NotCH 2: A Digits Puzzle

OK, hands up who thought there was ever gonna be a second NotCH?

We’re not really a puzzle sort of guy, and base ten puzzles in particular tend to bore us. So, this is unlikely to be a regular thing. Still, the following question came up in some non-puzzle reading (upon which we plan to post very soon), and it struck us as interesting, for a couple reasons. And, a request to you smart loudmouths who comment frequently:

Please don’t give the game away until non-regular commenters have had time to think and/or comment. 

Start by writing out a few terms of the standard doubling sequence:

1, 2, 4, 8, 16, 32, 64, etc.  Continue reading “NotCH 2: A Digits Puzzle”

NotCH 1: Maths Masters Quizzes

More or less by accident, this post is the beginning of a new series: Not Crap Here.

A couple of people have suggested that we could occasionally include Dr. Jekyll material on this blog. You know, helpful stuff. It’s a decent idea, if our current thoughts weren’t so influenced by misanthropic disgust and murderous rage. Still, we’ve received two specific requests for the same old Jekyll material,* and which entailed some digging. Having finally dug, we’ve decided to post the material here, for whomever is interested. Whether or not there will ever be a NotCH 2 is anybody’s guess.

Continue reading “NotCH 1: Maths Masters Quizzes”

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 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.

WitCH 57: Tunnel Vision

The following is just a dumb exercise, and so is probably more of a PoSWW. It seems so lemmingly stupid, however, that it comes around full cycle to be a WitCH. It is an exercise from Maths Quest Mathematical Methods 11. The exercise appears in a pre-calculus, CAS-permitted chapter, Cubic Polynomials. The suggested answers are (a) \boldsymbol{y = -\frac{1}{32}x^2(x-6)} , and (b) 81/32 km.

ACARA is Confronted With the Big Ideas

In this column, ACARA will be playing the role of the Good Guy.

Now that we have your attention, we’ll confess that we were exaggerating. ACARA is, of course, always the Bad Guy. But this column also contains a Worse Guy, a bunch of grifters called Center for Curriculum Redesign. ACARA appears to be fighting them, and fighting themselves.

Last week, The Australian‘s education reporter, Rebecca Urban, wrote a column on ACARA’s current attempts to revise the Australian Curriculum (paywalled, and don’t bother, and it’s Murdoch). The article, titled Big ideas for mathematics curriculum fails the test, begins as follows:

Plans for a world-class national school curriculum to arrest Australia’s declining academic results are in disarray after a proposal to base the teaching of mathematics around “big ideas” was rejected twice.

So, apparently Australia has plans for a world-class curriculum.1 Who knew? At this stage we’d be happy with plans for a second rate curriculum, and we’d take what we got. But a curriculum based upon “big ideas”? It’s a fair bet that that’s not aiming within cooee of first or second. We’ll get to these “big ideas”, and some much worse little ideas, but first, some background.

The sources of this nonsense are two intertwined and contradictory undertakings within ACARA. The first undertaking is a review of the Australian Curriculum, which ACARA began last year, with a particular emphasis on mathematics. On ACARA’s own terms, the Review makes some sense; if nothing else, the Australian Curriculum is unarguably a tangled mess, with “capabilities” and “priorities” and “learning areas” and “strands” and “elaborations” continually dragging teachers this way and that. The consequence, independent of the Curriculum being good or bad, is that is difficult to discern what the Curriculum is, what it really cares about. As such, the current Review is looking for simplification of the Curriculum, with emphasis on “refining” and “decluttering”, and the like.

This attempt to tidy the Australian Curriculum, to give it a trim and a manicure, is natural and will probably do some good. Not a lot of good: the current Review is fundamentally too limited, even on its own terms, and so appears doomed to timidity.2 But, some good. The point, however, is the current Review is definitively not seeking a major overhaul of the Curriculum, much less a revolution. Of course we would love nothing more than a revolution, but “revolution” does not appear in the Terms of Reference.

The hilarious problem for ACARA is the second, contradictory undertaking: ACARA have hired themselves a gang of revolutionaries. In 2018, ACARA threw a bunch of money at the Center for Curriculum Redesign, for CCR “to develop an exemplar world-class mathematics curriculum”. ACARA’s “oh, by the way” announcement suggests that they weren’t keen on trumpeting this partnership, but CCR went the full brass band. Their press release proudly declared the project a “world’s first”, and included puff quotes from then ACARA CEO, Bob the Blunder, and from PISA king, Andreas Schleicher. And the method to produce this exemplar world-class, ACARA-PISA-endorsed masterpiece? CCR would be

“applying learnings from recent innovations in curriculum design and professional practice …”

And the driving idea?

“… the school curriculum needs to allow more time for deeper learning of discipline-specific content and 21st century competencies.”

This grandiose, futuristic snake oil was an idiot step too far, even for the idiot world of Australian education, and as soon as the ACARA-CCR partnership became known there was significant pushback. In an appropriately snarky report (paywalled, Murdoch), Rebecca Urban quoted ex-ACARA big shots, condemning the ACARA-CCR plan as “the latest in a long line of educational fads” and “a rather stealthy shift in approach”. Following Urban’s report, there was significant walking back, both from Bob the Blunder, and from the then federal education minister, Dan “the Forger” Tehan. But revolutionaries will do their revolutionary thing, and CCR seemingly went along their merry revolutionising way. And, here we are.

Urban notes that the proposal that ACARA has just rejected – for a second time – placed a “strong focus on developing problem-solving skills”, and she quotes from the document presented to ACARA, on the document’s “big ideas”:3

Core concepts in mathematics centre around the three organising ideas of mathematics structures approaches and mathematising [emphasis added] …Knowledge and conceptual understanding of mathematical structures and approaches enables students to mathematise situations, making sense of the world.”

Mathematising? Urban notes that this uncommon term doesn’t appear in ACARA’s literature, but is prominent in CCR’s work. She quotes the current proposal as defining mathematising as

“the process of seeing the world using mathematics by recognising, interpreting situations mathematically.”

So, all this big ideas stuff appears to amount to the standard “work like a mathematician”, problem-centred idiocy, ignoring the fact that the learning of the fundamentals of mathematics has very, very little to do with being a mathematician.4 Really, not a fresh hell, just some variation of the current, familiar hell.

So, why write on this latest version of the familiar problem-solving nonsense? Because what has reportedly been presented to ACARA may be far, far worse.

Most sane people realise that before tackling some big idea it is somewhat useful to get comfortable with relevant small ideas. In this vein, before the grand adventure of mathematising one would reasonably want kids to engage in some decent numbering and algebra-ing. You want the kids to do some mathematising nonsense? Ok, it’s dumb, but at least make sure that the kids first know some arithmetic and can handle an equation or two.  And this is where the proposal just presented to ACARA seems to go from garden-variety nonsense to full-blown lunacy.

Recall that the stated, non-revolutionary goal of the current Review is to clarify and refine and declutter the Australian Curriculum. Along these lines, the proposal presented to ACARA contained a number of line-item suggestions to accompany the big ideas. Urban quotes some small beer suggestions, such as the appropriate stage to be recognising coin denominations, the ordering of the months and the like. But, along with the small beer, Urban documents some big poison, such as the following:

\color{red}\mbox{\bf Year 4} \quad\boldsymbol{-}\quad  \left(\aligned&\mbox{\bf recognise represent and order numbers}\\ &\mbox{\bf to at least tens of thousands}\endaligned\right) \quad \mbox{\bf Not essential at year level}

Christ. If students don’t have a handle on ten-ing by the end of Year 4 then something is seriously screwed. At that stage the students should be happily be zooming into the zillions, but some idiots – the same idiots hell bent on real world problem-solving – imagine tens of thousands is some special burden.

The next poison:

\color{red}\mbox{\bf Year 5} \quad\boldsymbol{-}\quad \mbox{\bf Using fractions to represent probabilities} \quad  \left(\aligned&\mbox{\bf students are not ready,}\\ &\mbox{\bf promotes procedural knowledge} \\ & \mbox{\bf over conceptual understanding} \endaligned\right)

Here, the idiots are handed a gun on a platter, which they grab by the muzzle and then shoot themselves. There is absolutely zero need to cover probability, or statistics, in primary school. Its inclusion is exactly the kind of thoughtless and cumbersome numeracy bloat that makes the Australian Curriculum such a cow. But, if one is going to cover probability in primary school, the tangible benefit is that it provides novel and natural contexts to represent with fractions. Take away the fractions, and what is this grand “conceptual understanding” remaining? That some things happen less often often than other things? Wonderful.

One last swig of poison, strong enough to down an elephant:

\color{red}\mbox{\bf Years 7-10} \quad\boldsymbol{-}\quad \mbox{\bf Solving equations algebraically} \quad  \left(\aligned&\mbox{\bf Not essential for all students,}\\ &\mbox{\bf especially for more complex equations.} \\ & \mbox{\bf Technology can be used as a support} \endaligned\right)

On the scale of pure awfulness, this one scores an 11, maybe a 12. It is as bad as it can be, and then worse.

PISA types really have a thing about algebra. They hate it. And, this hatred of algebra demonstrates the emptiness of their grand revolutionary plans. Algebra is the fundamental mechanics of mathematical thought. Without a solid sense of and facility with algebra, all that mathematising and problem-solving is fantasy; it can amount to no more than trivial and pointless number games.

The teaching of algebra is already in an appalling, tokenistic state in Australia. It is woefully, shamefully underemphasised in lower secondary school, which is then the major source of students’ problems in middle school, and why so many students barely crawl across the finish line of senior mathematics, if they make it at all.

What is “more complex equations” supposed to mean for 7 – 10 algebra? The material gets no more complicated than quadratics, so presumably they mean quadratics, the hobgoblin of little saviours. True, this material tends to be taught pointlessly and poorly. But “complex”? Simply, no. It amounts to little more than AB = 0 implying that either A or B is 0, a simple and powerful idea that many students never solidly get. The rest is detail, not much detail, and the detail is just not that hard.

Of course, a significant reason why algebra is taught so, so badly is that it is almost universally taught and tested with “technology”, from calculators to nuclear CAS weapons, to online gaming of the kind that that asshole Tudge is promoting. And all of this is “used as a support”? That idea of “support”, just as stated, is bad enough, bringing forth images of kids limping through the material. But all this technology is much worse than a crutch; it is an opiate.

It is a minimal relief if ACARA has rejected the current proposal, but we have no real idea what is going on or what will happen next. We don’t how much much poison the proposal contained, or even who concocted it. We don’t know if the rejection of this proposal amounts to a war between CCR and a new, more enlightened ACARA, or a civil war within ACARA itself.5 We should find out soon enough, however. ACARA has promised to release a draft curriculum by the end of April, giving them a month or so to come to terms with the truly idiotic ideas that they are being presented. ACARA has a month or so to avoid becoming, yet again and still, Australia’s educational laughing stock.

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

1) We really wanted to slip “Urban myth” into the title of this post, but decided it would have been unfair. Yes, “world class” required quotation marks, or something. It seems, however, that Rebecca Urban was just carelessly, or perhaps snidely, repeating a piece of ACARA puffery, which is not the focus of her report. In general, Urban tends to be less stenographic than other education (all) reporters; she is opinionated and, from what we’ve seen, she seems critical of the right things. We haven’t seen evidence that Urban knows about mathematics education, or is aware of just how awful things now are, but we also haven’t seen her repeat any of the common idiocies.

2) We hope to write on the Curriculum Review in the next week or so, give or take a Mathologer task.

3) The proposal just presented to ACARA is not publicly available, and Urban appears to have only viewed snippets of it. It is not even clear, at least to us, who are the authors of the proposal. We’re accepting that Urban’s report is accurate as far as it goes, while trying to avoid speculating on the much missing information.

4) Urban’s report includes some good and critical, but not sufficiently critical, quotes from teacher and writer, Greg Ashman.

5) David de Carvalho, ACARA’s new CEO, appears to be an intelligent and cultured man. Maybe insufficiently intelligent or cultured, or insufficiently honest, to declare the awfulness of NAPLAN and the Australian Curriculum, but a notable improvement over the past.