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.

****************************************

*) 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”

ACARA Crash 3: Fool’s Gold

This is another quick one, but it keeps the bullets flying while we prepare a more substantial post for tomorrow(ish). It can be considered a companion to the previous ACARA Crash a Content-Elaborations for Year 8 Number.

CONTENT

recognise and investigate irrational numbers in applied contexts including certain square roots and π

ELABORATIONS

recognising that the real number system includes irrational numbers which can be approximately located on the real number line, for example, the value of π lies somewhere between 3.141 and 3.142 such that 3.141 < π < 3.142

using digital tools to explore contexts or situations that use irrational numbers such as finding length of hypotenuse in right angle triangle with sides of 1 m or 2 m and 1 m or given area of a square find the length of side where the result is irrational or the ratio between paper sizes A0, A1, A2, A3, A4

investigate the Golden ratio as applied to art, flowers (seeds) and architecture

ACARA Crash 2: Shell Game

We’re still desperately trying to find the time to properly go through the Daft Curriculum, and we hope to have some longer posts in the coming week. Until then we’ll try to keep things ticking over, sniping a little each day.

This is a short one, and can be thought of as a WitCH or a PoSWW. It is a Content-Elaboration for Year 6 Algebra. We won’t comment now, except to note that we cannot see how any competent and attentive mathematician (or grammaticist) would sign off on this. The consideration of possible corollaries is left for the reader.

CONTENT

recognise and distinguish between patterns growing additively and multiplicatively and connect patterns in one context to a pattern of the same form in another context

ELABORATION

investigating patterns on-Country/Place and describing their sequence using a rule to continue the sequence such as Fibonacci patterns in shells and in flowers.

ACARA Crash: The Very Beginning

Doh!

Let’s start at the very beginning
A very good place to start
When you read you begin with A B C
When you count you begin with 1 2 3, establishing an understanding of the language and processes, and which you use to quantify, compare, order and make correspondences between collections.

It’s just possible that Julie Andrews will single-handedly take down ACARA.

We’re desperately trying to find the time to give the Daft Australian Curriculum the comprehensive hammering it deserves. Until we can get to that, we’ll keep things rolling with a series of short(ish) posts: ACARA Crash (pronounced with a thick Italian accent). We’ll start at the very beginning … with Foundation, the Prep year or whatever you want to call it.

What’s the very first thing you want to teach (or confirm the knowledge of)? Yep, you want the kids to know the numbers, their symbols and their names, and you want them to be able to count. You want the kids to have a sense of ordering, and the language to capture it.

It’s pretty straight-forward, and fun. Sing some counting songs, practise writing the words and the numerals, watch out for those reversed 5s. Maybe go to an antique shop and grab some coloured blocks to play with. There are things to be learned, practice to be done, the understanding of ordering to be gained. But there’s just not a whole lot need be said about this practice. ACARA, of course, believes otherwise.

Following on from its (bloated) content on number names and its (badly misplaced and bloated) content on quantifying, ACARA has their content descriptor on counting:

“establish understanding of the language and processes of counting to quantify, compare, order and make correspondences between collections, initially to 20, and explain reasoning (AC9MFN03)”

There is a fundamental rule of teaching: say less. Make every word count, and if nothing needs to be said then say nothing. It is a rule that ACARA (and all education princelings) desperately needs to learn.

What is the point of ACARA’s word swamp? In what conceivable sense can it be considered “refining” or “decluttering”? What does it clarify to anyone? How is it in any way better than a bare bones content description:

Teach the little monsters to count.

Ok, you might want to tweak the wording, but for content, that is pretty much it. None of ACARA’s blather is required or remotely helpful. It is much, much worse than the corresponding content in the current Australian Curriculum.

Content is meant to be the bones, the clear and solid structure. If you want meat then sure, have meat. But if you don’t want to lose sight of the bones — and you really, really don’t — then put the meat in the damn meat section, in the elaborations. And of course, ACARA has plenty more meat in the elaborations; it will come as no surprise that their meat is off.

Most of ACARA’s counting elaborations are benign, just standard classroom exercises and token (but ok) Aboriginal material. Ten elaborations is more a textbook chapter than elaboration, but individually they’re not intrinsically bad. The problem is with what there isn’t, and it is a massive problem.

After counting, what sense of number do you want Foundation kids to attain? The answer is in the question: you want them to begin to develop a sense of number. Beyond the Four Horsemen and the Third Man, you want the kids to develop a sense of fourness and threeness, numbers as quantity, and the prelude to proper arithmetic. That is abstract and not easy, which is why it is important to begin with the hints early on.

The current Australian Curriculum, is not strong on this, but the draft curriculum is way, way worse. The stage is set right at the start, by ACARA’s “Level description” for Foundation mathematics:

The Australian Curriculum: Mathematics focuses on the development of 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. As students engage in learning mathematics in Foundation year, they:

    • explore situations, sparking curiosity to investigate and solve everyday problems using physical and virtual materials to model, sort, quantify and compare 
    • begin to bring some mathematical meaning to their use of familiar terms and language when they pose and respond to questions and explain their
    •  look for and make connections between number names, numerals and quantities and through active learning experiences, compare quantities and shapes, using elementary mathematical reasoning
    • build confidence and autonomy in being able to make and justify mathematical decisions based on quantification and direct comparisons
    • learn to recognise repetition and apply this to creatively build repeating patterns in a wide range of contexts
    • begin to build a sense of chance and variability when they engage in play-based activities, imagine and think about familiar chance events.

For Foundation year? “Deep knowledge and conceptual understanding”? Have you gotten them to count backwards yet? Have you bothered to try to explain what a number is?

This is nonsense, of course, but it is also poison. Pretty much all of the Number strand in the Foundation Year is on “modelling” and “Problems” and “practical situations”, and there’s a lot of it. There’s barely a hint of numbers as numbers, and what hint there is, is certain to be dissolved and forgotten in the ocean of inquiry.

Sure, you expect young kids to be playing with things more than ideas. They will add three blocks to four blocks many times before they add 3 + 4. But there are better and worse activities to suggest and encourage this abstraction; ACARA’s are much worse, and deliberately so. The writers don’t want it. Fundamentally, they don’t want mathematics.

What the draft curriculum makes clear, already at the Foundation level, is that the curriculum writers, deep in their hearts, hate mathematics. They hate the abstraction at the heart of mathematics and proper mathematical thought. They might love what mathematics can do, assigning numbers here and there, but their sense of mathematics is wade-pool deep. Real mathematics, they hate.

This hatred glows brightly from almost every line. Almost never is the opportunity grasped to display the internal beauty and power of mathematics. Almost never is mathematics promoted as its own end, as its own good. It is clearly unimaginable to the writers. Mathematics is just a tool, the annoying but necessary “m” in STEm.

This is not just cultural philistinism, it is ACARA shooting its own philistine feet. Without a proper appreciation of mathematics and the source of its power, all of ACARA’s real-world games are, well, just games. 13 years of pointless games, that’s what’s on offer.

There is plenty more nonsense in Foundation draft: we haven’t even mentioned the “Algebra”. But that’ll do. We’ve never paid much attention to the Foundation curriculum. We figured the damage mostly began around Year 2, and up until now that is probably true. But no more: ACARA’s draft begins with a perfectly awful Foundation for the greater awfulness to come.