## ACARA Crash 18: Errors in the Draft Mathematics Curriculum

The following is a list of errors – and possible/arguable errors – in the draft mathematics curriculum. Commenters are invited and encouraged to suggest additions, and deletions.

By “error” we mean a statement or instruction that is factually wrong or that makes no logical/mathematical/everyday sense. Some of the listed “errors” are clear-cut, while others are less so. Of course the fact that a statement/instruction made no sense to us does not prove that it makes no sense; we’ve attempted to be fair, being tough on the improper use of technical terms while giving weird phrasings a good-faith pondering in context. Nonetheless, there may well be reasonable interpretations that we have missed. (Of course phrasing that is difficult to interpret has no place in a curriculum document, but that is a separate category of sin.) As well, it is not always clear whether to characterise a statement as an error or simply a really dumb idea, but we’ve tried to stick pretty closely to “error”, leaving the noting of really dumb ideas to our other ninety-eight posts.

The list follows. The majority are elaborations. There are a few content descriptors, for which associated elaborations are indicated by a further indentation. Again, commenters are encouraged both to suggest additions to the list, and to argue for deletions from the list.

## ACARA CRASH 17: Algebraic Fractures

The following are Year 10 Number-Algebra content-elaborations in the current curriculum:

CONTENT

Apply the four operations to simple algebraic fractions with numerical denominators

ELABORATIONS

expressing the sum and difference of algebraic fractions with a common denominator

using the index laws to simplify products and quotients of algebraic fractions

CONTENT

Solve linear equations involving simple algebraic fractions

ELABORATIONS

solving a wide range of linear equations, including those involving one or two simple algebraic fractions, and checking solutions by substitution

representing word problems, including those involving fractions, as equations and solving them to answer the question

And what does the draft curriculum do with these?

Removed

And, why?

Not essential for all students to learn in Year 10.

God only knows how one develops fluency with expressions that cease to exist.

## ACARA CRASH 16: Unlevel Playing Field

Each Year’s content in the draft curriculum begins with a Level description, and each of the thirteen Level description begins with the exact same sentences:

The Australian Curriculum: Mathematics focuses on the development of a deep knowledge and conceptual understanding of mathematical structures and fluency with procedures. Students learn through the approaches for working mathematically, including modelling, investigation, experimentation and problem solving, all underpinned by the different forms of mathematical reasoning. [emphasis added]

Yep, “an equal focus on building fluency“, no doubt about it.

## ACARA CRASH 15: Digital Insertion

Continuing to try to rid ourselves of ACARA irritants, the following are the “calculator” elaborations from Year 1 – Year 6 Number and Algebra (sic):

YEAR 1

using the constant function on a calculator to add ten to single digit numbers, recording the numbers to make, show and explore the patterns in a 0 – 100 chart

with the use of a calculator, exploring skip-counting sequences that start from different numbers, discussing patterns

modeling skip counting sequences using the constant function on a calculator, while saying, reading and recording the numbers as they go

## ACARA CRASH 14: Backward Thinking

This one we really don’t get. It concerns Foundation and Year 1 Number, and was pointed out to us by Mr. Big.

We began the Crash series by critiquing the draft curriculum’s approach to counting in Foundation. Our main concern was the painful verbosity and the real-world awfulness, but we also provided a cryptic hint of one specifically puzzling aspect. The draft curriculum’s content descriptor on counting is as follows:

“establish understanding of the language and processes of counting to quantify, compare, order and make correspondences between collections, initially to 20, and explain reasoning” (draft curriculum)

“explain reasoning”. Foundation kids.

OK, let’s not get distracted; we’ve already bashed this nonsense. Here, we’re interested in the accompanying elaborations. There are ten of them, which one would imagine incorporates any conceivable manner in which one might wish to elaborate on counting. One would be wrong.

The corresponding content descriptor in the current Mathematics Curriculum is as follows:

“Establish understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20, moving from any starting point” (current curriculum)

Notice how much more “cluttered” is the current descriptor… OK, OK stay focussed.

The current descriptor on counting has just (?) four elaborations, including the following two:

“identifying the number words in sequence, backwards and forwards, and reasoning with the number sequences, establishing the language on which subsequent counting experiences can be built” (current curriculum, emphasis added)

“developing fluency with forwards and backwards counting in meaningful contexts, including stories and rhymes” (current curriculum, emphasis added)

The point is, these elaborations also emphasise counting backwards, which seems an obvious idea to introduce and an obvious skill to master. And which is not even hinted at in any of the ten elaborations of the draft counting descriptor.

Why would the writers of the draft curriculum do that? Why would they consciously eliminate backward counting from Foundation? We’re genuinely perplexed. It is undoubtedly a stupid idea, but we cannot imagine the thought process that would lead to this stupid idea.

OK, we know what you’re thinking: it’s part of their dumbing down – maybe “dumbing forward” is a more apt expression – and they’ve thrown backward counting into Year 1. Well, no. In Year 1, students are introduce to the idea of skip-counting. And, yep, you know where this is going. So we’ll, um, skip to the end.

The current Curriculum has two elaborations of the skip-counting descriptor, one of which emphasises the straight, pure ability to count numbers backwards. And the draft curriculum? There are four elaborations on skip-counting, suggesting in turn the counting of counters in a jar, pencils, images of birds, and coins. Counting unadorned numbers? Forget it. And counting backwards? What, are you nuts?

OK, so eventually the draft curriculum seems, somehow, to get around to kids counting backwards, to look at “additive pattern sequences” and possibly to solve “subtraction problems”. The content descriptors are so unstructured and boneless, and the elaborations so vague and cluttered, it is difficult to tell. But how are the kids supposed to get there? Where is the necessary content description or elaboration:

Teach the little monsters to count backwards.

If it is there, somewhere in the draft curriculum, we honestly can’t see it. And if it is not there, that it is simply insane.

## 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 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 is equivalent to and and therefore 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, and , 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 or 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 of giving of 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 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”.