 

“The central role of computation and algorithms in mathematics”.

Clearly, the point is not to promote Computational Thinking. The point is to promote computing.

There is a strong push for this type of content, usually under the title “Algorithmic Thinking”. It can, rarely, refer to nice investigations of algorithms for solving mathematical problems. In this form, and only in this form, Algorithmic Thinking has a natural and minor place in a mathematics curriculum.

But that is not what is going on here. What is going on here is the turning of mathematics into an experimental subject and a computer science subject, in order to write crappy little programs to run on crappy little models. It is not a mathematics education and it is not remotely good.

We’re done. Thank Christ.

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*) It’s not.

 **) It’s a minor complaint in the scheme of things, but it is worth noting that the section labels and subsection labels have almost indistinguishable fonts, making it almost impossible to keep track of where one is. How a bunch of guys who cornered the market on Neon could even stuff this up, God only knows.

***) The fourth proficiency as well, if one has a Singaporean view of “problem-solving”