The ACARA Page

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

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

 

Open Letter

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

 

The Draft Curriculum

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

Warm Up for the Draft Curriculum

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

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

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

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

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

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

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

 

Curriculum Review

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

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

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

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

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

 

The Current Australian Curriculum

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

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

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

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

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

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

 

NAPLAN

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

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

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

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

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

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

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

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

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

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

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 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 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.

A Simple Message to Primary Schools About Multiplication Tables

Dear Primary Schools,

If your students are not learning their multiplication tables, up to 12, by heart, then you are fucking up.

If you think giving your students a grab-bag of tricks replaces multiplication tables, then you are fucking up.

If you think orchestrating play-based, student-centred theatricalities replaces multiplication tables, then you are fucking up.

If you think quoting the Australian Curriculum gives you license to not teach multiplication tables, then you are fucking up.

If you think quoting some education twat gives you license to not teach multiplication tables, then you are fucking up.

Thank you for your attention.

BODMAS v USBB

Yesterday, I received an email from Stacey, a teacher and good friend and former student. Stacey was asking for my opinion of “order of operations”, having been encouraged to contact me by Dave, also a teacher and good friend and former student. Apparently, Dave had suggested that I had “strong opinions” on the matter. I dashed off a response which, in slightly tidied and toned form, follows. 

Strong opinions? Me? No, just gentle suggestions. I assume they’re the same as Dave’s, but this is it:

1) The general principle is that if mathematicians don’t worry about something then there is good reason to doubt that students or teachers should. It’s not an axiom, but it’s a very good principle.

2) Specifically, if I see something like
3 x 5 + 2 x -3
my response is

a) No mathematician would ever, ever write that.

b) I don’t know what the Hell the expression means. Honestly.

c) If I don’t know what it means, why should I expect anybody else to know?

3) The goal in writing mathematics is not to follow God-given rules, but to be clear. Of course clarity can require rules, but it also requires common sense. And in this case common sense dictates

USE 

SOME 

BLOODY 

BRACKETS

For Christ’s sake, why is this so hard for people to understand? Just write (3 x 5) + 2 or 3 x (5 + 2), or whatever. It is almost always trivial to deambiguousize something, so do so.

The fact that schools don’t instruct this first and foremost, that demonstrates that BODMAS or whatever has almost nothing to do with learning or understanding. It is overwhelmingly a meaningless ritual to see which students best follow mindless rules and instruction. It is not in any sense mathematics. In fact, I think this all suggests a very worthwhile and catchy reform: don’t teach BODMAS, teach USBB.

[Note: the original acronym, which is to be preferred, was USFB]

4) It is a little more complicated than that, because mathematicians also write arguably ambiguous expressions, such ab + c and ab2 and a/bc. BUT, the concatenation/proximity and fractioning is much, much less ambiguous in practice. (a/bc is not great, and I would always look to write that with a horizontal fraction line or as a/(bc).)

5) Extending that, brackets can also be overdone, if people jump to overinterpret every real or imagined ambiguousness. The notation sin(x), for example, is truly idiotic; in this case there is no ambiguity that requires clarification, and so the brackets do nothing but make the mathematics ugly and more difficult to read.

6) The issue is also more complicated because mathematicians seldom if ever use the signs ÷ or x. That’s partially because they’re dealing with algebra rather than arithmetic, and partially because “division” is eventually not its own thing, having been replaced by making the fraction directly, by dealing directly with the result of the division rather than the division.

So, this is a case where it is perfectly reasonable for schools to worry about something that mathematicians don’t. Arithmetic obviously requires a multiplication sign. And, primary students must learn what division means well before fractions, so of course it makes sense to have a sign for division.  I doubt, however, that one needs a division sign in secondary school.

7) So, it’s not that the order of operations issues don’t exist. But they don’t exist nearly as much as way too many prissy teachers imagine. It’s not enough of a thing to be a tested thing.